Differential Equations

01

PYQ 2021

medium

mathematics ID: cuet-ug-

The general solution of the differential equation is ________.

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4

02

PYQ 2021

medium

mathematics ID: cuet-ug-

The solution of the differential equation, where P and Q are constants or functions of y, is given by ________.

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03

PYQ 2023

medium

mathematics ID: cuet-ug-

Match List I with List II List I List II A. I. order 2, degree 1 B. II. order 2, degree not defined C. III. order 2, degree 4 D. IV. order 1, degree 2 Choose the correct answer from the options given below :

1

A-I, B-II, C-III, D-IV

2

A-III, B-IV, C-I, D-II

3

A-III, B-II, C-IV, D-I

4

A-III, B-I, C-II, D-IV

04

PYQ 2023

medium

mathematics ID: cuet-ug-

The sum of the order and degree of differential equation is :

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4

2

5

3

6

4

not defined

05

PYQ 2023

medium

mathematics ID: cuet-ug-

The integrating factor of the differential equation (1 +y 2)dx - (tan -1 y - x)dy = 0, is :

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tan -1 y

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3

4

06

PYQ 2023

medium

mathematics ID: cuet-ug-

If and,then value of K is:

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2

2

3

0

4

1

07

PYQ 2023

medium

mathematics ID: cuet-ug-

If, then the sum of order and degree of the differential equation is:

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3

2

2

3

5

4

4

08

PYQ 2023

medium

mathematics ID: cuet-ug-

If then value of is:

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2

3

4

2x+log2

09

PYQ 2023

medium

mathematics ID: cuet-ug-

The differential equation whose solution is Ax 2 +By 2 =1 where A and B are arbitrary constant is of: (A) first order and first degree (B) second order and first degree (C) second order and second degree (D) second order Choose the correct answer from the options given below:

1

(D) Only

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(C) and (D) Only

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(B) and (D) Only

4

(A) Only

10

PYQ 2023

medium

mathematics ID: cuet-ug-

Integrating factor of the differential equation is:

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4

11

PYQ 2023

medium

mathematics ID: cuet-ug-

If, then the value of at x = 1 is :

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0

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3

4

12

PYQ 2023

medium

mathematics ID: cuet-ug-

Match List I with List II List I List II I. order 2, degree 3 II. order 2, degree 1 III. order 1, degree 2 IV. order 3, degree 2 Choose the correct answer from the options given below:

1

А-І, В-II, C-IV, D-II

2

А-III, В-I, C-II, D-IV

3

A-IV, B-II, C-III, D-I

4

A-II, B-IV, C-I, D-III

13

PYQ 2023

medium

mathematics ID: cuet-ug-

The integrating factor of differential equation is:

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2

3

4

14

PYQ 2024

medium

mathematics ID: cuet-ug-

The degree and order of the differential equation are:

1

Degree 2, Order 5

2

Degree 5, Order 1

3

Degree 20, Order 2

4

Degree 4, Order 2

Official Solution

Correct Option: (4)

The given differential equation is . To determine the degree and order, we follow these steps: Order: The order of a differential equation is the highest derivative present. Here, the highest derivative is , which is the second derivative. Therefore, the order is 2.Degree: The degree of a differential equation is the highest power of the highest order derivative, after removing any fractional or negative powers. The given equation is .To express the equation in a polynomial form, eliminate the fractional power by raising both sides to the power of 5: .This results in: .Now, the degree of the differential equation is the highest power of , which is 4.Therefore, the degree is 4 and the order is 2.AttributeValueDegree4Order2The correct answer is: Degree 4, Order 2.

Attribute	Value
Degree	4
Order	2

15

PYQ 2024

medium

mathematics ID: cuet-ug-

If then is:

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4

16

PYQ 2024

easy

mathematics ID: cuet-ug-

If and, then is equal to:

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2

3

4

17

PYQ 2024

medium

mathematics ID: cuet-ug-

The particular solution of the differential equation with, is:

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4

18

PYQ 2024

easy

mathematics ID: cuet-ug-

If the solution of the differential equation represents a circle, then is equal to:

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3

2

-3

3

-2

4

5

19

PYQ 2024

medium

mathematics ID: cuet-ug-

For the differential equation : (A) Degree of the given differential equation is 1. (B) It is a homogeneous differential equation. (C) Solution is, where is an arbitrary constant. (D) Solution is, where is an arbitrary constant.

1

(A) and (C) only

2

(A), (B), and (C) only

3

(A), (B), and (D) only

4

(A) and (D) only

20

PYQ 2024

medium

mathematics ID: cuet-ug-

The integrating factor of the differential equation is:

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2

3

4

21

PYQ 2025

medium

mathematics ID: cuet-ug-

Which of the following are linear first order differential equations? (A) (B) (C) (D) Choose the correct answer from the options given below:

1

(A), (B) and (D) only

2

(A) and (B) only

3

(A), (B) and (C) only

4

(A), (B), (C) and (D)

22

PYQ 2025

medium

mathematics ID: cuet-ug-

$The integrating factor of the differential equation is$

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2

$x$

3

4

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: The given differential equation is a linear first-order differential equation. To solve it, we first need to convert it to the standard form and then find the integrating factor (I.F.). Step 2: Key Formula or Approach: The standard form of a linear differential equation is . The integrating factor is given by the formula: I.F. = . Step 3: Detailed Explanation: The given differential equation is: To convert this to the standard form, divide the entire equation by : Comparing this with the standard form, we have: Now, we calculate the integrating factor: To evaluate the integral, we use the substitution method. Let . Then, . The integral becomes: Now, substitute this back into the formula for the integrating factor: Using the property, we get: Step 4: Final Answer: The integrating factor of the given differential equation is .$

23

PYQ 2025

medium

mathematics ID: cuet-ug-

$If, then which of the following is true?$

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Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: The problem requires finding a differential equation that is satisfied by the given function. This involves finding the first and second derivatives of y with respect to x using implicit differentiation and then substituting them into the given options to find the correct relationship. Step 2: Detailed Explanation: Given the equation: First Derivative: Differentiate both sides with respect to x: Using the chain rule on the left side: Rearranging this equation, we get: Second Derivative: Now, differentiate equation (1) with respect to x. We use the product rule for three functions (u.v.w)' = u'vw + uv'w + uvw' on the left side, where u=x, v=, w=dy/dx. Since is always positive, we can divide the entire equation by : Step 3: Final Answer: Rearranging the terms to match the options, we get: This matches option (D).$

24

PYQ 2025

medium

mathematics ID: cuet-ug-

$Consider the differential equation,, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is, where C is constant of integration (D) Its general solution is, where C is constant of integration (E) If y(1) = 1, then its particular solution is y = x Choose the correct answer from the options given below:$

1

$(A), (D) and (E) only$

2

$(A) and (D) only$

3

$(B) and (C) only$

4

$(B), (C) and (E) only$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: We need to analyze the given differential equation based on its type (linear, homogeneous), find its general solution, and then determine a particular solution given an initial condition. Step 3: Detailed Explanation: The given equation is . Using properties of logarithms, we can rewrite it as: (A) Is it linear? A linear DE is of the form . Due to the term, this equation is not linear. So, (A) is false . (B) Is it homogeneous? A DE is homogeneous if it can be written in the form . Our equation is in this form. So, (B) is true . (C) & (D) General Solution: Since it is homogeneous, we use the substitution . This implies . Substituting into the equation: Separating the variables: Integrating both sides: . For the left integral, let, so . For the right integral, . Equating the results: Substitute back : This matches statement (C). So, (C) is true and (D) is false . (E) Particular Solution: Given the initial condition . Using the general solution from (C): . Substitute and : The particular solution is . Taking the exponential of both sides: So, (E) is true . Step 4: Final Answer: The true statements are (B), (C), and (E). This corresponds to option (4).$

25

PYQ 2025

hard

mathematics ID: cuet-ug-

$The general solution of the differential equation is:$

1

$, Where is constant of integration$

2

$, Where is constant of integration$

3

$, Where is constant of integration$

4

$, Where is constant of integration$

Official Solution

Correct Option: (2)

$Given the differential equation: Rewrite it as: Let and . The equation is . Check if the equation is exact: Since, the equation is exact. Find the potential function such that: Integrate w.r.t. : Differentiate w.r.t. : Set equal to : Therefore, Rewrite to match options: Since integrating would give the term. Thus, the general solution is:$

26

PYQ 2025

medium

mathematics ID: cuet-ug-

$Consider the differential equation,, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is, where C is constant of integration (D) Its general solution is, where C is constant of integration (E) If y(1) = 1, then its particular solution is y = x Choose the correct answer from the options given below:$

1

$(A), (D) and (E) only$

2

$(A) and (D) only$

3

$(B) and (C) only$

4

$(B), (C) and (E) only$

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: We need to analyze the given differential equation based on its type (linear, homogeneous), find its general solution, and then determine a particular solution given an initial condition. Step 3: Detailed Explanation: The given equation is . Using properties of logarithms, we can rewrite it as: (A) Is it linear? A linear DE is of the form . Due to the term, this equation is not linear. So, (A) is false . (B) Is it homogeneous? A DE is homogeneous if it can be written in the form . Our equation is in this form. So, (B) is true . (C) & (D) General Solution: Since it is homogeneous, we use the substitution . This implies . Substituting into the equation: Separating the variables: Integrating both sides: . For the left integral, let, so . For the right integral, . Equating the results: Substitute back : This matches statement (C). So, (C) is true and (D) is false . (E) Particular Solution: Given the initial condition . Using the general solution from (C): . Substitute and : The particular solution is . Taking the exponential of both sides: So, (E) is true . Step 4: Final Answer: The true statements are (B), (C), and (E). This corresponds to option (4).$

27

PYQ 2025

medium

mathematics ID: cuet-ug-

$Let, where be a particular solution of the differential equation and passes through the point (1, 1). The value of is$

1

2

3

4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: This problem requires solving a first-order differential equation using the method of separation of variables. After finding the general solution, we use the given point (1, 1) to find the particular solution. Finally, we compare the coefficients of our solution with the given form to determine the values of, and C, and compute the required expression. Step 2: Key Formula or Approach: The given differential equation is: We will separate the variables (terms with y and dy on one side, terms with x and dx on the other) and then integrate both sides. Step 3: Detailed Explanation: Start by rearranging the equation: Now, group the y-terms on the left and x-terms on the right. Divide by : Simplify both sides: Integrate both sides: where K is the constant of integration. To match the given form, let's move all terms to one side: Now, compare this with the given form . By comparing the coefficients, we get: The solution passes through the point (1, 1). Substitute x=1, y=1 into our solution to find K: So, . The constant C in the given form is, so: We need to find the value of : Step 4: Final Answer: The value of the expression is .$

28

PYQ 2025

medium

mathematics ID: cuet-ug-

$The solution of the differential equation is given by$

1

$, where C is constant of integration$

2

$, where C is constant of integration$

3

$, where C is constant of integration$

4

$, where C is constant of integration$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This is a first-order differential equation that can be solved using the method of separation of variables. Step 2: Key Formula or Approach: The method involves rearranging the equation so that all terms involving y and dy are on one side, and all terms involving x and dx are on the other. Then, integrate both sides. Step 3: Detailed Explanation: The given differential equation is: To remove the logarithm, we take the exponential of both sides: Using the property of exponents , we can write: Now, we separate the variables by moving all y-terms to the left side and all x-terms to the right side: Integrate both sides of the equation: Performing the integration: where is the constant of integration. To match the form of the options, we can rearrange the terms. Move the x-term to the left side: Multiply the entire equation by -12 to eliminate the fractions: Let . Since is an arbitrary constant, C is also an arbitrary constant. This can be written as: Since C is an arbitrary constant, we can replace -C with a new constant, which we can also call C, giving the final form: Step 4: Final Answer: The solution to the differential equation is }.$

29

PYQ 2025

medium

mathematics ID: cuet-ug-

$Let, where be a particular solution of the differential equation and passes through the point (1, 1). The value of is$

1

2

3

4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept: This problem requires solving a first-order differential equation using the method of separation of variables. After finding the general solution, we use the given point (1, 1) to find the particular solution. Finally, we compare the coefficients of our solution with the given form to determine the values of, and C, and compute the required expression. Step 2: Key Formula or Approach: The given differential equation is: We will separate the variables (terms with y and dy on one side, terms with x and dx on the other) and then integrate both sides. Step 3: Detailed Explanation: Start by rearranging the equation: Now, group the y-terms on the left and x-terms on the right. Divide by : Simplify both sides: Integrate both sides: where K is the constant of integration. To match the given form, let's move all terms to one side: Now, compare this with the given form . By comparing the coefficients, we get: The solution passes through the point (1, 1). Substitute x=1, y=1 into our solution to find K: So, . The constant C in the given form is, so: We need to find the value of : Step 4: Final Answer: The value of the expression is .$

30

PYQ 2025

medium

mathematics ID: cuet-ug-

$The solution of the differential equation is given by$

1

$, where C is constant of integration$

2

$, where C is constant of integration$

3

$, where C is constant of integration$

4

$, where C is constant of integration$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: This is a first-order differential equation that can be solved using the method of separation of variables. Step 2: Key Formula or Approach: The method involves rearranging the equation so that all terms involving y and dy are on one side, and all terms involving x and dx are on the other. Then, integrate both sides. Step 3: Detailed Explanation: The given differential equation is: To remove the logarithm, we take the exponential of both sides: Using the property of exponents , we can write: Now, we separate the variables by moving all y-terms to the left side and all x-terms to the right side: Integrate both sides of the equation: Performing the integration: where is the constant of integration. To match the form of the options, we can rearrange the terms. Move the x-term to the left side: Multiply the entire equation by -12 to eliminate the fractions: Let . Since is an arbitrary constant, C is also an arbitrary constant. This can be written as: Since C is an arbitrary constant, we can replace -C with a new constant, which we can also call C, giving the final form: Step 4: Final Answer: The solution to the differential equation is }.$

31

PYQ 2025

medium

mathematics ID: cuet-ug-

$Which of the following are linear first order differential equations? (A) (B) (C) (D) Choose the correct answer from the options given below:$

1

$(A), (B) and (D) only$

2

$(A) and (B) only$

3

$(A), (B) and (C) only$

4

$(A), (B), (C) and (D)$

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: A first-order differential equation is called linear if it can be expressed in the standard form or . In this form, the dependent variable (y in the first case, x in the second) and its derivative appear only to the first power and are not multiplied together. Step 3: Detailed Explanation: Let's analyze each equation: (A) :} This is the very definition of a linear first-order differential equation with y as the dependent variable. So, (A) is linear. (B) :} This is the standard form of a linear first-order differential equation with x as the dependent variable. So, (B) is linear. (C) :} We can rewrite this as . This equation cannot be arranged into either of the standard linear forms. The terms involve products of y and, and it's a homogeneous equation, not linear. (D) :} To check if this is linear, we try to put it in the standard form. Divide the entire equation by : This equation is exactly in the form, where and . Therefore, (D) is a linear differential equation. Step 4: Final Answer: The equations (A), (B), and (D) are linear first-order differential equations. This corresponds to option (1).$

32

PYQ 2025

medium

mathematics ID: cuet-ug-

$The integrating factor of the differential equation is$

1

2

$x$

3

4

Official Solution

Correct Option: (1)

$Step 1: Understanding the Concept: The given differential equation is a linear first-order differential equation. To solve it, we first need to convert it to the standard form and then find the integrating factor (I.F.). Step 2: Key Formula or Approach: The standard form of a linear differential equation is . The integrating factor is given by the formula: I.F. = . Step 3: Detailed Explanation: The given differential equation is: To convert this to the standard form, divide the entire equation by : Comparing this with the standard form, we have: Now, we calculate the integrating factor: To evaluate the integral, we use the substitution method. Let . Then, . The integral becomes: Now, substitute this back into the formula for the integrating factor: Using the property, we get: Step 4: Final Answer: The integrating factor of the given differential equation is .$

33

PYQ 2025

medium

mathematics ID: cuet-ug-

$If, then which of the following is true?$

1

2

3

4

Official Solution

Correct Option: (4)

$Step 1: Understanding the Concept: The problem requires finding a differential equation that is satisfied by the given function. This involves finding the first and second derivatives of y with respect to x using implicit differentiation and then substituting them into the given options to find the correct relationship. Step 2: Detailed Explanation: Given the equation: First Derivative: Differentiate both sides with respect to x: Using the chain rule on the left side: Rearranging this equation, we get: Second Derivative: Now, differentiate equation (1) with respect to x. We use the product rule for three functions (u.v.w)' = u'vw + uv'w + uvw' on the left side, where u=x, v=, w=dy/dx. Since is always positive, we can divide the entire equation by : Step 3: Final Answer: Rearranging the terms to match the options, we get: This matches option (D).$

34

PYQ 2025

hard

mathematics ID: cuet-ug-

$The general solution of the differential equation is:$

1

$, Where is constant of integration$

2

$, Where is constant of integration$

3

$, Where is constant of integration$

4

$, Where is constant of integration$

Official Solution

Correct Option: (2)

$Given the differential equation: Rewrite it as: Let and . The equation is . Check if the equation is exact: Since, the equation is exact. Find the potential function such that: Integrate w.r.t. : Differentiate w.r.t. : Set equal to : Therefore, Rewrite to match options: Since integrating would give the term. Thus, the general solution is:$

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34 MCQs

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List I		List II
A.	$%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%2B(%5Cfrac%7Bdy%7D%7Bdx%7D)%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D$	I.	order 2, degree 1
B.	$%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7Bx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%7By%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D(1%2Bx)%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D$	II.	order 2, degree not defined
C.	$%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%3D%5Ccos3x%2B%5Csin3x$	III.	order 2, degree 4
D.	$%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%2B2%5Cfrac%7Bdy%7D%7Bdx%7D%2By%3D%5Clog(%5Cfrac%7Bdy%7D%7Bdx%7D)$	IV.	order 1, degree 2

List I	List II
$A.%5C%20%5B1%20%2B%20(%5Cfrac%7Bdy%7D%7Bdx%7D)%5E2%5D%20%3D%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D$	I. order 2, degree 3
$B.%20%5C%20(%5Cfrac%7Bd%5E3y%7D%7Bdx%5E2%7D)%5E2%20-%203%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%2B%202(%5Cfrac%7Bdy%7D%7Bdx%7D)%5E4%20%3D%20y%5E4$	II. order 2, degree 1
$C.%20%5C%20(%5Cfrac%7Bdy%7D%7Bdx%7D)%5E2%20%2B%20(%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D)%5E3%20%3D%200$	III. order 1, degree 2
$D.%5C%20(%5Cfrac%7Bdy%7D%7Bdx%7D)%5E2%20%2B%206y%5E3%20%3D%200$	IV. order 3, degree 2

Differential Equations

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Chapter Questions 34 MCQs

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Chapter Questions
34 MCQs