How to separate variables in this PDE?

[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??

 

[pasmith]

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…

 

[hutchphd]

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.

 

[hutchphd]

What is $F_{,x,t}$?

 

[BB711]

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

 

[hutchphd]

Wow I've never seen it with the commas. OK thanks.