# Solutions to $\Delta f - \partial_z f = 0$

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?

$$f(x,y,z)=X(x)Y(y)Z(z)$$
$$Z''+Z'-\lambda Z=0$$ $$X''+X(\mu-\lambda)=0$$ $$Y''+\mu Y=0$$