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

Proof
$$x=e^t$$

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

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

Base case
When $$n=1$$ we have:


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

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

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

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

$$ $$ $$ $$

$$

$$

$$

Hence the result by the Principle of Mathematical Induction.