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 [itex](\zeta, \eta) = (x, x + t)[/itex]. Then the Ansatz [itex]F = H(\eta)Z(\zeta)[/itex] yields [tex]
H'(\eta)Z'(\zeta) + H''(\eta)Z(\zeta) + A(\zeta)H(\eta)Z(\zeta)\eta^{-3/2} = 0.[/tex] 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]

[tex]F_{,x,t} + \frac{A(x)}{(x+t)^{3/2}} F(x,t) = 0[/tex]

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

 

[hutchphd]

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

 

[BB711]

Common notation, [itex]F_{,x,t} \equiv \frac{\partial ^{2} F(x,t)}{\partial x \partial t}[/itex]

 

[hutchphd]

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