# NCERT solution class 12 chapter 3 Differential Equations exercise 3.3 mathematics part 2

## EXERCISE 3.3

#### Question 1:  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:  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:  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:  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:  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.

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.

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.

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.

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 of in 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.

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 has as the general solution?

A. B. C. D. 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 has as one of its particular solution?

A. B. C. D. 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.

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