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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?

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    @Américo: I meant derivatives. I hope that helps, is there anything else unclear?2011-06-09

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Define g(x) = a(x) f''(x). We then have \begin{pmatrix} f''(x) \\ g''(x) \end{pmatrix} = \begin{pmatrix} 0 & 1/a(x) \\ -\lambda^2 & 0 \end{pmatrix} \begin{pmatrix} f(x) \\ g(x) \end{pmatrix}.

We can try to integrate this, it's not easy (very difficult for arbitrary $a$). Like the constant $a$ case, one has solutions for pretty much any value of $\lambda$.

(It is in fact a bit easier to expand this to a 4x4 first-order matrix ODE by defining auxiliary variables for f'(x) and g'(x), but I'll leave that to you.)