# NCERT Solutions for Class 12 Mathemetics Chapter 9 - Differential Equations

##### Question 1:

Determine order and degree (if defined) of differential equations given in

Order     Degree
4     Not a polynomial in derivatives

##### Question 2:

Determine order and degree (if defined) of differential equations given in y' + 5y = 0.

Order     Degree
1              1

##### Question 3:

Determine order and degree (if defined) of differential equations given in

Order     Degree
2             1

##### Question 4:

Determine order and degree (if defined) of differential equations given in

Order     Degree
2             Not a polynomial in derivatives

##### Question 5:

Determine order and degree (if defined) of differential equations given in

Order     Degree
2             1

##### Question 6:

Determine order and degree (if defined) of differential equations given in
(y''')2 + (y'')3 + (y')4 + y5 = 0.

Order     Degree
3             2

##### Question 7:

Determine order and degree (if defined) of differential equations given in
y''' + 2y'' + y' = 0.

Order     Degree
3             1

##### Question 8:

Determine order and degree (if defined) of differential equations given in
y'+ y = 0.

Order     Degree
1             1

##### Question 9:

Determine order and degree (if defined) of differential equations given in
y''+ (y')2 + 2y = 0.

Order     Degree
2             1

##### Question 10:

Determine order and degree (if defined) of differential equations given in
y''+ 2y'+ sin y = 0.

Order     Degree
2             1

##### Question 11:

The degree of the differential equation :

1. 3
2. 2
3. 1
4. not defined

not defined

##### Question 12:

The order of the differential equation :

1. 2
2. 1
3. 0
4. Not defined.

2

##### Question 13:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
y = ex + 1 : y''– y'= 0

We have : y = ex + 1.
$\therefore$ y' = ex and y'' = ex ... (1)
Now y''– y' = ex – ex [Using (1)]
= 0, which is true.

##### Question 14:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
y = x2 + 2x + C : y'– 2x – 2 = 0.

We have : y = x2 + 2x + C.
$\therefore$ y'= 2x + 2
$⇒$ y'– 2x – 2 = 0, which is true.

##### Question 15:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
y = cos x + C : y' sin x = 0

We have : y = cos x + C.
$\therefore$ y' = – sin x + C
$⇒$ y' + sin x = 0, which is true.

##### Question 16:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :

##### Question 17:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
y = Ax : xy'= y (x $\ne$ 0)

We have : y= Ax ...(1)
$\therefore$ y'= A.
Putting the value of A in (1),
y = y' x $⇒$ xy' = y,
which is true.

##### Question 18:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :

We have : y = x sin x ...(1)
$\therefore$ y' = x cos x + sin x . 1
$⇒$ y'= sin x + x cos x ...(2)

##### Question 19:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :

##### Question 20:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
y – cos x = x : (y sin y + cos y + x)y'= y.

We have : y – cos y = x ...(1)
Diff. w.r.t. x, y'+ sin y . y'= 1
$⇒$ (1 + sin y) y'= 1
$⇒$ (y + y sin y)y' = y ...(2) [Multiplying by y]
From (1), y = x + cos y.
Putting in (2), (x + cos y + y sin y) y'= y
$⇒$(y sin y + cos y + x) y'= y, which is true.

##### Question 21:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :
x + y = tan–1 y : y2y'+ y2 + 1 = 0

We have : x + y = tan–1 y.

$⇒$ (1 + y2) (1 + y') = y'
$⇒$ 1 + y2 + y'(1 + y2) = y'
$⇒$ 1 + y2 + y'(1 + y2 – 1) = 0
$⇒$ 1 + y2 + y'y2 = 0
$⇒$ y2y' + y2 + 1 = 0, which is true.

##### Question 22:

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation :

##### Question 23:

The number of arbitrary constants in the general solution of a differential equation of fourth order is :

1. 0
2. 2
3. 3
4. 4

4

##### Question 24:

The number of arbitrary constants in the particular solution of a differential equation of third order is :

1. 3
2. 2
3. 1
4. 0

0

##### Question 25:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

##### Question 26:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :
y2 = a (b2 – x2)

We have : y2 = a (b2 – x2) ...(1)
Diff. w.r.t. x, 2yy' = a (0 – 2x)
$⇒$ yy' = – ax ...(2)
Again diff. w.r.t. x, yy'+ y'2 = – a ...(3)
Dividing (3) by (2),

$⇒$ x (yy''+ y'2) = yy'
$⇒$ xyy''+ x y'2 – yy' = 0,

##### Question 27:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :
y = ae3x + be–2x

We have : y = ae3x + be– 2x ...(1)
Diff. w.r.t. x, y' = 3ae3x – 2b e–2x ...(2)
Again diff. w.r.t. x, y' = 9ae3x + 4be– 2x ...(3)
Multiplying (1) by 2,
2y = 2ae3x + 2be–2x ...(4)
Adding (2) and (4), y''+ 2y = 5ae3x

##### Question 28:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :
y = e2x (a + bx)

We have : y = e2x (a + bx) ...(1)
Diff. w.r.t. x, y'= e2x (b) + 2e2x (a + bx)
$⇒$ y'= e2x (2a + b + 2bx) ...(2)
Multiplying (1) by 2, 2y = e2x (2a + 2bx) ...(3)
Subtracting (3) from (2),
y'– 2y = be2x ...(4)
Again diff. w.r.t x, y''– 2y'= 2be2x ...(5)

##### Question 29:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :
y = ex (a cos x + b sin x).

We have : y = ex (a cos x + b sin x) ...(1)
Diff. w.r.t. x,
y''= ex (– a sin x + b cos x)
+ ex (a cos x + b sin x)
y''= ex [(a + b) cos x – (a – b) sin x] ...(2)
Again diff. w.r.t. x,
y''= ex [– (a + b) sin x – (a – b) cos x]
+ ex [(a + b) cos x – (a – b) sin x]
$⇒$ y'' = ex [2b cos x – 2 a sin x]

##### Question 30:

Form the differential equation of the family of circles touching the y-axis at origin.

Let ($\mathrm{\alpha }$, 0) be the centre of any member of the circles. Then the equation of the family of circles is :
(x – $\mathrm{\alpha }$)2 + y2 =$\mathrm{\alpha }$2 $⇒$x2 + y2 – 2x = 0 ...(1)

##### Question 31:

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Let the equation of parabolas be :
x2 = 4ay ...(1)
Diff. w.r.t. x,
2x = 4ay'

##### Question 32:

Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

##### Question 33:

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

##### Question 34:

Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Let the equation of the family of circles be :
x2 + (y – $\mathrm{\alpha }$)2 = 9 ...(1)

##### Question 35:

Which of the following differential equation has y = C1 ex + C2 e–x as the general solution ?

##### Question 36:

Which of the following differential equations has y = x as one of its particular solution ?

##### Question 37:

For each of the differential equations, find the general solution :

##### Question 38:

For each of the differential equations, find the general solution :

##### Question 39:

For each of the differential equations, find the general solution :

##### Question 40:

For each of the differential equations, find the general solution :
sec2 x tan y dx + sec2 y tan x dy = 0.

$⇒$ log | tan x tan y | = log | C |
$⇒$ tan x tan y = C,
which is the reqd. solution.

##### Question 41:

For each of the differential equations, find the general solution :
(ex + e– x) dy – (ex – e– x) dx = 0.

We have : (ex + e–x) dy – (ex – e–x) dx = 0

##### Question 42:

For each of the differential equations, find the general solution :

##### Question 43:

For each of the differential equations, find the general solution :
y log y dx – x dy = 0.

##### Question 44:

For each of the differential equations, find the general solution :

##### Question 45:

For each of the differential equations, find the general solution :

##### Question 46:

For each of the differential equations, find the general solution :
ex tan y dx + (1 – ex) sec2 y dy = 0.

##### Question 47:

For each of the differential equations find a particular solution satisfying the given condition :

##### Question 48:

For each of the differential equations find a particular solution satisfying the given condition :

##### Question 49:

For each of the differential equations find a particular solution satisfying the given condition :

##### Question 50:

For each of the differential equations find a particular solution satisfying the given condition :

##### Question 51:

Find the equation of a curve passing through the point (0, 0) and whose differential equation is :
y' = ex sin x.

##### Question 53:

Find the equation of a curve passing through the point (0, – 2), given that at any point (x, y) on the curve the product of the slope of its tangent and y – co-ordinate of the point is equal to the x-co-ordinate of the point.

##### Question 54:

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line joining the point of contact to the point (– 4, – 3). Find the equation of the curve, given that it passes through (– 2, 1).

By the question,

##### Question 55:

The volume of a spherical balloon is 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 the balloon after ‘t’ seconds.

Let ‘r’ be the radius of spherical balloon after time ‘t’.

##### Question 56:

In a bank, principal increases at the rate of r% per year. Find the value of ‘r’ if ₹ 100 double itself in10 years (loge 2 = 0·6931).

Let ‘P’ be the principal at any time t.

##### Question 57:

In a bank, principal, increases continuously at the rate of 5% per year. An amount of ₹ 1000 is deposited with this bank, how much will it worth after 10 years. (e0·5 = 1·648)

Let ‘P’ be the principal at any time t.

##### Question 58:

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 culture reach 2,00,000, if the rate of growth of bacteria is proportional to the number present ?

##### Question 59:

1. ex + e–y = C
2. ex + ey = C
3. e–x + ey = C
4. e–x + e–y = C

ex + e–y = C

##### Question 60:

Show that the given differential equation is homogeneous and solve each of them :
(x2 + xy) dy = (x2 + y2) dx.

##### Question 61:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 62:

Show that the given differential equation is homogeneous and solve each of them :
(x – y) dy – (x + y) dx = 0

##### Question 63:

Show that the given differential equation is homogeneous and solve each of them :
(x2 – y2) dx + 2xy dy = 0.

##### Question 64:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 65:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 66:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 67:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 68:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 69:

Show that the given differential equation is homogeneous and solve each of them :

##### Question 70:

For each of the differential equations find the particular solution satisfying the given condition :
(x + y) dy + (x – y) dx = 0 ;
y = 1 when x = 1.

##### Question 71:

For each of the differential equations find the particular solution satisfying the given condition :
x2 dy + (xy + y2) dx = 0 ;
y = 1 when x = 1.

##### Question 72:

For each of the differential equations find the particular solution satisfying the given condition :

##### Question 73:

For each of the differential equations find the particular solution satisfying the given condition :

##### Question 74:

For each of the differential equations find the particular solution satisfying the given condition :

1. y = vx
2. y = yx
3. x = vy
4. x = y

x = vy

##### Question 76:

Which of the following is a homogeneous differential equation ?

1. (4x + 6y + 5) dy – (3xy + 2x + 4) dx
2. (xy) dx – (x3 + y3) dy = 0
3. (x3 + 2y2) dx + 2xy dy = 0
4. y2dx + (x2 – xy – y2) dy = 0.

y2dx + (x2 – xy – y2) dy = 0.

##### Question 77:

For each of the differential equations find the general solutions :

##### Question 78:

For each of the differential equations find the general solutions :

##### Question 79:

For each of the differential equations find the general solutions :

##### Question 80:

For each of the differential equations find the general solutions :

##### Question 81:

For each of the differential equations find the general solutions :

##### Question 82:

For each of the differential equations find the general solutions :

##### Question 83:

For each of the differential equations find the general solutions :

##### Question 84:

For each of the differential equations find the general solutions :
(1 + x2) dy + 2xy dx = cot x dx (x $\ne$ 0).

##### Question 85:

For each of the differential equations find the general solutions :

##### Question 86:

For each of the differential equations find the general solutions :

##### Question 87:

For each of the differential equations find the general solutions :
y dx + (x – y2) dy = 0.

##### Question 88:

For each of the differential equations find the general solutions :

##### Question 89:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Question 90:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Question 91:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Question 92:

Find the equation of a curve passing through the origin, given that the slope of the tangent to the curve at any point (x, y) is the equal to the sum of the co-ordinates of the point.

##### Question 93:

Find the equation of a curve passing through the point (0, 2), given that the sum of the co-ordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

By the question,

1. e–x
2. e–y
3. x

##### Question 95:

The Integrating Factor of the differential equation :

##### Question 96:

The degree of the differential equation :

1. 3
2. 2
3. 1
4. not defined

not defined

##### Question 97:

The order of the differential equation :

1. 2
2. 1
3. 0
4. not defined

2

##### Question 98:

The number of arbitrary constants in the general solution of a differential equation of fourth order is ;

1. 0
2. 2
3. 3
4. 4

4

##### Question 99:

The number of arbitrary constants in the particular solution of a differential equation of third order is :

1. 3
2. 2
3. 1
4. 0

0

##### Question 100:

Which of the following differential equations has y = c1 ex + c2 e–x as the general solution ?

##### Question 101:

Which of the following differential equations has y = x as one of its particular solutions ?

##### Question 102:

1. ex + e–y = c
2. ex + ey = c
3. e–x + ey = c
4. e–x + e–y = c

ex + e–y = c

##### Question 103:

Which of the following differential equations cannot be solved, using variable separable method ?

1. (y2 – 2xy) dx = (x2 – 2xy) dy

(y2 – 2xy) dx = (x2 – 2xy) dy

1. y = vx
2. v = yx
3. x = vy
4. x = v.

x = vy

##### Question 105:

Which of the following is a homogeneous differential equation ?

1. (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
2. xy dx – (x3 + y3) dy = 0
3. (x3 + 2y2) dx + 2xy dy = 0
4. y2 dx + (x2 – xy – y2) dy = 0.

y2 dx + (x2 – xy – y2) dy = 0.

1. e–x
2. e–y
3. x

1. xy = c
2. x = cy2
3. y = cx
4. y = cx2

y = cx

##### Question 110:

The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is :

1. x ey + x2 = c
2. x ey + y2 = c
3. y ex + x2 = c
4. y ey + x2 = c.

y ex + x2 = c

##### Question 111:

The degree of the differential equation representing the family of curves (x – a)2 + y2 = 16 is :

1. 0
2. 2
3. 3
4. 1