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NCERT Solutions class 12 Maths Differential Equations
1. For each of the differential equations given below, indicate its order and degree (if defined):
(i) 
(ii) 
(iii) 
Ans. (i) Given: Differential equation
The highest order derivative present in this differential equation is 
and hence order of this differential equation if 2.
The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative is 1.
Therefore, Order = 2, Degree = 1
(ii) Given: Differential equation
The highest order derivative present in this differential equation is 
and hence order of this differential equation if 1.
The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative is 3.
Therefore, Order = 1, Degree = 3
(iii) Given: Differential equation
The highest order derivative present in this differential equation is 
and hence order of this differential equation if 4.
The given differential equation is not a polynomial equation in derivatives therefore, degree of this differential equation is not defined.
Therefore, Order = 4, Degree not defined
2. For each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:
(i) 
(ii) 
(iii) 
(iv) 
Ans. (i) The given function is 
……….(i)
To verify: Function (i) is a solution of D.E. 
……….(ii)
Differentiating both sides of eq. (i) w.r.t. 
Again differentiating both sides w.r.t.
Putting 
from eq. (i), we have,
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
(ii) The given function is 
……….(i)
To verify: Function given by (i) is a solution of D.E. 
…….(ii)
From (i), 
[By eq. (i)] ……….(iii)
[Using eq. (iii) and (i)]
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
(iii) The given function is 
………(i)
To verify: Function given by eq. (i) is a solution of D.E. 
…(ii)
From eq. (i), 
[Using eq. (i)]
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
(iv) The given function is 
……….(i)
To verify: Function given by eq. (i) is a solution of D.E. 
……(ii)
Differentiating both sides of eq. (i) w.r.t. 
Putting 
from eq. (i), we get
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
3. Form the differential equation representing the family of curves 
where 
ia an arbitrary constant.
Ans. Equation of the given family of curves is 
……….(i)
Here number of arbitrary constants is one only 
So, we will differentiate both sides of equation only once, w.r.t.
……….(ii)
Dividing eq. (i) by eq. (ii), we have 
4. Prove that 
is the general equation of the differential equation 
where 
is a parameter.
Ans. Given: Differential equation 
……….(i)
Here each coefficient of 
and 
is of same degree, i.e., 3, therefore differential equation looks to be homogeneous.
……….(ii)
Therefore, the given differential equation is homogeneous.
Putting 
Putting these values in eq. (ii), 
[Separating variables]
Integrating both sides, 
……….(iii)
Now forming partial fraction of 
= 
= 
= 
……….(iv)
Comparing coefficients of like powers of
A – B – C = 1 ……….(v)
A + B – D = 0 ……….(vi)
A – B + C = –3 ……….(vii)
Constants A + B + D = 0 ………(viii)
Now eq. (v) – eq. (vii)
– 2C = 4 C = – 2
Eq. (vi) – eq. (viii)
– 2D = 0 D = 0
Putting C = – 2 in eq. (v), A – B + 2 = 1
A – B = –1 ……….(ix)
Putting D = 0 in eq. (vi) A + B = 0 ……….(x)
Adding eq. (ix) and (x) 2A = –1
A = 
From eq. (x), B = –A = 
Putting the values of A, B, C and D in eq. (iv), we have
= 
Putting this value in eq. (iii), 
Squaring both sides and cross-multiplying, 
Putting 
where 
5. For the differential equation of the family of the circles in the first quadrant which touch the coordinate axes.
Ans. We know that the circle in the first quadrant which touches the co-ordinates axes has centre 
where is the radius of the circle.
Equation of the circle is
……….(i)
Differentiating with respect to 
Substituting value of in eq. (i), 
6. Find the general solution of the differential equation 
Ans. Given: Differential Equation
Integrating both sides 
NCERT Solutions class 12 Maths Miscellaneous
7. Show that the general solution of the differential equation 
is given by 
Ans. Given: Differential equation
Integrating both sides, 
……….(i)
Now 
[Completing the squares]
Therefore, 
= 
Similarly, 
Putting these values in eq. (i), 
[Multiplying by 
]
where 
Multiplying every term in the numerator and denominator of L.H.S. by 3, and dividing every term by 
where A = 
NCERT Solutions class 12 Maths Miscellaneous
8. Find the equation of the curve passing through the point 
whose differential equation is 
Ans. Given: Differential equation 
Integrating both sides, 
……….(i)
Now, curve (i) passes through 
Therefore, putting 
in eq. (i), 
Putting in eq. (i), 
NCERT Solutions class 12 Maths Miscellaneous
9. Find the particular solution of the differential equation 
given that 
when 
Ans. Given: Differential equation
Dividing every term by 
we have 
Integrating both sides, 
……….(i)
Now to evaluate 
, putting 
= 
Putting this value in eq. (i), 
……….(ii)
Now putting 
in eq. (ii), 
Putting in eq. (ii), 
NCERT Solutions class 12 Maths Miscellaneous
10. Solve the differential equation: 
Ans. Given: Differential equation 
……….(i)
It is not a homogeneous differential equation because of presence of only 
as a factor, yet it can be solved by putting 
i.e., 
Putting these values in eq. (i), we get
NCERT Solutions class 12 Maths Miscellaneous
11. Find the particular solution of the differential equation 
given that 
when
Ans. Given: Differential equation
……….(i)
Putting 
Putting this value in eq. (i), 
= 
Integrating both sides, 
Putting , 
……….(ii)
Now putting 
in eq. (ii), 
Putting in eq. (ii), 
NCERT Solutions class 12 Maths Miscellaneous
12. Solve the differential equation: 
Ans. Given: Differential equation 
Comparing this equation with 
, P = 
and Q = 
I.F. = 
The general solution is (I.F.) = 
NCERT Solutions class 12 Maths Miscellaneous
13. Find the particular solution of the differential equation 
given that 
when 
Ans. Given: Differential equation 
Comparing this equation with , P = 
and Q = 
I.F. = 
The general solution is (I.F.) =
……….(i)
Now putting 
in eq. (i), 
Putting in eq. (i), 
NCERT Solutions class 12 Maths Miscellaneous
14. Find the particular solution of the differential equation 
given that when
Ans. Given: Differential equation
Integrating both sides, 
Putting 
Putting , 
where C = 
……….(i)
Putting 
in eq. (i), 
= C
C = 1
Putting C = 1 in eq. (i), 
This solution may be written as 
= 
where expresses as an explicit function of 
NCERT Solutions class 12 Maths Miscellaneous
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?
Ans. Let P be the population of the village at time 
According to the question, Rate of increase of population of the village is proportional to the number of inhabitants.
where 
> 0 because of increase and is the constant of proportionality
[Separating variables]
Integrating both sides, 
……….(i)
Now Population of the village was P = 20,000 in the year 1999.
Let us take the base year 1999 as 
Putting 
and P = 20000 in eq. (i), 
Now putting in eq. (i), 
………..(ii)
Again Population of the village was P = 25,000 in the year 2004, when 
Putting 
and P = 25000 in eq. (ii), 
Putting value of in eq. (ii), 
………..(iii)
To find the population in the year 2009, 
Putting 
in eq. (iii), 
25 x 1250 = 31250
Choose the correct answer:
16. The general solution of the differential equation 
is:
(A) 
= C
(B) 
(C) 
(D) 
Ans. Given: Differential equation
[Separating variables]
Integrating both sides, 
where C = 
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Miscellaneous
17. The general equation of a differential equation of the type 
is:
(A) 
(B) 
(C) 
(D) 
Ans. We know that general solution of differential equation of the type 
is
where 
= 
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Miscellaneous
18. The general solution of the differential equation 
is:
(A) 
(B) 
(C) 
(D)
Ans. Given: Differential equation
Comparing with 
P = 1 and Q = 
I.F. = 
Solution is 
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Miscellaneous
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