## EXERCISE 3.3

#### 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.