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

Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.

Its auxiliary equation is:
 * $(2): \quad m^2 - k^2 = 0$

From Solution to Quadratic Equation: Real Coefficients, the roots of $(2)$ are:
 * $m_1 = k$
 * $m_2 = -k$

These are real and unequal.

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
 * $y = C_1 e^{k x} + C_2 e^{-k x}$