Originally Posted by

**Chris L T521** Everything is perfect so far.

Now observe that you are equating two expressions that are defined in different variables. Thus, the only time they are equal is if they're equal to the same constant, call it $\displaystyle \lambda$.

Now, you can rewrite this as a pair of ODEs:

$\displaystyle x^2\varphi^{\prime\prime}(x)+5x\varphi^{\prime}(x) =\lambda\varphi(x) \implies x^2\varphi^{\prime\prime}(x)+5x\varphi^{\prime}(x)-\lambda\varphi(x)=0$

$\displaystyle y^2\psi^{\prime\prime}(y)-5y\psi^{\prime}(y)+4\psi(y)=-\lambda\psi(y) \implies y^2\psi^{\prime\prime}(y)-5y\psi^{\prime}(y)+\left(4+\lambda\right)\psi(y)=0$

Each of these ODEs are Cauchy-Euler equations. Do you know how to proceed?