Differential Equations MCQs

\[
\textbf{MCQs on Differential Equations with Answers}
\]

\[
\textbf{Q1: Which of the following is the general solution of the differential equation } \frac{d^2y}{dx^2} = 0 \text{?}
\]
\[
\text{(A) } y = C_1x + C_2 \quad
\text{(B) } y = C_1x^2 + C_2 \quad
\text{(C) } y = C_1e^x + C_2e^{-x} \quad
\text{(D) } y = \sin(x) + \cos(x)
\]
\[
\textbf{Answer: (A) } y = C_1x + C_2
\]

\[
\textbf{Q2: What is the order of the differential equation } \frac{d^3y}{dx^3} + 2\frac{dy}{dx} = 0 \text{?}
\]
\[
\text{(A) } 1 \quad
\text{(B) } 2 \quad
\text{(C) } 3 \quad
\text{(D) } 4
\]
\[
\textbf{Answer: (C) } 3
\]

\[
\textbf{Q3: Which of the following is a solution to the differential equation } \frac{dy}{dx} = y \text{?}
\]
\[
\text{(A) } y = e^x \quad
\text{(B) } y = x^2 \quad
\text{(C) } y = \ln(x) \quad
\text{(D) } y = \sin(x)
\]
\[
\textbf{Answer: (A) } y = e^x
\]

\[
\textbf{Q4: The general solution of the first-order linear differential equation } \frac{dy}{dx} + P(x)y = Q(x) \text{ is:}
\]
\[
\text{(A) } y = Ce^{\int P(x)dx} + \int Q(x)e^{\int P(x)dx}dx \quad
\text{(B) } y = Ce^{\int Q(x)dx} + \int P(x)e^{\int Q(x)dx}dx \quad
\text{(C) } y = Ce^{\int P(x)dx} \quad
\text{(D) } y = \int P(x)dx + \int Q(x)dx
\]
\[
\textbf{Answer: (A) } y = Ce^{\int P(x)dx} + \int Q(x)e^{\int P(x)dx}dx
\]

\[
\textbf{Q5: Which of the following is a solution to the differential equation } \frac{d^2y}{dx^2} + y = 0 \text{?}
\]
\[
\text{(A) } y = \cos(x) \quad
\text{(B) } y = e^x \quad
\text{(C) } y = \ln(x) \quad
\text{(D) } y = x^2
\]
\[
\textbf{Answer: (A) } y = \cos(x)
\]

\[
\textbf{Q6: What is the particular solution of the differential equation } \frac{dy}{dx} = 3x^2 \text{ with the initial condition } y(1) = 4 \text{?}
\]
\[
\text{(A) } y = x^3 + 4 \quad
\text{(B) } y = x^3 + 3 \quad
\text{(C) } y = x^3 – 4 \quad
\text{(D) } y = 3x^3 + 4
\]
\[
\textbf{Answer: (A) } y = x^3 + 4
\]

\[
\textbf{Q7: The method of solving a non-homogeneous linear differential equation is known as:}
\]
\[
\text{(A) } Separation of variables \quad
\text{(B) } Method of undetermined coefficients \quad
\text{(C) } Variation of parameters \quad
\text{(D) } Both (B) and (C)
\]
\[
\textbf{Answer: (D) } Both (B) and (C)
\]

\[
\textbf{Q8: Which of the following represents the solution of the homogeneous differential equation } \frac{dy}{dx} = y \text{?}
\]
\[
\text{(A) } y = Ce^x \quad
\text{(B) } y = Cx^2 \quad
\text{(C) } y = \sin(x) \quad
\text{(D) } y = \cos(x)
\]
\[
\textbf{Answer: (A) } y = Ce^x
\]

\[
\textbf{Q9: The equation } \frac{dy}{dx} = x + y \text{ is:}
\]
\[
\text{(A) } Homogeneous \quad
\text{(B) } Non-homogeneous \quad
\text{(C) } Linear \quad
\text{(D) } Both (B) and (C)
\]
\[
\textbf{Answer: (D) } Both (B) and (C)
\]

\[
\textbf{Q10: A second-order differential equation is:}
\]
\[
\text{(A) } \frac{dy}{dx} = y \quad
\text{(B) } \frac{d^2y}{dx^2} = 0 \quad
\text{(C) } \frac{d^2y}{dx^2} + y = 0 \quad
\text{(D) } Both (B) and (C)
\]
\[
\textbf{Answer: (D) } Both (B) and (C)
\]