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?

no, for a general $p$. $\endgroup$