# First order PDE help

1. Apr 12, 2010

### 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!

Last edited: Apr 12, 2010
2. Apr 12, 2010

### gabbagabbahey

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

3. Apr 12, 2010

### gato_

You need to variables for the plane:

$$z=x+y, t=x-y$$

your equation only depends on one of them

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

Last edited by a moderator: Apr 15, 2010