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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

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

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