Question

Prove that y(x) = C1 sin2x + C2 cos2x is a solution of y

(2) + 4y = 0

Answer 1

$\text{Given } y(x) = c_1 \sin 2x + c_2 \cos 2x \\ y'(x) = 2c_1 \cos 2x - 2c_2 \sin 2x \\ y''(x)= -4c_1 \cos 2x - 4c_2 \cos 2x \\ \implies y''(x) + 4y(x) = -4c_1 \cos 2x - 4c_2 \cos 2x + 4c_1 \sin 2x + 4c_2 \cos 2x \\ \qquad \qquad \qquad \qquad \quad = -4c_1 \cos 2x + 4c_1 \cos 2x -4c_1 \sin 2x + 4c_1 \cos 2x\\ \qquad \qquad \qquad \qquad \quad = 0 \\ \therefore y(x) = c_1 \sin 2x + c_2 \cos 2x \text{ is a solution of } y''(x) + 4y(x) = 0$