Hi guys so I have this differential of order 2 that I want to get to the Sturm-Liouville form by first finding the $p(x)$. The form itself is : $(p(x).y'(x))'+q(x).y(x)=0$ And of course, it has equivalent other forms and definitions And the exercise is : $x^3.y''(x)-x.y'(x)+2y(x)=0$ Is to be turned to the S-L form by finding the $p$ polynomial first as I said.
Thanx much in advance
$$(p(x).y'(x))'+q(x).y(x)=0$$ $$py''+p'y'+qy=0$$ $$y''+\frac{p'}{p}y'+\frac{q}{p}y=0$$ to be compared with $$y''-\frac{1}{x^2}y'+\frac{2}{x^3}y=0$$ Thus you have to solve : $$\begin{cases} \frac{p'}{p}=-\frac{1}{x^2} \\ \frac{2}{x^3}=\frac{q}{p} \end{cases}$$ For a more rigorous manner see : https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory .