What method should I use to solve this partial differential equation?

Question

I have the following PDE to solve:

(mu - ny)u_x + (nx - lu)u_y = ly - mx; l,m,n constant

I can't find a starting point to solve this. Separation of variables doesn't work and whatever I do I keep having all three variables (u,x,y) present in my starting eqns.

Has anyone got any tips?

u_x means du/dx (or u subscript x) - sorry.

Answer 1

First of all remember that not all PDEs have closed form solutions. A numerical approach involving discretization may be the best option.

It may also help to rewrite some terms
eg u*u_x = 1/2 * (u^2)_x
u*u_y = 1/2 * (u^2)_y
y*u_x = (u*y)_x
etc

Otherwise, seeing that you have a linear relation between x and y on the RHS, maybe you could try assuming a linear form for u(x,y).

u(x,y) = a*x + b*y
u_x = a, u_y = b
substituting into the pde gives
(ma^2 + nb - lab)*x + (mab - na - lb^2)*y = l*y - m*x
Equating coefficients yields a = -l/n, b = -m/n as ksoileau, below, shows.

But this is by no means the only possible solution. As you know, the boundary conditions greatly affect the actual form of the solution of PDEs. You didn't give any.

Answer 2

Shhhh The tip is second horse in the fifth race. Where is that PDE?

Answer 3

One solution is given by

u(x,y)=-l/n*x-m/n*y