Consider the differential equation $\frac{d^2 y}{dx^2} + 4y = 0$ with initial conditions $y(0) = 3$ and $y'(0) = 8$. What is the value of $y(\pi/4)$?

Answer Key:

4

### Solution: Characteristic roots: $r^2 + 4 = 0 \implies r = \pm 2i$. General solution: $y(x) = A\cos(2x) + B\sin(2x)$. $y(0) = A = 3$. $y'(x) = -2A\sin(2x) + 2B\cos(2x) \implies y'(0) = 2B = 8 \implies B = 4$. $y(x) = 3\cos(2x) + 4\sin(2x)$. At $x = \pi/4$: $2x = \pi/2$. Thus $y(\pi/4) = 3\cos(\pi/2) + 4\sin(\pi/2) = 3(0) + 4(1) = 4$. Correct: 4.

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