I need the eigenfunctions $f$ and eigenvalues $\lambda$ of $(a(x) f^{II}(x))^{II}= - \lambda^2f$ for a given $a(x)$. For $a(x)$ constant the solution is a combination of sin, cos, sinh and cosh. For $a(x)$ not constant one gets $af^{IV}+2a^{I}f^{III}+a^{II}f^{II}+\lambda^2f=0$
How could i find a way to find $\lambda$ and $f(x)$ if $a(x)$ is not constant?