Linear Second Order ODE/y'' - k^2 y = 0

Theorem
The second order ODE:
 * $(1): \quad y'' - k^2 y = 0$

has the solution:
 * $y = C_1 e^{k x} + C_2 e^{k x}$

Proof
Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:

Setting $C_1 = \dfrac C 2$ and $C_2 = - \dfrac {a^2} {2 C}$:
 * $y = C_1 e^{\pm k x} + C_2 e^{\mp k x}$

which is the same thing as:
 * $y = C_1 e^{k x} + C_2 e^{k x}$

by allowing for the constants to be interchanged.