Self-adjoint forms of the Laguerre, Hermite and Chebyshev equations

Given the eigenvalue problem \({ \mathcal { L } } ( x ) \psi ( x ) = \lambda \psi ( x )\) with \({ \mathcal L } ( x ) = p _ { 0 } ( x ) \frac { d ^ { 2 } } { d x ^ { 2 } } +\) \(p _ { 1 } ( x ) \frac { d } { d x } + p _ { 2 } ( x )\) , is self-adjoint if \(p _ { 0 } ^ { \prime } = p _ { 1 }\) . Find a function \(f ( x )\) such that, by multiplying the following ODE, it makes them self adjoint:

1. Laguerre’s ODE: \(x y ^ { \prime \prime } + ( 1 - x ) y ^ { \prime } + a y = 0\)

2. Hermite’s ODE: \(y ^ { \prime \prime } - 2 x y ^ { \prime } + 2 \alpha y = 0\)

3. Chabyshev’s ODE: \(( 1 - x ^ { 2 } ) y ^ { \prime \prime } - x y ^ { \prime } + n ^ { 2 } y = 0\)