# How to separate variables in this PDE?

• A
BB711
TL;DR Summary
Can anyone suggest a way to separate the variables in this PDE, so I can solve it analytically?
My PDE:
F,x,t + A(x)*F(x,t)*[(x+t)^(-3/2)] = 0

A(x) is a known function of x.
Trying to separate F(x,t) like
F(x,t) = F1(x)*F2(t)*F3(x+t).

I’m getting desperate to solve,
any suggestions??

Homework Helper
2022 Award
Make the change of variable $(\zeta, \eta) = (x, x + t)$. Then the Ansatz $F = H(\eta)Z(\zeta)$ yields $$H'(\eta)Z'(\zeta) + H''(\eta)Z(\zeta) + A(\zeta)H(\eta)Z(\zeta)\eta^{-3/2} = 0.$$ I'm not sure this is separable.

BB711
Thanks pasmith,
I did try something like that, and basically got that same type of equation, but I’m not sure what to do with it next. Still searching…

Homework Helper
2022 Award
It wwould surely be easier to parse in ##LaTeX##

BB711
$$F_{,x,t} + \frac{A(x)}{(x+t)^{3/2}} F(x,t) = 0$$

$A(x)$ is messy but known; $F(x,t)$ must be solved for analytically, through separability or any other way.

Last edited:
Homework Helper
2022 Award
What is ##F_{,x,t} ##?

malawi_glenn
BB711
Common notation, $F_{,x,t} \equiv \frac{\partial ^{2} F(x,t)}{\partial x \partial t}$