Solve homogeneous first order PDE's

[Dragonfall]

$$u_x+u_y+u=e^{x+2y}, y(x,0)=0$$

I have no idea how to do this. We were only taught how to solve homogeneous first order PDE's.

 

[Dr Transport]

try separating variables...

 

[Dragonfall]

What do you mean?

 

[Dragonfall]

I am getting nowhere with this. I tried using the method of characteristics, but it doesn't work out.

 

[J77]

What do you mean?

Write $$u(x,y)=X(x)Y(y)$$, plug this into you equation and...

Note also: $$e^{x+2y}=e^xe^{2y}$$

 

[Dragonfall]

How do you know that u actually can be written as X(x)Y(y)?

 

[d_leet]

How do you know that u actually can be written as X(x)Y(y)?

I think that it's more or less an assumption we make when dealing with homogenous PDEs, but I'm probably completely wrong on this.