Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE/General Result

Theorem
Let $n \in \Z_{>0}$ be a strictly positive integer.

The ordinary differential equation:


 * $a_n x^n f^{\left({n}\right)} \left({x}\right) + \cdots + a_1 x f' \left({x}\right) + a_0 f \left({x}\right) = 0$

can be transformed to linear differential equations by substitution $x = e^t$.

Proof
Let $y = f \left({x}\right)$.

First the following are established:

The proof now proceeds by induction.

For all $n \in \Z_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $a_n x^n f^{\left({n}\right)} \left({x}\right) + \cdots + a_1 x f' \left({x}\right) + a_0 f \left({x}\right) = 0$

can be transformed to a linear differential equation by substitution $x = e^t$.

Basis for the Induction
$P \left({1}\right)$ is the case:

Thus:
 * $a_1 x f' \left({x}\right) + a_0 f \left({x}\right) = 0$

has been transformed into:
 * $a_1 \dfrac {\mathrm d} {\mathrm d t} f \left({e^t}\right) + a_0 f \left({e^t}\right) = 0$

which is a linear differential equation.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $a_k x^k f^{\left({k}\right)} \left({x}\right) + \cdots + a_1 x f' \left({x}\right) + a_0 f \left({x}\right) = 0$

can be transformed to a linear differential equation by substitution $x = e^t$ thus:

Then we need to show that:
 * $a_{k+1} x^{k+1} f^{\left({k+1}\right)} \left({x}\right) + \cdots + a_1 x f' \left({x}\right) + a_0 f \left({x}\right) = 0$

can be transformed to a linear differential equation by substitution $x = e^t$ thus:

Induction Step
This is our induction step:

Hence the result by the Principle of Mathematical Induction.