Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0

Theorem
The second order ODE:
 * $(1): \quad x^2 y'' + 2 x y' - 12 y = 0$

has the solution:
 * $y = C_1 x^3 + C_2 x^{-4}$

Proof
It can be seen that $(1)$ is an instance of the Cauchy-Euler Equation:
 * $x^2 y'' + p x y' + q y = 0$

where:
 * $p = 2$
 * $q = -12$

By Conversion of Cauchy-Euler Equation to Linear Equation, this can be expressed as:
 * $\dfrac {\mathrm d^2 y} {\mathrm d t^2} + \left({p - 1}\right) \dfrac {\mathrm d y} {\mathrm d t^2} + q y = 0$

by making the substitution:
 * $x = e^t$

Hence it can be expressed as:
 * $(2): \quad \dfrac {\mathrm d^2 y} {\mathrm d t^2} + \dfrac {\mathrm d y} {\mathrm d t^2} - 12 y = 0$

From Second Order ODE: $y'' + y' - 12 y = 0$, this has the general solution: