NCERT Solutions: Differential Equations- 1 - Notes | Study Mathematics (Maths) Class 12 - JEE

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 – 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 Cis 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   e^–x) = t.

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.