{"id":4535,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/"},"modified":"2018-05-07T12:36:28","modified_gmt":"2018-05-07T07:06:28","slug":"ncert-solutions-class-12-maths-miscellaneous-8","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Differential_Equations\" >NCERT Solutions class 12 Maths Differential Equations<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#2_For_each_of_the_exercises_given_below_verify_that_the_given_function_implicit_or_explicit_is_a_solution_of_the_corresponding_differential_equation\" >2. For each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#3_Form_the_differential_equation_representing_the_family_of_curves_where_ia_an_arbitrary_constant\" >3. Form the differential equation representing the family of curves \u00a0where \u00a0ia an arbitrary constant.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#4_Prove_that_is_the_general_equation_of_the_differential_equation_where_is_a_parameter\" >4. Prove that \u00a0is the general equation of the differential equation \u00a0where \u00a0is a parameter.<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#Therefore_the_given_differential_equation_is_homogeneous\" >Therefore, the given differential equation is homogeneous.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#5_For_the_differential_equation_of_the_family_of_the_circles_in_the_first_quadrant_which_touch_the_coordinate_axes\" >5. For the differential equation of the family of the circles in the first quadrant which touch the coordinate axes.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#6_Find_the_general_solution_of_the_differential_equation\" >6. Find the general solution of the differential equation \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#7_Show_that_the_general_solution_of_the_differential_equation_is_given_by\" >7. Show that the general solution of the differential equation \u00a0is given by \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#8_Find_the_equation_of_the_curve_passing_through_the_point_whose_differential_equation_is\" >8. Find the equation of the curve passing through the point \u00a0whose differential equation is \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#9_Find_the_particular_solution_of_the_differential_equation_given_that_when\" >9. Find the particular solution of the differential equation \u00a0given that \u00a0when \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#10_Solve_the_differential_equation\" >10. Solve the differential equation: \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#11_Find_the_particular_solution_of_the_differential_equation_given_that_when\" >11. Find the particular solution of the differential equation \u00a0given that \u00a0when<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#12_Solve_the_differential_equation\" >12. Solve the differential equation: \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#13_Find_the_particular_solution_of_the_differential_equation_given_that_when\" >13. Find the particular solution of the differential equation \u00a0given that \u00a0when \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#14_Find_the_particular_solution_of_the_differential_equation_given_that_when\" >14. Find the particular solution of the differential equation \u00a0given that \u00a0when<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-9\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#15_The_population_of_a_village_increases_continuously_at_the_rate_proportional_to_the_number_of_its_inhabitants_present_at_any_time_If_the_population_of_the_village_was_20000_in_1999_and_25000_in_the_year_2004_what_will_be_the_population_of_the_village_in_2009\" >15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25,000 in the year 2004, what will be the population of the village in 2009?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#Choose_the_correct_answer\" >Choose the correct answer:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-10\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#17_The_general_equation_of_a_differential_equation_of_the_type_is\" >17. The general equation of a differential equation of the type \u00a0is:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-11\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#18_The_general_solution_of_the_differential_equation_is\" >18. The general solution of the differential equation \u00a0is:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#NCERT_Solutions_class_12_Maths_Miscellaneous-12\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-differential-equations\/1293\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Differential Equations \u00a0<\/a>as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20Part-02%20.jpg\" alt=\"NCERT Solutions class 12 Maths Miscellaneous\" width=\"178\" height=\"228\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Differential_Equations\"><\/span>NCERT Solutions class 12 Maths Differential Equations<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. For each of the differential equations given below, indicate its order and degree (if defined):<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><strong><img decoding=\"async\" style=\"height: 49px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image001.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><strong><img decoding=\"async\" style=\"height: 49px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image002.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><strong><img decoding=\"async\" style=\"height: 51px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image003.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image001.png\" \/><\/p>\n<p style=\"text-align: justify;\">The highest order derivative present in this differential equation is <img decoding=\"async\" style=\"height: 44px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image004.png\" \/>\u00a0and hence order of this differential equation if 2.<\/p>\n<p style=\"text-align: justify;\">The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative <img decoding=\"async\" style=\"height: 44px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image004.png\" \/>\u00a0is 1.<\/p>\n<p style=\"text-align: justify;\">Therefore, Order = 2, Degree = 1<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image002.png\" \/><\/p>\n<p style=\"text-align: justify;\">The highest order derivative present in this differential equation is <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image005.png\" \/>\u00a0and hence order of this differential equation if 1.<\/p>\n<p style=\"text-align: justify;\">The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image005.png\" \/>\u00a0is 3.<\/p>\n<p style=\"text-align: justify;\">Therefore, Order = 1, Degree = 3<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 51px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image003.png\" \/><\/p>\n<p style=\"text-align: justify;\">The highest order derivative present in this differential equation is <img decoding=\"async\" style=\"height: 44px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image006.png\" \/>\u00a0and hence order of this differential equation if 4.<\/p>\n<p style=\"text-align: justify;\">The given differential equation is not a polynomial equation in derivatives therefore, degree of this differential equation is not defined.<\/p>\n<p style=\"text-align: justify;\">Therefore, Order = 4, Degree not defined<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"2_For_each_of_the_exercises_given_below_verify_that_the_given_function_implicit_or_explicit_is_a_solution_of_the_corresponding_differential_equation\"><\/span><strong>2. For each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><img decoding=\"async\" id=\"Picture 8\" style=\"height: 138px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image007.jpg\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 324px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image008.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 288px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image009.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image010.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 235px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image011.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> The given function is <img decoding=\"async\" style=\"height: 24px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image012.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">To verify: Function (i) is a solution of D.E. <img decoding=\"async\" style=\"height: 44px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image013.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Differentiating both sides of eq. (i) w.r.t. <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 41px; width: 177px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image015.png\" \/><\/p>\n<p style=\"text-align: justify;\">Again differentiating both sides w.r.t. <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image014.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 44px; width: 221px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image016.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image018.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 24px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image019.png\" \/>\u00a0from eq. (i), we have,<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image020.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image013.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, Function given by eq. (i) is a solution of D.E. (ii).<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> The given function is <img decoding=\"async\" style=\"height: 27px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image021.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">To verify: Function given by (i) is a solution of D.E. <img decoding=\"async\" style=\"height: 44px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image022.png\" \/>\u00a0\u00a0\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">From (i), <img decoding=\"async\" style=\"height: 41px; width: 340px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image023.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 309px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image025.png\" \/>\u00a0[By eq. (i)]\u00a0\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 360px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 351px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image027.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image028.png\" \/>\u00a0 [Using eq. (iii) and (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image029.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image022.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, Function given by eq. (i) is a solution of D.E. (ii).<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> The given function is <img decoding=\"async\" style=\"height: 21px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image030.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026(i)<\/p>\n<p style=\"text-align: justify;\">To verify: Function given by eq. (i) is a solution of D.E. <img decoding=\"async\" style=\"height: 44px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image031.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026(ii)<\/p>\n<p style=\"text-align: justify;\">From eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image033.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 288px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image034.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 280px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image035.png\" \/>\u00a0\u00a0 [Using eq. (i)]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image031.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, Function given by eq. (i) is a solution of D.E. (ii).<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> The given function is <img decoding=\"async\" style=\"height: 24px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image036.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">To verify: Function given by eq. (i) is a solution of D.E. <img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image037.png\" \/>\u00a0\u00a0\u2026\u2026(ii)<\/p>\n<p style=\"text-align: justify;\">Differentiating both sides of eq. (i) w.r.t. <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 48px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image038.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image039.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 47px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image041.png\" \/>\u00a0from eq. (i), we get<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 72px; width: 287px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image042.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image043.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image044.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image037.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, Function given by eq. (i) is a solution of D.E. (ii).<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_Form_the_differential_equation_representing_the_family_of_curves_where_ia_an_arbitrary_constant\"><\/span><strong>3. Form the differential equation representing the family of curves <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image045.png\" \/><\/strong><strong>\u00a0where <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image046.png\" \/><\/strong><strong>\u00a0ia an arbitrary constant.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Equation of the given family of curves is <img decoding=\"async\" style=\"height: 29px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image047.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image048.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image049.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image050.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Here number of arbitrary constants is one only <img decoding=\"async\" style=\"height: 27px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image051.png\" \/><\/p>\n<p style=\"text-align: justify;\">So, we will differentiate both sides of equation only once, w.r.t. <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image014.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image052.png\" \/>\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image053.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Dividing eq. (i) by eq. (ii), we have\u00a0 <img decoding=\"async\" style=\"height: 64px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image054.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 64px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image055.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image056.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image057.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image059.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_Prove_that_is_the_general_equation_of_the_differential_equation_where_is_a_parameter\"><\/span><strong>4. Prove that <\/strong><strong><img decoding=\"async\" style=\"height: 32px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image060.png\" \/><\/strong><strong>\u00a0is the general equation of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image061.png\" \/><\/strong><strong>\u00a0where <\/strong><strong><img decoding=\"async\" style=\"height: 15px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image062.png\" \/><\/strong><strong>\u00a0is a parameter.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image063.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Here each coefficient of <img decoding=\"async\" style=\"height: 19px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image064.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image065.png\" \/>\u00a0is of same degree, i.e., 3, therefore differential equation looks to be homogeneous.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image066.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 56px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image067.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 93px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image068.png\" \/>\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Therefore_the_given_differential_equation_is_homogeneous\"><\/span>Therefore, the given differential equation is homogeneous.<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 41px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image069.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image070.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image071.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting these values in eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image073.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image074.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image075.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image076.png\" \/>\u00a0 [Separating variables]\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image077.png\" \/>\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p style=\"text-align: justify;\">Now forming partial fraction of <img decoding=\"async\" style=\"height: 44px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image078.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image078.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 52px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image079.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image078.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 52px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image080.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image078.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image081.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iv)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 397px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image082.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 427px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image083.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing coefficients of like powers of<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image084.png\" \/>\u00a0\u00a0 A \u2013 B \u2013 C = 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(v)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 21px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image085.png\" \/>\u00a0\u00a0 A + B \u2013 D = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(vi)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image086.png\" \/>\u00a0\u00a0 A \u2013 B + C = \u20133\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(vii)<\/p>\n<p style=\"text-align: justify;\">Constants A + B + D = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026(viii)<\/p>\n<p style=\"text-align: justify;\">Now eq. (v) \u2013 eq. (vii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0\u2013 2C = 4 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0C = \u2013 2<\/p>\n<p style=\"text-align: justify;\">\u00a0 Eq. (vi) \u2013 eq. (viii)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0\u2013 2D = 0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0D = 0<\/p>\n<p style=\"text-align: justify;\">\u00a0 Putting C = \u2013 2 in eq. (v), A \u2013 B + 2 = 1<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0A \u2013 B = \u20131\u00a0 \u2026\u2026\u2026.(ix)<\/p>\n<p style=\"text-align: justify;\">\u00a0 Putting D = 0 in eq. (vi) A + B = 0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(x)<\/p>\n<p style=\"text-align: justify;\">Adding eq. (ix) and (x) 2A = \u20131<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0A = <img decoding=\"async\" style=\"height: 41px; width: 23px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image087.png\" \/><\/p>\n<p style=\"text-align: justify;\">From eq. (x),\u00a0\u00a0 B = \u2013A = <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image088.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting the values of A, B, C and D in eq. (iv), we have<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 60px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image089.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 348px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image090.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 339px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image091.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 340px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image092.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 305px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image093.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 253px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image094.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 56px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image095.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting this value in eq. (iii),\u00a0 <img decoding=\"async\" style=\"height: 56px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image096.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image097.png\" \/><\/p>\n<p style=\"text-align: justify;\">Squaring both sides and cross-multiplying,\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image099.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 53px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image100.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 52px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image101.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 52px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image102.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image103.png\" \/>\u00a0 where <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image104.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_For_the_differential_equation_of_the_family_of_the_circles_in_the_first_quadrant_which_touch_the_coordinate_axes\"><\/span><strong>5. For the differential equation of the family of the circles in the first quadrant which touch the coordinate axes.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>We know that the circle in the first quadrant which touches the co-ordinates axes has centre <img decoding=\"async\" style=\"height: 27px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image105.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image046.png\" \/>\u00a0is the radius of the circle.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image066.png\" \/>\u00a0 Equation of the circle is<\/p>\n<p style=\"text-align: justify;\">\u00a0\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image106.png\" \/>\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image107.png\" \/><\/p>\n<p style=\"text-align: justify;\">Differentiating with respect to <img decoding=\"async\" style=\"height: 17px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image014.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 21px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image108.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image110.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image111.png\" \/><\/p>\n<p style=\"text-align: justify;\">Substituting value of <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image046.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 52px; width: 283px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 52px; width: 347px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image114.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image116.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_Find_the_general_solution_of_the_differential_equation\"><\/span><strong>6. Find the general solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 48px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image117.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential Equation\u00a0 <img decoding=\"async\" style=\"height: 48px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image117.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image118.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image120.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image122.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image123.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_Show_that_the_general_solution_of_the_differential_equation_is_given_by\"><\/span><strong>7. Show that the general solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image124.png\" \/><\/strong><strong>\u00a0is given by <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image125.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image124.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 51px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image126.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image127.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image128.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image129.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now <img decoding=\"async\" style=\"height: 41px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image130.png\" \/>\u00a0\u00a0\u00a0[Completing the squares]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 56px; width: 291px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image131.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, <img decoding=\"async\" style=\"height: 76px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image132.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 91px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image133.png\" \/><\/p>\n<p style=\"text-align: justify;\">Similarly, <img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image134.png\" \/><img decoding=\"async\" style=\"height: 44px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image135.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting these values in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 235px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image136.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 48px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image137.png\" \/>\u00a0\u00a0[Multiplying by <img decoding=\"async\" style=\"height: 45px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image138.png\" \/>]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 85px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image139.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 45px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image140.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 52px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image141.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 28px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image142.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 28px; width: 236px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image143.png\" \/><\/p>\n<p style=\"text-align: justify;\">Multiplying every term in the numerator and denominator of L.H.S. by 3, and dividing every term by <img decoding=\"async\" style=\"height: 24px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image144.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image146.png\" \/>\u00a0 where A = <img decoding=\"async\" style=\"height: 44px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image147.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_Find_the_equation_of_the_curve_passing_through_the_point_whose_differential_equation_is\"><\/span><strong>8. Find the equation of the curve passing through the point <\/strong><strong><img decoding=\"async\" style=\"height: 45px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image148.png\" \/><\/strong><strong>\u00a0whose differential equation is <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 201px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image149.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image150.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image151.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image152.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image153.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image154.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image155.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image156.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image157.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 17px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image158.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now, curve (i) passes through <img decoding=\"async\" style=\"height: 45px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image159.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, putting <img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image160.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image161.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image162.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image162.png\" \/>\u00a0in eq. (i),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 25px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image163.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image164.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image165.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Find_the_particular_solution_of_the_differential_equation_given_that_when\"><\/span><strong>9. Find the particular solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image166.png\" \/><\/strong><strong>\u00a0given that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image167.png\" \/><\/strong><strong>\u00a0when <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image168.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image166.png\" \/><\/p>\n<p style=\"text-align: justify;\">Dividing every term by <img decoding=\"async\" style=\"height: 29px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image169.png\" \/>\u00a0we have\u00a0 <img decoding=\"async\" style=\"height: 47px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image170.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 47px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image171.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image172.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now to evaluate <img decoding=\"async\" style=\"height: 44px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image173.png\" \/>, putting <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image174.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image175.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image176.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image066.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image173.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image177.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting this value in eq. (i),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image178.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now putting <img decoding=\"async\" style=\"height: 21px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image179.png\" \/>\u00a0in eq. (ii),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image180.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image181.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image182.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image183.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 41px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image183.png\" \/>\u00a0in eq. (ii),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image184.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Solve_the_differential_equation\"><\/span><strong>10. Solve the differential equation: <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image185.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image186.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 47px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image187.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image188.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">It is not a homogeneous differential equation because of presence of only <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image189.png\" \/>\u00a0as a factor, yet it can be solved by putting <img decoding=\"async\" style=\"height: 44px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image190.png\" \/>\u00a0i.e., <img decoding=\"async\" style=\"height: 17px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image191.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image192.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting these values in eq. (i), we get<\/p>\n<p style=\"text-align: justify;\">\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image193.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image194.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image195.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image196.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image197.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"11_Find_the_particular_solution_of_the_differential_equation_given_that_when\"><\/span><strong>11. Find the particular solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image198.png\" \/><\/strong><strong>\u00a0given that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image199.png\" \/><\/strong><strong>\u00a0when <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image168.png\" \/><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image198.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image200.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image201.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 204px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image202.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image203.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image204.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image205.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image206.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image207.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image208.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting this value in eq. (i), <img decoding=\"async\" style=\"height: 45px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image209.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image210.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image211.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image212.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image213.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image214.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image215.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image216.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image217.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image218.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image219.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image205.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image220.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image221.png\" \/>\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p style=\"text-align: justify;\">Now putting <img decoding=\"async\" style=\"height: 21px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image222.png\" \/>\u00a0in eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image223.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image224.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image225.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image225.png\" \/>\u00a0in eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image226.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_Solve_the_differential_equation\"><\/span><strong>12. Solve the differential equation: <\/strong><strong><img decoding=\"async\" style=\"height: 53px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image227.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 53px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image228.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image229.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image230.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing this equation with <img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image231.png\" \/>,\u00a0 P = <img decoding=\"async\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image232.png\" \/>\u00a0and Q = <img decoding=\"async\" style=\"height: 48px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image233.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 48px; width: 269px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image234.png\" \/>\u00a0\u00a0 I.F. = <img decoding=\"async\" style=\"height: 28px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image235.png\" \/><\/p>\n<p style=\"text-align: justify;\">The general solution is\u00a0 <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image189.png\" \/>\u00a0(I.F.) = <img decoding=\"async\" style=\"height: 29px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image236.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 255px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image237.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 75px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image238.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image239.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"13_Find_the_particular_solution_of_the_differential_equation_given_that_when\"><\/span><strong>13. Find the particular solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image240.png\" \/><\/strong><strong>\u00a0given that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image241.png\" \/><\/strong><strong>\u00a0when <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image242.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image243.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing this equation with <img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image231.png\" \/>,\u00a0 P = <img decoding=\"async\" style=\"height: 17px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image244.png\" \/>\u00a0and Q = <img decoding=\"async\" style=\"height: 19px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image245.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 29px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image246.png\" \/>\u00a0\u00a0\u00a0\u00a0 I.F. = <img decoding=\"async\" style=\"height: 28px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image247.png\" \/><\/p>\n<p style=\"text-align: justify;\">The general solution is\u00a0 <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image189.png\" \/>\u00a0(I.F.) = <img decoding=\"async\" style=\"height: 29px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image236.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image248.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 208px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image249.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image250.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image251.png\" \/>\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now putting <img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image252.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image253.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image254.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 44px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image254.png\" \/>\u00a0in eq. (i),\u00a0\u00a0\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image255.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"14_Find_the_particular_solution_of_the_differential_equation_given_that_when\"><\/span><strong>14. Find the particular solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image256.png\" \/><\/strong><strong>\u00a0given that <\/strong><strong><img decoding=\"async\" style=\"height: 21px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image241.png\" \/><\/strong><strong>\u00a0when <\/strong><strong><img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image168.png\" \/><\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image256.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image257.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image258.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image259.png\" \/><\/p>\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image260.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image261.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image262.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image263.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image066.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image264.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image265.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image261.png\" \/>,\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image266.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image267.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image268.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 115px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image269.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image270.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image271.png\" \/>\u00a0where C = <img decoding=\"async\" style=\"height: 21px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image272.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 21px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image273.png\" \/>\u00a0in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image274.png\" \/>\u00a0= C<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0C = 1<\/p>\n<p style=\"text-align: justify;\">Putting C = 1 in eq. (i),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image275.png\" \/><\/p>\n<p style=\"text-align: justify;\">This solution may be written as <img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image276.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image277.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image278.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image279.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image280.png\" \/><\/p>\n<p style=\"text-align: justify;\">where expresses <img decoding=\"async\" style=\"height: 17px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image189.png\" \/>\u00a0as an explicit function of <img decoding=\"async\" style=\"height: 15px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image281.png\" \/><\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-9\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"15_The_population_of_a_village_increases_continuously_at_the_rate_proportional_to_the_number_of_its_inhabitants_present_at_any_time_If_the_population_of_the_village_was_20000_in_1999_and_25000_in_the_year_2004_what_will_be_the_population_of_the_village_in_2009\"><\/span><strong>15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25,000 in the year 2004, what will be the population of the village in 2009?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Let P be the population of the village at time <img decoding=\"async\" style=\"height: 16px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image282.png\" \/><\/p>\n<p style=\"text-align: justify;\">According to the question, Rate of increase of population of the village is proportional to the number of inhabitants.<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image283.png\" \/>\u00a0\u00a0where <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image284.png\" \/>\u00a0&gt; 0 because of increase and is the constant of proportionality<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image285.png\" \/>\u00a0\u00a0[Separating variables]\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image286.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image287.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Now Population of the village was P = 20,000 in the year 1999.<\/p>\n<p style=\"text-align: justify;\">Let us take the base year 1999 as <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image288.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image289.png\" \/>\u00a0and P = 20000 in eq. (i),\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image290.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now putting <img decoding=\"async\" style=\"height: 21px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image290.png\" \/>\u00a0in eq. (i),\u00a0\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image291.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image292.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image293.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026..(ii)<\/p>\n<p style=\"text-align: justify;\">Again Population of the village was P = 25,000 in the year 2004, when <img decoding=\"async\" style=\"height: 19px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image294.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image295.png\" \/>\u00a0and P = 25000 in eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image296.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image297.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image298.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting value of <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image284.png\" \/>\u00a0in eq. (ii),\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image299.png\" \/>\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026..(iii)<\/p>\n<p style=\"text-align: justify;\">To find the population in the year 2009, <img decoding=\"async\" style=\"height: 19px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image300.png\" \/><\/p>\n<p style=\"text-align: justify;\">Putting <img decoding=\"async\" style=\"height: 19px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image301.png\" \/>\u00a0in eq. (iii),\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image302.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 49px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image303.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image304.png\" \/>\u00a025 x 1250 = 31250<\/p>\n<hr \/>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Choose_the_correct_answer\"><\/span><strong>Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>16. The general solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image305.png\" \/><\/strong><strong>\u00a0is:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image306.png\" \/>\u00a0= C\u00a0 <\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image307.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image308.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image309.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 44px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image305.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image310.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image311.png\" \/>\u00a0\u00a0 [Separating variables]\n<p style=\"text-align: justify;\">Integrating both sides,\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image312.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image313.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 20px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image314.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image315.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image308.png\" \/>\u00a0 where C = <img decoding=\"async\" style=\"height: 41px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image316.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-10\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"17_The_general_equation_of_a_differential_equation_of_the_type_is\"><\/span><strong>17. The general equation of a differential equation of the type <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image317.png\" \/><\/strong><strong>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <\/strong><strong><img decoding=\"async\" style=\"height: 43px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image318.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <\/strong><strong><img decoding=\"async\" style=\"height: 43px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image319.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <\/strong><strong><img decoding=\"async\" style=\"height: 43px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image320.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <\/strong><strong><img decoding=\"async\" style=\"height: 43px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image321.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>We know that general solution of differential equation of the type <img decoding=\"async\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image322.png\" \/>\u00a0is<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 29px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image323.png\" \/>\u00a0\u00a0 where <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image324.png\" \/>= <img decoding=\"async\" style=\"height: 28px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image325.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image066.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 43px; width: 181px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image326.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-11\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"18_The_general_solution_of_the_differential_equation_is\"><\/span><strong>18. The general solution of the differential equation <\/strong><strong><img decoding=\"async\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image327.png\" \/><\/strong><strong>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" style=\"height: 21px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image328.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" style=\"height: 24px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image329.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" style=\"height: 24px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image330.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" style=\"height: 24px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image330.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: Differential equation\u00a0 <img decoding=\"async\" style=\"height: 29px; width: 164px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image327.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image331.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image332.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image333.png\" \/><\/p>\n<p style=\"text-align: justify;\">Comparing with <img decoding=\"async\" style=\"height: 41px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image334.png\" \/>\u00a0\u00a0P = 1 and Q = <img decoding=\"async\" style=\"height: 61px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image335.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 29px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image336.png\" \/>\u00a0\u00a0\u00a0\u00a0 I.F. = <img decoding=\"async\" style=\"height: 28px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image337.png\" \/><\/p>\n<p style=\"text-align: justify;\">Solution is <img decoding=\"async\" style=\"height: 29px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image338.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image339.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image340.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image341.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image342.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image017.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/09%20Misc\/image330.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (C) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-12\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-8\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4535","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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