Q. Find the general solution of dy/dx = 3x^2. (2020)
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x + C
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Solution
Integrating 3x^2 gives y = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Find the general solution of the equation y' = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = C/x^3
D.
y = Cx^3
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Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer:
A
— y = Ce^(x^3)
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Q. Find the general solution of the equation y' = 5y + 3.
A.
y = Ce^(5x) - 3/5
B.
y = Ce^(5x) + 3/5
C.
y = 3/5 + Ce^(-5x)
D.
y = 5x + C
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Solution
The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Correct Answer:
A
— y = Ce^(5x) - 3/5
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Q. Find the general solution of the equation y'' - 3y' + 2y = 0.
A.
y = C1 e^(2x) + C2 e^(x)
B.
y = C1 e^(x) + C2 e^(2x)
C.
y = C1 e^(3x) + C2 e^(0)
D.
y = C1 e^(0) + C2 e^(3x)
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Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r-1)(r-2)=0. Thus, the general solution is y = C1 e^(x) + C2 e^(2x).
Correct Answer:
B
— y = C1 e^(x) + C2 e^(2x)
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Q. Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.
A.
y = 2e^(4x)
B.
y = e^(4x)
C.
y = 4e^(x)
D.
y = 2e^(x)
Show solution
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the particular solution of dy/dx = 4y, given y(0) = 2.
A.
y = 2e^(4x)
B.
y = e^(4x)
C.
y = 4e^(2x)
D.
y = 2e^(x/4)
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Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
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Q. Find the solution of the differential equation dy/dx = y^2.
A.
y = 1/(C - x)
B.
y = C/(x - 1)
C.
y = Cx
D.
y = e^(x)
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Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer:
A
— y = 1/(C - x)
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Q. Find the solution of the differential equation y' = 3y + 6.
A.
y = Ce^(3x) - 2
B.
y = Ce^(3x) + 2
C.
y = 2e^(3x)
D.
y = 3Ce^(x)
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Solution
This is a linear first-order equation. The integrating factor is e^(3x). The solution is y = Ce^(3x) + 2.
Correct Answer:
B
— y = Ce^(3x) + 2
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Q. Find the solution of the equation dy/dx = y^2 - 1.
A.
y = tan(x + C)
B.
y = C/(1 - Cx)
C.
y = 1/(C - x)
D.
y = C/(x + 1)
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Solution
This is a separable equation. The solution is y = tan(x + C).
Correct Answer:
A
— y = tan(x + C)
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Q. Find the solution of the equation y' + 2y = 0.
A.
y = Ce^(-2x)
B.
y = Ce^(2x)
C.
y = 2Ce^x
D.
y = Ce^x
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Solution
This is a first-order linear differential equation. The solution is y = Ce^(-2x).
Correct Answer:
A
— y = Ce^(-2x)
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Q. Solve the differential equation dy/dx = 2x + 1.
A.
y = x^2 + x + C
B.
y = x^2 + 2x + C
C.
y = 2x^2 + x + C
D.
y = x^2 + C
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Solution
Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.
Correct Answer:
A
— y = x^2 + x + C
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Q. Solve the differential equation dy/dx = 2y + 3. (2023)
A.
y = Ce^(2x) - 3/2
B.
y = Ce^(-2x) + 3/2
C.
y = 3e^(2x)
D.
y = 2e^(2x) + C
Show solution
Solution
Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.
Correct Answer:
A
— y = Ce^(2x) - 3/2
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Q. Solve the differential equation dy/dx = 6x^2y.
A.
y = Ce^(2x^3)
B.
y = Ce^(3x^2)
C.
y = Ce^(6x^2)
D.
y = Ce^(x^6)
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Solution
This is a separable equation. Integrating gives y = Ce^(2x^3).
Correct Answer:
A
— y = Ce^(2x^3)
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Q. Solve the differential equation dy/dx = y/x. (2023)
A.
y = Cx
B.
y = Cx^2
C.
y = C/x
D.
y = C ln(x)
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Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer:
A
— y = Cx
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Q. Solve the differential equation y' = 5 - 2y.
A.
y = 5/2 + Ce^(-2x)
B.
y = 5 + Ce^(-2x)
C.
y = 2 + Ce^(2x)
D.
y = 5/2 - Ce^(-2x)
Show solution
Solution
This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).
Correct Answer:
A
— y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation y' = 5y + 3.
A.
y = (3/5) + Ce^(5x)
B.
y = (5/3) + Ce^(5x)
C.
y = Ce^(5x) - 3
D.
y = Ce^(3x) + 5
Show solution
Solution
Using the integrating factor method, we find the solution y = (3/5) + Ce^(5x).
Correct Answer:
A
— y = (3/5) + Ce^(5x)
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Q. Solve the differential equation y'' - 3y' + 2y = 0.
A.
y = C1e^(2x) + C2e^(x)
B.
y = C1e^(x) + C2e^(2x)
C.
y = C1e^(-x) + C2e^(-2x)
D.
y = C1e^(3x) + C2e^(x)
Show solution
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).
Correct Answer:
B
— y = C1e^(x) + C2e^(2x)
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Q. Solve the equation y' = 6y + 12.
A.
y = 2 - Ce^(-6x)
B.
y = Ce^(6x) - 2
C.
y = 2 + Ce^(6x)
D.
y = 6Ce^(-x)
Show solution
Solution
This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).
Correct Answer:
A
— y = 2 - Ce^(-6x)
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Q. Solve the first-order differential equation dy/dx = y/x.
A.
y = Cx
B.
y = Cx^2
C.
y = C/x
D.
y = C ln(x)
Show solution
Solution
This is a separable equation. Separating variables and integrating gives y = Cx.
Correct Answer:
A
— y = Cx
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Q. Solve the first-order linear differential equation dy/dx + 2y = 6.
A.
y = 3 - Ce^(-2x)
B.
y = 3 + Ce^(-2x)
C.
y = 6 - Ce^(-2x)
D.
y = 6 + Ce^(-2x)
Show solution
Solution
Using an integrating factor e^(2x), we solve to get y = 3 - Ce^(-2x).
Correct Answer:
A
— y = 3 - Ce^(-2x)
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Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
A.
y = x + C/x
B.
y = Cx - x
C.
y = Cx + x
D.
y = C/x + x
Show solution
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.
Correct Answer:
A
— y = x + C/x
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Q. Solve the first-order linear differential equation dy/dx = y/x.
A.
y = Cx
B.
y = Cx^2
C.
y = C/x
D.
y = C ln(x)
Show solution
Solution
This is separable: dy/y = dx/x. Integrating gives ln|y| = ln|x| + C, thus y = Cx.
Correct Answer:
A
— y = Cx
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Q. What is the general solution of the equation y' + 4y = 0?
A.
y = Ce^(-4x)
B.
y = Ce^(4x)
C.
y = 4x + C
D.
y = Cx^4
Show solution
Solution
This is a separable equation, and integrating gives y = Ce^(-4x).
Correct Answer:
A
— y = Ce^(-4x)
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Q. What is the general solution of the equation y' = 4y + 3?
A.
y = Ce^(4x) - 3/4
B.
y = Ce^(4x) + 3/4
C.
y = 3e^(4x)
D.
y = Ce^(3x) + 4
Show solution
Solution
The integrating factor is e^(-4x). The solution is y = Ce^(4x) + 3/4.
Correct Answer:
B
— y = Ce^(4x) + 3/4
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Q. What is the general solution of the equation y'' - 3y' + 2y = 0?
A.
y = C1 e^(x) + C2 e^(2x)
B.
y = C1 e^(2x) + C2 e^(x)
C.
y = C1 e^(3x) + C2 e^(0)
D.
y = C1 e^(0) + C2 e^(3x)
Show solution
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. Thus, the general solution is y = C1 e^(2x) + C2 e^(x).
Correct Answer:
B
— y = C1 e^(2x) + C2 e^(x)
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Q. What is the general solution of the equation y'' - 4y' + 4y = 0?
A.
y = (C1 + C2x)e^(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 e^(4x) + C2 e^(-4x)
D.
y = C1 cos(2x) + C2 sin(2x)
Show solution
Solution
The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).
Correct Answer:
A
— y = (C1 + C2x)e^(2x)
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Q. What is the integrating factor for the equation dy/dx + (1/x)y = 2?
A.
x
B.
e^(ln(x))
C.
e^(ln(x^2))
D.
1/x
Show solution
Solution
The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.
Correct Answer:
C
— e^(ln(x^2))
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Q. What is the integrating factor for the equation dy/dx + 2y = 3?
A.
e^(2x)
B.
e^(-2x)
C.
e^(3x)
D.
e^(-3x)
Show solution
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer:
A
— e^(2x)
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Q. What is the integrating factor for the equation dy/dx + 2y = 6?
A.
e^(2x)
B.
e^(-2x)
C.
e^(6x)
D.
e^(-6x)
Show solution
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer:
A
— e^(2x)
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Q. What is the particular solution of dy/dx = 4y with the initial condition y(0) = 2?
A.
y = 2e^(4x)
B.
y = e^(4x)
C.
y = 4e^(4x)
D.
y = 2e^(x)
Show solution
Solution
The general solution is y = Ce^(4x). Using the initial condition y(0) = 2, we find C = 2, thus y = 2e^(4x).
Correct Answer:
A
— y = 2e^(4x)
Learn More →
Showing 1 to 30 of 47 (2 Pages)
Differential Equations MCQ & Objective Questions
Differential Equations play a crucial role in mathematics and are essential for students preparing for various school and competitive exams. Mastering this topic not only enhances your understanding of mathematical concepts but also boosts your confidence in solving objective questions. Practicing MCQs and important questions on Differential Equations can significantly improve your exam preparation and help you score better.
What You Will Practise Here
Basic definitions and types of Differential Equations
Methods of solving first-order Differential Equations
Higher-order Differential Equations and their solutions
Applications of Differential Equations in real-world problems
Graphical representation of solutions
Initial value problems and boundary value problems
Common Differential Equations used in physics and engineering
Exam Relevance
Differential Equations are a significant part of the curriculum for CBSE, State Boards, NEET, and JEE. Students can expect questions that test their understanding of concepts, methods of solving equations, and applications in various scenarios. Common question patterns include direct problem-solving, conceptual applications, and theoretical explanations, making it essential to practice a variety of Differential Equations MCQ questions.
Common Mistakes Students Make
Confusing different types of Differential Equations and their solutions
Overlooking initial conditions in problems
Misapplying methods for solving higher-order equations
Neglecting the importance of graphical interpretations
Failing to check the validity of solutions
FAQs
Question: What are Differential Equations?Answer: Differential Equations are mathematical equations that relate a function with its derivatives, representing various physical phenomena.
Question: How can I prepare effectively for Differential Equations in exams?Answer: Regular practice of MCQs and understanding key concepts through objective questions will enhance your preparation.
Start solving practice MCQs today to test your understanding of Differential Equations and boost your confidence for upcoming exams. Remember, consistent practice is the key to success!
