Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE

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

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

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

Proof
We have:

Then:

and:

Substituting back into $(1)$: