Linear Second Order ODE/y'' + y = 0/Proof 2

Proof
$(1)$ can be seen to be a special case of:
 * $(2): \quad$ Second Order ODE: $y'' + k^2 y = 0$

with $k = 1$.

$(2)$ has the solution:
 * $y = C_1 \sin k x + C_2 \cos k x$

Hence setting $k = 1$:
 * $y = C_1 \sin x + C_2 \cos x$