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

I try to let $x'=x+y, y'=x-y$ and reduced to $2u_{x'}+u=e^{0.5(3x'-y)}$

How to proceed to the next step? Any other methods to solve?

Thank you!

mathshungry
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    $$2u_{x'} + u = \exp \bigg( \frac{1}{2} (3x'-y) \bigg)$$ is an ODE in $x'$. You could use an integrating factor. Alternatively, you could solve the problem using the method of characteristics. – Matthew Cassell Dec 01 '16 at 14:07

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If you are familiar with the method of characteristics, see below.

A preliminary change of function makes easier the solving of the second characteristic ODE.

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(A typo corrected).

JJacquelin
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