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:
 * $(2): \quad: m^2 + y^2 = 0$

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

These are complex and unequal.

So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
 * $y = C_1 \cos k x + C_2 \sin k x$

or, by disposing the constants differently:
 * $y = C_1 \sin k x + C_2 \cos k x$