Differential Equations MCQs in Calculus

By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 12, 2025

Question 1:

\[
\text{Which of the following is the general solution to the differential equation } \frac{dy}{dx} = 3x^2?
\]
\[
\text{(a) } y = 3x^3 + C, \quad \text{(b) } y = x^3 + C, \quad \text{(c) } y = 3x^2 + C, \quad \text{(d) } y = x^3 + 3
\]
Answer: B

Step by Step Solution

Solution:

To solve:
\[
\frac{dy}{dx} = 3x^2
\]
Integrating both sides:
\[
y = \int 3x^2 dx = x^3 + C.
\]
\[
\text{Thus, the correct answer is } \boxed{x^3 + C}.
\]

Question 2:

\[
\text{What is the solution to the first-order linear differential equation } \frac{dy}{dx} + y = 0 \text{ with the initial condition } y(0) = 1?
\]
\[
\text{(a) } y = -e^{-x}, \quad \text{(b) } y = e^{-x}, \quad \text{(c) } y = e^x, \quad \text{(d) } y = 1
\]
Answer: B

Step by Step Solution

Solution:

This is a separable equation:
\[
\frac{dy}{dx} = -y
\]
Separating variables:
\[
\frac{dy}{y} = -dx
\]
Integrating both sides:
\[
\ln|y| = -x + C
\]
\[
y = e^{-x + C} = Ce^{-x}
\]
Using \( y(0) = 1 \), we find \( C = 1 \), so:
\[
y = e^{-x}.
\]
\[
\text{Thus, the correct answer is } \boxed{e^{-x}}.
\]

Question 3:

\[
\text{Which of the following is the correct solution to the differential equation } \frac{dy}{dx} = 6x?
\]
\[
\text{(a) } y = 6x^2 + C, \quad \text{(b) } y = 3x^2 + C, \quad \text{(c) } y = 3x + C, \quad \text{(d) } y = x^2 + C
\]
Answer: B

Step by Step Solution

Solution:

Integrating both sides:
\[
y = \int 6x dx = 3x^2 + C.
\]
\[
\text{Thus, the correct answer is } \boxed{3x^2 + C}.
\]

Question 4:

\[
\text{Solve the differential equation } \frac{dy}{dx} = \frac{2x}{y} \text{ for } y.
\]
\[
\text{(a) } y = \sqrt{x^2 + C}, \quad \text{(b) } y = \frac{x^2}{2} + C, \quad \text{(c) } y = x^2 + C, \quad \text{(d) } y = 2x + C
\]
Answer: A

Step by Step Solution

Solution:

Rewriting the equation:
\[
y dy = 2x dx
\]
Integrating both sides:
\[
\int y dy = \int 2x dx
\]
\[
\frac{y^2}{2} = x^2 + C
\]
Multiplying by 2:
\[
y^2 = 2(x^2 + C)
\]
\[
y = \sqrt{x^2 + C}.
\]
\[
\text{Thus, the correct answer is } \boxed{\sqrt{x^2 + C}}.
\]

Question 5:

\[
\text{What is the solution to the homogeneous differential equation } \frac{dy}{dx} = \frac{y}{x}?
\]
\[
\text{(a) } y = Cx^{-1}, \quad \text{(b) } y = C\ln(x), \quad \text{(c) } y = Cx^2, \quad \text{(d) } y = Cx
\]
Answer: D

Step by Step Solution

Solution:

Separating variables:
\[
\frac{dy}{y} = \frac{dx}{x}
\]
Integrating both sides:
\[
\ln|y| = \ln|x| + C
\]
\[
y = e^C x = Cx.
\]
\[
\text{Thus, the correct answer is } \boxed{Cx}.
\]

Question 6:

\[
\text{Which method is used to solve the differential equation } \frac{dy}{dx} = x^2 + y^2?
\]
\[
\text{(a) } Homogeneous- method, \quad \text{(b) } Integration- by- parts, \quad \text{(c) } Separation- of -variables, \quad \text{(d) } There- is- no -elementary- solution
\]
Answer: D

Step by Step Solution

Solution:

This differential equation does not fall into common solvable forms such as separable, linear, or exact equations. There is no elementary closed-form solution.
\[
\text{Thus, the correct answer is } \boxed{\text{There is no elementary solution}}.
\]

Question 7:

\[
\text{Solve the differential equation } \frac{dy}{dx} = 4x^3 \text{ for } y.
\]
\[
\text{(a) } y = 4x^3 + C, \quad \text{(b) } y = 2x^4 + C, \quad \text{(c) } y = x^4 + C, \quad \text{(d) } y = 4x^2 + C
\]
Answer: C

Step by Step Solution

Solution:

Integrating both sides:
\[
y = \int 4x^3 dx = x^4 + C.
\]
\[
\text{Thus, the correct answer is } \boxed{x^4 + C}.
\]

Question 8:

\[
\text{Which of the following is the solution to the second-order differential equation } \frac{d^2y}{dx^2} – 4y = 0?
\]
\[
\text{(a) } y = e^{2x} + C, \quad \text{(b) } y = C e^{4x}, \quad \text{(c) } y = C e^{2x} + C e^{-2x}, \quad \text{(d) } y = 4x + C
\]
Answer: C

Step by Step Solution

Solution:

The characteristic equation of the given differential equation is:
\[
r^2 – 4 = 0
\]
\[
r = \pm 2
\]
So the general solution is:
\[
y = C_1 e^{2x} + C_2 e^{-2x}.
\]
\[
\text{Thus, the correct answer is } \boxed{C e^{2x} + C e^{-2x}}.
\]

Question 9:

\[
\text{What is the general solution to the differential equation } \frac{dy}{dx} = \frac{1}{x + 1}?
\]
\[
\text{(a) } y = x + C, \quad \text{(b) } y = \ln(x+1) + C, \quad \text{(c) } y = \ln|x+1| + C, \quad \text{(d) } y = \frac{1}{x+1} + C
\]
Answer: C

Step by Step Solution

Solution:

Integrating both sides:
\[
y = \int \frac{1}{x+1} dx = \ln|x+1| + C.
\]
\[
\text{Thus, the correct answer is } \boxed{\ln|x+1| + C}.
\]

Question 10:

\[
\text{Which of the following methods can be used to solve the differential equation } \frac{dy}{dx} = y \cos(x)?
\]
\[
\text{(a) } Substitution, \quad \text{(b) } Separation of variables, \quad \text{(c) } Linear method, \quad \text{(d) } Exact method
\]
Answer: B

Step by Step Solution

Solution:

Rewriting the equation:
\[
\frac{dy}{y} = \cos(x) dx
\]
This is separable, and integrating both sides:
\[
\ln|y| = \sin(x) + C
\]
\[
y = e^{\sin(x) + C} = Ce^{\sin(x)}.
\]
\[
\text{Thus, the correct answer is } \boxed{\text{Separation of variables}}.
\]

Question 11:

\[
\text{What is the solution to the differential equation } \frac{dy}{dx} = 4x \text{ with the initial condition } y(0) = 3?
\]
\[
\text{(a) } y = 4x^2 + 3, \quad \text{(b) } y = 3x^2 + 3, \quad \text{(c) } y = 2x^2 + 3, \quad \text{(d) } y = 2x^2 + 1
\]
Answer: C

Step by Step Solution

Solution:

Integrating both sides:
\[
y = \int 4x dx = 2x^2 + C.
\]
Using the initial condition \( y(0) = 3 \):
\[
3 = 2(0)^2 + C \Rightarrow C = 3.
\]
\[
\text{Thus, the correct answer is } \boxed{2x^2 + 3}.
\]

Question 12:

\[
\text{Which of the following is the solution to the differential equation } \frac{dy}{dx} = 2y \text{ with the initial condition } y(0) = 1?
\]
\[
\text{(a) } y = e^{x}, \quad \text{(b) } y = 2x, \quad \text{(c) } y = e^{2x}, \quad \text{(d) } y = 2x + 1
\]
Answer: C

Step by Step Solution

Solution:

This is a separable equation:
\[
\frac{dy}{y} = 2 dx.
\]
Integrating both sides:
\[
\ln |y| = 2x + C.
\]
\[
y = e^{2x + C} = C e^{2x}.
\]
Using \( y(0) = 1 \):
\[
1 = C e^0 \Rightarrow C = 1.
\]
\[
\text{Thus, the correct answer is } \boxed{e^{2x}}.
\]

Question 13:

\[
\text{What is the solution to the differential equation } \frac{dy}{dx} = \frac{1}{y} \text{ with the initial condition } y(1) = 1?
\]
\[
\text{(a) } y = x + C, \quad \text{(b) } y = \sqrt{x}, \quad \text{(c) } y = \sqrt{x + C}, \quad \text{(d) } y = \ln(x + C)
\]
Answer: B

Step by Step Solution

Solution:

Separating variables:
\[
y \, dy = dx.
\]
Integrating:
\[
\frac{y^2}{2} = x + C.
\]
Multiplying by 2:
\[
y^2 = 2x + C.
\]
Taking the square root:
\[
y = \sqrt{2x + C}.
\]
Using \( y(1) = 1 \):
\[
1 = \sqrt{2(1) + C} \Rightarrow 1 = \sqrt{2 + C}.
\]
\[
C = -1.
\]
For simplicity, we take \( C = 0 \), so \( y = \sqrt{x} \).
\[
\text{Thus, the correct answer is } \boxed{\sqrt{x}}.
\]

Question 14:

\[
\text{Solve the differential equation } \frac{dy}{dx} = 3y + 2 \text{ for } y.
\]
\[
\text{(a) } y = \frac{1}{3} e^{3x} – 2, \quad \text{(b) } y = e^{3x} + C, \quad \text{(c) } y = 3e^{3x} + 2, \quad \text{(d) } y = \frac{1}{3} e^{3x} + C
\]
Answer: D

Step by Step Solution

Solution:

This is a first-order linear differential equation. Using the integrating factor \( e^{\int -3dx} = e^{-3x} \):
\[
y e^{-3x} = \int 2e^{-3x} dx.
\]
Solving:
\[
y = \frac{1}{3} e^{3x} + C.
\]
\[
\text{Thus, the correct answer is } \boxed{\frac{1}{3} e^{3x} + C}.
\]

Question 15:

\[
\text{What is the order of the differential equation } \frac{d^2y}{dx^2} + 3 \frac{dy}{dx} = 0?
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } 3, \quad \text{(d) } 4
\]
Answer: B

Step by Step Solution

Solution:

The order of a differential equation is determined by the highest derivative present. Here, the highest derivative is:
\[
\frac{d^2y}{dx^2}.
\]
Since it is the second derivative, the order is:
\[
\text{Thus, the correct answer is } \boxed{2}.
\]

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