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Consider a function $f$ that satisfies the following differential equation:

\begin{equation} \Delta f - \lambda^2 ((\Delta f)^2) = 0, \end{equation}

where $\lambda$ is some real constant. Expressing $\Delta$ in spherical coordinates, does the differential equation admit separable solutions of the type $f(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi)$ or does this only makes sense if $\lambda = 0$?

Mathphys meister
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  • When you use method of separation of variables, you are just looking for solutions in a particular form, So why not? Try and see if you can solve the equation. – Vasili Mar 12 '24 at 15:32
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    The equation reads $$ \Delta f (1- \lambda^2 \Delta f)=0.$$ Both possible zeros, the Laplace equation and the Helmholtz equation can be separated in spherical coordinates. – Roland F Mar 12 '24 at 20:38

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