Writing an equation in self-adjoint form

Question

I'm trying to solve a S-L problem and I need to write this equation in self adjoint form. How do I go about doing this?

My equation is: $X''(x)+ 2X(x) +\lambda X(x)=0$

Answer 1

The self-adjoint form is $$ (p(x)X'(x))'+q(x)X(x)=0. $$ In its expanded form $$ p(x)X''(x)+p'(x)X'(x)+q(x)X(x)=0 $$ this has to be a multiple of the given equation. That is, $(p(x),p'(x),q(x))$ has to be proportional to $(1,2,λ)$. From $p'=2p$ one gets $p(x)=e^{2x}$. $q=λp$ gives the second coefficient function.