# One class Sturm-Liouville differential equation

Let $\phi_n(x), \psi_n(x)$ be solution Sturm-Liouville differential equation

$$p(x) y''(x) - 2n p'(x)y'(x)+2n(2n+1)y(x)=0$$ $$\phi_{n}(0)=0, \hspace{3mm} \phi'_{n}(0)=1;$$ $$\psi_{n}(0)=1, \hspace{3mm} \psi'_{n}(0)=0;$$ where p(x) is given.

Question: Is the set $\{\phi_n(x),\psi_n(x)\}_{n=0}^{\infty}$ orthogonal set?

• It's not really a well defined question in this form (orthogonal in what space?), but ignoring that, the answer has to be no, for a general $p$. Jan 1, 2018 at 21:33