Suppose that we have a function $f(x,y,z)$ that satisfies:
\begin{align}
\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}=0
\end{align}
which can be solved be using separation of variables method. If we add a term to the equation which has a mixed derivatives as this example:
\begin{align}
\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}+ \frac{\partial^2 f}{\partial y \partial z}=0
\end{align}
How can we solve this equation?
What is the appropriate method to solve a nonlinear equation similar to the above equation. As example; consider a function $f(\theta,\gamma,\zeta)$ that satisfies:
\begin{align}
\cot\theta \frac{\partial f}{\partial \theta} + \frac{\partial^2 f}{\partial\theta^2}+\frac{1}{\sin^2\theta}\frac{\partial^2 f}{\partial\gamma^2} + \frac{1}{\sin^2\theta}\frac{\partial^2 f}{\partial\zeta^2} -\frac{2\cos\theta}{\sin^2\theta} \frac{\partial^2 f}{\partial \gamma \partial\zeta}=0
\label{eq:2.22}
\end{align}
The first four terms are Laplace equation given in Euler angles but the last term makes it difficult to use separation method. Any suggestion or method how to tackle this problem would be highly appreciated.