First order linear non-homogeneous PDE

kingwinner

1. The problem statement, all variables and given/known data
Find the general solution to the PDE and solve the initial value problem:
y² (u_x) + x² (u_y) = 2 y², initial condition u(x, y) = -2y on y³ = x³ - 2


2. Relevant equations/concepts
First order linear non-homogeneous PDEs


3. The attempt at a solution
I know that the general solution to the non-homogeneous PDE = a particular soltuion to it + the general solution to the assoicated homogenous PDE, so I first consider to assocatied homogeneous equation:
y² (u_x) + x² (u_y) = 0

The characteristic equation is dy/dx = x²/y²
And the solution to this ODE is y³ - x³ = C where C is an arbitrary constant.
So the general solution to the homogeneous equation is u = f(y³ - x³) where f is arbitrary differentiable function of one variable.
But then I am stuck. How can I find a particular solution to the original non-homogeneous PDE? Is there a systematic way to find one such solution?

Any help is appreciated! :)

kingwinner

Can someone please help? I am still stuck on this problem... :(

LCKurtz

I don't know about any systematic ways, but just looking at it, if you could find a u such that u_x = 2 and u_y = 0 that would do it. And that's easy.

kingwinner

Then, I think u=2x will work as a particular solution.

But really, can anyone suggest a more general/systematic way to solve it that would work also for more complicated first order linear PDEs?

Thanks!!!