# Partial Differential Equation equal to a function [duplicate]

Hi the question I have to solve is:

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

First, I tried to solve the question of the form:

$$u_x + u_y + u = 0$$

We know that $$\dot x = 1 \$$ and $$\dot y = 1 \$$. From that we can introduce s where $$x(s) = s +x_0$$ and $$y(s) = s$$. Then if $$z = u(x(s), y(s))$$ we have that $$\dot z + z = 0$$.

By setting $$s = y$$ we have then that $$x_0 = y-x$$.

Then we can get that $$u(x,y) = e^{-y} \ g(y-x)$$.

Now, to get $$\dot z + z = e^{x+2y}$$ I am completely stuck

• Substitution $u(x,y)=f(x,y)e^{x+2y}$ seems useful here. – StephenG Oct 3 '19 at 4:21
• This question has been asked here, here, here, here. – mattos Oct 3 '19 at 4:33
• And here, here, here and here. – mattos Oct 3 '19 at 4:36
• Here too. – mattos Oct 3 '19 at 4:38