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Question 1:

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

Solution 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

Comments

Sadika

Question 2:

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

Solution 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.

Comments

Sadika

Question 3:

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

Solution 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.

Comments

Sadika

Question 4:

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

Solution 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= $e^{2x}$ (2a+2bx+b)− $e^{2x}$ (2a+2bx)

⇒ y'−2y= $b^{^{2x}}$ ...(3)

Differentiating both sides with respect to x, we get:

y''-2y'= $2be^{2x}}}$ ...(4)

Dividing equation (4) by equation (3), we get:

This is the required differential equation of the given curve.

Comments

Sadika

Question 5:

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

Solution 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.

Comments

Sadika

Question 6:

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

Solution 6:

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.

Comments

Sadika

Question 7:

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

Solution 7:

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.

Comments

Sadika

Question 8:

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

Solution 8:

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.

Comments

Sadika

Question 9:

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

Solution 9:

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.

Comments

Sadika

Question 10:

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

Solution 10:

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.

Comments

Sadika

Question 11:

Which of the following differential equations hasas the general solution?

Solution 11:

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.

Comments

Sadika

Question 12:

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

Solution 12:

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.

Comments

Tasneem

All Exercises - Chapter 9 - Differential Equations

Exercise 9.3

Miscellaneous Exercise 9

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Learn Exercise 9.3 with Free Lessons & Tips

All Exercises - Chapter 9 - Differential Equations

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