What method/steps 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?

to clarify, u_x means du/dx (or u subscript x).

Answer 1

all derivatives are partial here and du/dy means x constant
and vice versa)
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(m u – n y) d u/ d x + (n x – l u) d u /d y = l y – m x
(m u du/d x - l u du/d y) + (n x du/dy - n y du/dx ) = l y – m x
multiply both sides by 2
[d/dx (mu^2) - d/dy (lu^2)]+2n [d/dy (u*x)-d/dx (u*y)]=2(l y-m x)
[d/dx (mu^2 – 2n u y) - d/dy (lu^2 – 2n ux) ] = (2l y – 2m x)
[d/dx (mu^2 - 2n u y)-d/dy (lu^2-2n ux)]=[d/dy (ly^2)-d/dx (mx^2)

[d/dx (mu^2 – 2n u y + mx^2) - d/dy (lu^2 – 2n ux – ly^2)] = 0
[d/dx (mu^2 – 2n u y + mx^2) = d/dy (lu^2 – 2n u x – ly^2)
integrating partially
(mu^2 – 2n u y + mx^2) - (lu^2 – 2n u x – ly^2)] = C (constant)
u^2 (m-l) - u (2 n y + 2 n x) + (mx^2 – ly^2) = C ---(1)
now you can find the value to u = f (x,y,l,m,n,C)
by taking 2 roots of this equation.
Or you can leave it like that saying u will be given by (1)

Hope it helps

Answer 2

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