Third Order PDE written as a System of (Linear) First Order PDEs

Question

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs.

I have no idea how to tackle this problem. Any form of help will be appreciated. Are there any methodologies for such questions?

Thanks, Jay.

Answer 1

Hint: $g=f_x$ so $f_{xx}=g_x$. Now $h=g_x$ so $f_{xxx}=?$. Just add more dependent variables (unknown functions)