Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE

Theorem
Consider the Cauchy-Euler equation:
 * $(1): \quad x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$

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

it is possible to convert $(1)$ into a constant coefficient homogeneous linear second order ODE:
 * $\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q y = 0$

Proof
We have:

Then:

and:

Substituting back into $(1)$: