Solving 1st Order PDE with Initial Condition - Help Needed

[BustedBreaks]

I'm trying to solve this equation:

Ux + Uy + U = e^-(x+y) with the initial condition that U(x,0)=0


I played around and and quickly found that U = -e^-(x+y) solves the equation, but does not hold for the initial condition. For the initial condition to hold, I think there needs to be some factor of y in the equation for U, but after trying a few equations, I can't find one that satisfies both the initial condition and the equation.

Can someone throw me a hint?

Thanks!

 

[gabbagabbahey]

You might try the substitution u(x,y)=v(x,y)e^{-(x+y)}...

 

[gato_]

You need to variables for the plane:

z=x+y, t=x-y

your equation only depends on one of them

2u_{,z}+u=e^{-z}