"Euler's equation" (also called an "Euler type equation" or "equipotential equation") is a differential equation of the form \(\displaystyle a_nx^n\frac{d^ny}{dx^n}+ a_{n-1}x^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+ \cdot\cdot\cdot+ a_1x\frac{dy}{dx}+ a_0y= 0\). Since we are talking about a double root, it is sufficient to assume second order: \(\displaystyle a_2x^2\frac{d^2y}{dx^2}+ a_1x\frac{dy}{dx}+ a_0y= 0\).

Using the substitution \(\displaystyle v= ln(x)\), Euler's equation reduces to an equation with constant coefficients:

\(\displaystyle \frac{dy}{dx}= \frac{dy}{dv}\frac{dv}{dx}= \frac{1}{x}\frac{dy}{dv}\) and \(\displaystyle \frac{d^2y}{dx^2}= \frac{d}{dx}\left(\frac{1}{x}\frac{dy}{dv}\right)= -\frac{1}{x^2}\frac{dy}{dv}+ \frac{1}{x^2}\frac{d^2y}{dv^2}\) so the equation becomes \(\displaystyle a_2x^2\left(\frac{1}{x^2}\frac{d^2y}{dv^2}- \frac{1}{x^2}\frac{dy}{dv}\right)+ a_1x\left(\frac{1}{x}\frac{dy}{dv}\right)+ a_0y= a_2\frac{d^2y}{dv^2}+ (a_1- a_2)\frac{dy}{dv}+ a_0y= 0\).

That "constant coefficients" differential equation has characteristic equation \(\displaystyle a_2r^2+ (a_1- a_2)r+ a_0= 0\). If it has a double root, say \(\displaystyle r= r_0\) then the differential equation has solutions of the form \(\displaystyle y= e^{r_0v}\) and \(\displaystyle y= ve^{r_0v}\). Since \(\displaystyle v= ln(x)\), those solutions are \(\displaystyle y= e^{r_0 ln(x)}= e^{ln(x^{r_0})}= x^{r_0}\) and \(\displaystyle y= ln(x)e^{r_0ln(x)}= ln(x)e^{ln(x^{r_0})}= x^{r_0}ln(x)\).