Does exist an analytic solution for this PDE

Question

i'm trying to solve this PDE : $ \frac{1}{g(x,y)}\frac{\partial g(x,y)}{\partial y} = d(x)\frac{1}{h(x,z)}\frac{\partial^2h(x,z)}{\partial z^2} $ Actually it's almost the heat equation but how do you solve this ? i would like to know if i can express $h(x,z) = ...$ et $g(x,y) = ...$ I don't know if there are some analytic solutions.

Thanks :)

Answer 1

Solving the separable equation, $$g_y'(y)=\mu g(y),\\g(y)=c(z)e^{\mu y};$$

$$h_{zz}(z)=\frac\mu d h(z),\\ h(z)=c'(y)e^{\sqrt{\mu/d}z}+c''(y)e^{-\sqrt{\mu/d}z},$$

(for $\mu<0$, consider imaginary exponentials, i.e. $\sin/\cos$) and the general solution is a sum of terms

$$f_\mu(y,z)=c_\mu e^{\mu y}e^{\pm\sqrt{\mu/d}z},$$ i.e.

$$f_\mu(x,y,z)=c_\mu(x)e^{\mu y}e^{\pm\sqrt{\mu/d(x)}z}.$$