NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.3

NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.3

Page No 391:

Question 1:

Answer:

Differentiating both sides of the given equation with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Hence, the required differential equation of the given curve is

Question 2:

Answer:

Differentiating both sides with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Dividing equation (2) by equation (1), we get:

This is the required differential equation of the given curve.

Question 3:

Answer:

Differentiating both sides with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Multiplying equation (1) with (2) and then adding it to equation (2), we get:

Now, multiplying equation (1) with 3 and subtracting equation (2) from it, we get:

Substituting the values of in equation (3), we get:

This is the required differential equation of the given curve.

Question 4:

Answer:

Differentiating both sides with respect to x, we get:

Multiplying equation (1) with 2 and then subtracting it from equation (2), we get:

y’-2y=e2x2a+2bx+b-e2x2a+2bx⇒y’-2y=be2x                                      …(3)

Differentiating both sides with respect to x, we get:

y”-2y’=2be2x                        …4Dividing equation (4) by equation (3), we get:

This is the required differential equation of the given curve.

Question 5:

Answer:

Differentiating both sides with respect to x, we get:

Again, differentiating with respect to x, we get:

Adding equations (1) and (3), we get:

This is the required differential equation of the given curve.

Question 6:

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

Answer:

The centre of the circle touching the y-axis at origin lies on the x-axis.

Let (a, 0) be the centre of the circle.

Since it touches the y-axis at origin, its radius is a.

Now, the equation of the circle with centre (a, 0) and radius (a) is

Differentiating equation (1) with respect to x, we get:

Now, on substituting the value of a in equation (1), we get:

This is the required differential equation.

Question 7:

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

Answer:

The equation of the parabola having the vertex at origin and the axis along the positive y-axis is:

Differentiating equation (1) with respect to x, we get:

Dividing equation (2) by equation (1), we get:

This is the required differential equation.

Question 8:

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

Answer:

The equation of the family of ellipses having foci on the y-axis and the centre at origin is as follows:

Differentiating equation (1) with respect to x, we get:

Again, differentiating with respect to x, we get:

Substituting this value in equation (2), we get:

This is the required differential equation.

Question 9:

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

Answer:

The equation of the family of hyperbolas with the centre at origin and foci along the x-axis is:

Differentiating both sides of equation (1) with respect to x, we get:

Again, differentiating both sides with respect to x, we get:

Substituting the value ofin equation (2), we get:

This is the required differential equation.

Question 10:

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

Answer:

Let the centre of the circle on y-axis be (0, b).

The differential equation of the family of circles with centre at (0, b) and radius 3 is as follows:

Differentiating equation (1) with respect to x, we get:

Substituting the value of (y – b) in equation (1), we get:

This is the required differential equation.

Question 11:

Which of the following differential equations hasas the general solution?

A. 

B. 

C. 

D. 

Answer:

The given equation is:

Differentiating with respect to x, we get:

Again, differentiating with respect to x, we get:

This is the required differential equation of the given equation of curve.

Hence, the correct answer is B.

Question 12:

Which of the following differential equation hasas one of its particular solution?

A. 

B. 

C. 

D. 

Answer:

The given equation of curve is y = x.

Differentiating with respect to x, we get:

Again, differentiating with respect to x, we get:

Now, on substituting the values of y,  from equation (1) and (2) in each of the given alternatives, we find that only the differential equation given in alternative C is correct.

Hence, the correct answer is C.