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What are the solutions of $$\Delta f - \partial_z f = \partial_{xx} f+ \partial_{yy} f+ \partial_{zz} f- \partial_z f = 0?$$

I could find that $f = e^z g(x,y) + h(x,y)$ is a solution if $h$ and $g$ satisfy $\partial_{xx} g+ \partial_{yy} g = 0 = \partial_{xx} h + \partial_{yy} h$. What are the rest of the solutions?

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Have you tried separation of variables? The PDE is linear and homogenous, so separation of variables should give a result.

Assuming

$$f(x,y,z)=X(x)Y(y)Z(z)$$

gives the three ODEs

$$Z''+Z'-\lambda Z=0$$ $$X''+X(\mu-\lambda)=0$$ $$Y''+\mu Y=0$$

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