Question 1:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} = 0.
\]
\[
\text{(a) } y = C_1 x + C_2, \quad \text{(b) } y = C_1 x^2 + C_2, \quad \text{(c) } y = C_1 e^x + C_2, \quad \text{(d) } y = C_1 \cos(x) + C_2 \sin(x)
\]
Answer: A
Question 2:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} + 4y = 0.
\]
\[
\text{(a) } y = C_1 e^{2x} + C_2 e^{-2x}, \quad \text{(b) } y = C_1 \cos(2x) + C_2 \sin(2x), \quad \text{(c) } y = C_1 e^{x} + C_2 e^{-x}, \quad \text{(d) } y = C_1 x + C_2
\]
Answer: B
Question 3:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} – 5y = 0.
\]
\[
\text{(a) } y = C_1 e^{5x} + C_2 e^{-5x}, \quad \text{(b) } y = C_1 \cos(5x) + C_2 \sin(5x), \quad \text{(c) } y = C_1 e^{x} + C_2 e^{-x}, \quad \text{(d) } y = C_1 x^2 + C_2
\]
Answer: A
Question 4:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = 0.
\]
\[
\text{(a) } y = C_1 e^{x} + C_2 e^{-x}, \quad \text{(b) } y = (C_1 + C_2 x) e^{x}, \quad \text{(c) } y = (C_1 + C_2 x) e^{-x}, \quad \text{(d) } y = C_1 \cos(x) + C_2 \sin(x)
\]
Answer: A
Question 5:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} = 3 \frac{dy}{dx} + 2y.
\]
\[
\text{(a) } y = C_1 e^{x} + C_2 e^{2x}, \quad \text{(b) } y = C_1 e^{3x} + C_2 e^{-x}, \quad \text{(c) } y = C_1 e^{x} + C_2 e^{-x}, \quad \text{(d) } y = C_1 x + C_2
\]
Answer: A
Question 6:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = \sin(x).
\]
\[
\text{(a) } y = C_1 e^{-x} + C_2 e^{x} + \sin(x), \quad \text{(b) } y = C_1 e^{x} + C_2 e^{-x} + \cos(x), \quad \text{(c) } y = C_1 e^{2x} + C_2 e^{-2x} + \sin(x), \quad \text{(d) } y = C_1 \cos(x) + C_2 \sin(x) + \sin(x)
\]
Answer: A
Question 7:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} + y = \cos(x).
\]
\[
\text{(a) } y = C_1 \cos(x) + C_2 \sin(x) + \cos(x), \quad \text{(b) } y = C_1 e^{x} + C_2 e^{-x} + \cos(x), \quad \text{(c) } y = C_1 e^{2x} + C_2 e^{-2x} + \cos(x), \quad \text{(d) } y = C_1 \cos(x) + C_2 \sin(x)
\]
Answer: A
Question 8:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} = 0 \text{ with initial conditions } y(0) = 3, y'(0) = 2.
\]
\[
\text{(a) } y = 3x + 2, \quad \text{(b) } y = 2x + 3, \quad \text{(c) } y = 3x^2 + 2, \quad \text{(d) } y = x^2 + 2
\]
Answer: A
Question 9:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} + 3y = 0.
\]
\[
\text{(a) } y = C_1 e^{3x} + C_2 e^{-3x}, \quad \text{(b) } y = C_1 \cos(\sqrt{3}x) + C_2 \sin(\sqrt{3}x), \quad \text{(c) } y = C_1 \cos(3x) + C_2 \sin(3x), \quad \text{(d) } y = C_1 e^{x} + C_2 e^{-x}
\]
Answer: B
Question 10:
\[
\text{Solve the differential equation } \frac{d^2y}{dx^2} – 6y = 0.
\]
\[
\text{(a) } y = C_1 e^{\sqrt{6}x} + C_2 e^{-\sqrt{6}x}, \quad \text{(b) } y = C_1 e^{6x} + C_2 e^{-6x}, \quad \text{(c) } y = C_1 \cos(\sqrt{6}x) + C_2 \sin(\sqrt{6}x), \quad \text{(d) } y = C_1 e^{x} + C_2 e^{-x}
\]
Answer: A
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