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

Theorem

The second order ODE:

$(1): \quad y + y = 0$

has the general solution:

$y = C_1 \sin x + C_2 \cos x$

Proof

$(1)$ can be seen to be a special case of:

$(2): \quad$ Linear 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 $k = 1$:

$y = C_1 \sin x + C_2 \cos x$

$\blacksquare$