First-order linear partial differential equation with boundary conditions

Question

I am trying to solve this boundary-value problem:

$$u_x(x,y) + u_y(x,y) + u(x,y) = 0$$ $$u(0,y) = 1$$ $$u(x,0) = 1$$

Image version

I tried to use the method of characteristics, but it seems that it is only for initial-value problems. I would thank any help

By using the method of characteristics I solve the corresponding ODEs system and get the general solution

u(x,y) = f(x-y)exp(-x)

but, when I try to obtain f for one boundary condition, the answer don't match with the other boundary condition. I don't know what is really happening

Answer 1

On a line $x-y = c$, your pde says $u(x,x-c) e^{x}$ is constant. If that line intersects both of your boundaries and the values you get there are not consistent with each other, it says your boundary value problem has no solution.