Cauchy-Euler Equation

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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a_n x^{n}\frac{d^{n}y}{dx^n}\)	\(=\)	\(\displaystyle a_n e^{(k+1)t}\frac{d}{dt}\left( \frac{d^{k}y}{dx^k}\right) \frac{dt}{dx}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a_n e^{(k+1)t}\frac{d}{dt}\left( \frac{d^{k}y}{dx^k}\right) e^{-t}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a_n e^{kt}\frac{d}{dt}\left( \frac{d^{k}y}{dx^k}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a_n e^{kt}\frac{d}{dt}\left( c_k \frac{d^k y}{dt^k}e^{-tk}+c_{k-1}\frac{d^{k-1}y}{dt^{k-1} }e^{-tk}+...+c_1 \frac{dy}{dt}e^{-tk}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a_n e^{kt}\left( c_{k}\frac{d^{k+1}y}{dt^{k+1} } e^{-tk}-kc_k \frac{d^k y}{dt^k}e^{-tk}+c_{k-1}\frac{d^k y}{dt^k}e^{-tk}-kc_{k-1}\frac{d^{k-1}y}{dt^{k-1} }e^{-tk}+...+c_1 \frac{d^2 y}{dt^2}e^{-tk}-kc_1 \frac{dy}{dt}e^{-tk}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle a_n c_{k}\frac{d^{k+1}y}{dt^{k+1} }-a_n kc_k \frac{d^k y}{dt^k}+c_{k-1}\frac{d^k y}{dt^k}-kc_{k-1}\frac{d^{k-1}y}{dt^{k-1} }+...+a_n c_1 \frac{d^2 y}{dt^2}-a_n k c_1\frac{dy}{dt}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle b_n\frac{d^n y}{dt^n}+b_{n-1}\frac{d^{n-1}y}{dt^{n-1} }+...+b_1 \frac{dy}{dt}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Hence the result by the Principle of Mathematical Induction.

$\blacksquare$

Source of Name

This entry was named for Augustin Louis Cauchy and Leonhard Paul Euler.