The document NCERT Solutions (Part - 1) - Differential Equations JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.

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**Differential Equations**

**Question 1: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

**Question 2: Determine order and degree(if defined) of differential equation**

**ANSWER : -** The given differential equation is:

The highest order derivative present in the differential equation is. Therefore, its order is one.

It is a polynomial equation in. The highest power raised tois 1. Hence, its degree is one.

**Question 3: Determine order and degree(if defined) of differential equation**

**ANSWER : -**

The highest order derivative present in the given differential equation is. Therefore, its order is two.

It is a polynomial equation inand. The power raised tois 1.

Hence, its degree is one.

**Question 4: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the given differential equation is. Therefore, its order is 2.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

**Question 5: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is two.

It is a polynomial equation inand the power raised tois 1.

Hence, its degree is one.

**Question 6: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is three.

The given differential equation is a polynomial equation in.

The highest power raised tois 2. Hence, its degree is 2.

**Question 7: Determine order and degree(if defined) of differential equation**

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is three.

It is a polynomial equation in. The highest power raised tois 1. Hence, its degree is 1.

**Question 8: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is one.

The given differential equation is a polynomial equation inand the highest power raised tois one. Hence, its degree is one.

**Question 9: Determine order and degree(if defined) of differential equation**

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is two.

The given differential equation is a polynomial equation inandand the highest power raised tois one.

Hence, its degree is one.

**Question 10: Determine order and degree(if defined) of differential equation **

**ANSWER : -**

The highest order derivative present in the differential equation is. Therefore, its order is two.

This is a polynomial equation inandand the highest power raised tois one. Hence, its degree is one.

**Question 11: The degree of the differential equation is**

**(A)** 3

**(B)** 2

**(C)** 1

**(D)** not defined

**ANSWER : -**

The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.

Hence, the correct answer is D.

**Question 12: The order of the differential equationis**

**(A)** 2

**(B)** 1

**(C)** 0

**(D)** not defined

**ANSWER : -**

The highest order derivative present in the given differential equation is. Therefore, its order is two.

Hence, the correct answer is A.

**Question 13: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Now, differentiating equation (1) with respect to *x*, we get:

Substituting the values ofin the given differential equation, we get the L.H.S. as:

Thus, the given function is the solution of the corresponding differential equation.

**Question 14: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

L.H.S. == R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

**Question 15: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

L.H.S. == R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

**Question 16: **

**ANSWER : -**

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

L.H.S. = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

**Question 17: **

**ANSWER : -**

Differentiating both sides with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

**Question 18: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

**Question 19: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

L.H.S. = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

**Question 20: **

**ANSWER : -**

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

Substituting the value ofin equation (1), we get:

Hence, the given function is the solution of the corresponding differential equation.

**Question 21: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

**Question 21: **

**ANSWER : -**

Differentiating both sides of this equation with respect to *x*, we get:

Substituting the value ofin the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

**Question 22: The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:**

**(A)** 0

**(B)** 2

**(C)** 3

**(D)** 4

**ANSWER : -** We know that the number of constants in the general solution of a differential equation of order *n* is equal to its order.

Therefore, the number of constants in the general equation of fourth order differential equation is four.

Hence, the correct answer is D.

**Question 23: The numbers of arbitrary constants in the particular solution of a differential equation of third order are:**

**(A) 3 **

**(B) 2 **

**(C) 1 **

**(D) 0**

**ANSWER : -** In a particular solution of a differential equation, there are no arbitrary constants.

Hence, the correct answer is D.

**Question 24: **

**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 25: **

**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 26: **

**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 27: **

**ANSWER : -**

Differentiating both sides with respect to *x*, we get:

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

Differentiating both sides with respect to *x*, we get:

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

This is the required differential equation of the given curve.

**Question 28: **

**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 29: 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 30: 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 31: 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 32: 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 33: 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* â€“ *b*) in equation (1), we get:

This is the required differential equation.

**Question 34: 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 35: 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.

**Question 36: **

**ANSWER : -** The given differential equation is:

Now, integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

**Question 37: **

**ANSWER : -** The given differential equation is:

Now, integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

**Question 38: **

**ANSWER : -** The given differential equation is:

Now, integrating both sides, we get:

This is the required general solution of the given differential equation.

**Question 39: **

**ANSWER : -** The given differential equation is:

Integrating both sides of this equation, we get:

Substituting these values in equation (1), we get:

This is the required general solution of the given differential equation.

**Question 40: **

**ANSWER : -** The given differential equation is:

Integrating both sides of this equation, we get:

Let (*e ^{x}*

Differentiating both sides with respect to *x*, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

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