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**Download NCERT solutions for Differential Equations as PDF**

## NCERT Solutions class 12 Maths Differential Equations

**For each of the differential equations in Questions 1 to 4, find the general solution:**

**1. **** **

**Ans. **Given: Differential equation

Integrating both sides,

**2. **** **

**Ans. **Given: Differential equation

Integrating both sides,

**3. **** **

**Ans. **Given: Differential equation

Integrating both sides,

, where

**4. **** **

**Ans. **Given: Differential equation

Dividing by we have

**5. **** **

**Ans. **Given: Differential equation

Integrating both sides,

**6. **** **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides,

**7. **** **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides,

Putting on L.H.S., we get

Now eq. (i) becomes

[If all the terms in the solution of a differential equation involve log, it is better to use or instead of in the solution.]

where

**8. **** **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides

, where

**9. **** **

**Ans. **Given: Differential equation

Integrating both sides,

Applying product rule, I II

= ……….(i)

To evaluate

Putting, differentiate

=

Putting this value in eq. (i), the required general solution is

**10. **** **

**Ans. **Given: Differential equation

Dividing each term by we get

[Separating variables]

Integrating both sides,

**For each of the differential equations in Question 11 to 12, find a particular solution satisfying the given condition:**

**11. **** when **** **

**Ans. **Given:

[Separating variables]

Integrating both sides,

…(i)

Let [Partial fraction] ……….(ii)

Comparing the coefficients of on both sides, A + B = 2 ……….(iii)

Comparing the coefficients of on both sides, B + C = 1 ……….(iv)

Comparing constants on both sides, A + C = 0 ……….(v)

From eq. (iii) – (iv), we haveA – C = 1 ……….(vi)

Adding eq. (v) and (vi), we have 2A = 1

A =

From eq. (v), we have C = – A =

Putting the value of C in eq. (iv), B – = 1

B = 1 + =

Putting the values of A, B, and C in eq. (ii), we have

=

=

Putting this value in eq. (i),

……….(vii)

Now, when , putting these values in eq. (vii),

Putting value of in eq. (vii), the required general solution is

**12. **** when **** **

**For each of the differential equations in Question 13 to 14, find a particular solution satisfying the given condition.**

**Ans. **Given: Differential equation

Integrating both sides,

……….(i)

Let ……….(ii)

Comparing the coefficients of on both sides, A + B + C = 0 ……….(iii)

Comparing the coefficients of on both sides, B + C = 0

C = B ……….(iv)

Comparing constants on both sides, A = 1

A = 1 ……….(v)

Putting A = 1 and C = B in eq. (iii),

1 + B + B = 0 2B = 1 B =

From eq. (iv),C = B =

Putting the values of A, B and C in eq. (ii), we get

Putting this value in eq. (i),

……….(v)

Now, putting when in eq. (v), we get

Putting the value of in eq. (v), the required general solution is

[NOTE: You can also do, to evaluate = = Put ]

**13. **** when **** **

**Ans. **Given: Differential equation when

Integrating both sides,

……….(i)

Now putting when in eq. (i), we get

Putting in eq. (i),

**14. **** when **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides,

where ……….(i)

Now putting and in eq. (i), we get

Putting C = 1 in eq. (i), we get the required general solution

**NCERT Solutions class 12 Maths Exercise 9.4**

**15. Find the equation of the curve passing through the point (0, 0) and whose differential equation is **** **

**Ans. **Given: Differential equation

Integrating both sides

………(i)

where I = ……….(ii)

Applying product rule,

Again applying product rule,

[By eq. (ii)]

2I =

I =

Putting this value of I in eq. (i), we get

……….(iii)

Now putting and in eq. (iii)

Putting the value of in eq. (iii), we get the required general solution

**NCERT Solutions class 12 Maths Exercise 9.4**

**16. For the differential equation **** find the solution curve passing through the point **** **

**Ans. **Given: Differential equation

[Separating both sides]

Integrating both sides

……….(i)

Now putting in eq. (i),

Putting this value of in eq. (i) to get the required solution curve

**NCERT Solutions class 12 Maths Exercise 9.4**

**17. Find the equation of the curve passing through the point **** given that at any point **** on the curve the product of the slope of its tangent and ****coordinate of the point is equal to the ****coordinate of the point.**

**Ans. **Let P be any point on the required curve.

According to the question, Slope of the tangent to the curve at P

Integrating both sides,

where

Now it is given that curve passes through the point .

Therefore, putting and in this equation, we get C = 4

Putting the value of C in the equation ,

**NCERT Solutions class 12 Maths Exercise 9.4**

**18. At any point **** of a curve the slope of the tangent is twice the slope of the line segment joining the point of contact to the point **** Find the equation of the curve given that it passes through **** **

**Ans. **According to the question, slope of the tangent at any point P of the required curve

= 2. Slope of the line joining the point of contact P to the given point A

[Separating variables]

Integrating both side,

where ……….(i)

Now it is given that curve (i) passes through the point

Therefore, putting and in eq. (i),

4 = 4C C = 1

Putting C = 1 in eq. (i), we get the required solution,

**NCERT Solutions class 12 Maths Exercise 9.4**

**19. The volume of the spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after **** seconds.**

**Ans. **Let be the radius of the spherical balloon at time

Given: Rate of change of volume of spherical balloon is constant = (say)

[Separating variables]

Integrating both sides,

……….(i)

Now it is given that initially radius is 3 units, when

Therefore, putting in eq. (i),

……….(ii)

Again when sec, then units

Therefore, putting and in eq. (i),

[From eq. (ii)]

……….(iii)

Putting the value of and in eq. (i), we get

**NCERT Solutions class 12 Maths Exercise 9.4**

**20. In a bank principal increases at the rate of **** per year. Find the value of **** is ****`**** 100 double itself in 10 years. **** **

**Ans. **Let P be the principal (amount) at the end of years.

According to the given condition, rate of increase of principal per year = (of principal)

[Separating variables]

Integrating both sides, ……….(i)

[Since P being principal > 0, hence ]

Now initial principal = ` 100 (given), i.e., when then P = 100

Therefore, putting P = 100 in eq. (i),

Putting in eq. (i),

……….(ii)

Now putting P = double of itself = 2 x 100 = ` 200, when years (given)

**NCERT Solutions class 12 Maths Exercise 9.4**

**21. In a bank principal increases at the rate of **** per year. An amount of ****`**** 1000 is deposited with this bank, how much will it worth after 10 years? **** **

**Ans. **Let P be the principal (amount) at the end of years.

According to the given condition, rate of increase of principal per year = (of principal)

[Separating variables]

Integrating both sides, ……….(i)

[Since P being principal > 0, hence ]

Now initial principal = ` 1000 (given), i.e., when then P = 1000

Therefore, putting P = 1000 in eq. (i),

Putting in eq. (i),

……….(ii)

Now putting years (given)

P = 1000 x 1.648 = ` 1648

**NCERT Solutions class 12 Maths Exercise 9.4**

**22. In a culture the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present.**

**Ans. **Let be the bacteria present in the culture at time hours.

According to the question,

Rate of growth of bacteria is proportional to the number present

is proportional to

where is the constant of proportionality

Integrating both sides,

……….(i)

Now it is given that initially the bacteria count is (say) = 1,00,000

when then

Putting these values in eq. (i)

Putting in eq. (i), we get

……….(ii)

Now it is given also that the number of bacteria increased by 10% in 2 hours.

Therefore, increase in bacteria in 2 hours = = 10,000

the amount of bacteria at = 1,00,000 + 10,000 = 1,10,000 = (say)

Putting and in eq. (ii),

Putting this value of in eq. (ii), we get,

[when ]

hours

### NCERT Solutions class 12 Maths Exercise 9.4

**23. The general solution of the differential equation **** is:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides,

where which is required solution

Therefore, option (A) is correct.

## NCERT Solutions class 12 Maths Exercise 9.4

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

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