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

Theorem
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
$x=e^t$

$\displaystyle \frac{dx}{dt}=e^t=x$

$\displaystyle \frac{dt}{dx}=e^{-t}=x^{-1}$

Base case
When $n=1$ we have:


 * $\displaystyle a_{1}x\frac{dy}{dx}=a_{1}e^{t}\frac{dy}{dt}\frac{dt}{dx}=a_{1}e^{t}\frac{dy}{dt}e^{-t}=a_{1}\frac{dy}{dt}$

Induction Hypothesis
$\displaystyle a_{n}x^{n}\frac{d^{n}y}{dx^{n}}=b_{n}\frac{d^{n}y}{dt^{n}}+b_{n-1}\frac{d^{n-1}y}{dt^{n-1}}+...+b_{1}\frac{dy}{dt}$

$\displaystyle \frac{d^n y}{dx^{n}}=c_{n}\frac{d^{n}y}{dt^{n}}e^{-tn}+c_{n-1}\frac{d^{n-1}y}{dt^{n-1}}e^{-tn}+...+c_{1}\frac{dy}{dt}e^{-tn}$

Induction Step
When $n=k+1$ we have:

Hence the result by the Principle of Mathematical Induction.