I have a question about Sturm Liouville problem.
Given a SL equation, say $\frac{d^2 y}{dx^2}+p(x)\frac{d y}{dx}+(q(x)+\lambda r(x))y=0$, how to transform it to its normal form $\frac{d^2 \eta}{d\xi^2}+(\phi(\xi)+\lambda)\eta=0$? How one can think of the transformation? Note that I need to change both of independent and dependent variables. Indeed, I can find corresponding transformation but I would like to know how to think of the corresponding transformation.
What's more, if I do not consider changing independent variables, I can change to normal form as follow: Let $y=u(x)v(x)$, differentiating and let the coefficients involving first order term to be 0, one can solve $u(x)$ and get an equation of $v(x)$ in normal form. However, many authors use the transformation of both independent and dependent variables when I read their papers. The question is what are the benefits of considering transformation of both independent and dependent variables?