10 questions linked to/from Solve $u_x + u_y + u = e^{x+2y}$
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equation $u_x+u=e^{2y+x}$ (part of the solution to $u_x+u_y+u=e^{x+2y}$) [duplicate]

I solved/analyzed the below PDE $$\left\{\begin{matrix} u_x+u_y+u=e^{x+2y}\\ u(x,0)=0 \end{matrix}\right.$$ and have a question to the one of the steps involving the integration, see below Using ...
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Solve: $U_x+U_y+U=e^{x+2y}$ with $U(x,0)=0$ [duplicate]

So, I've been tasked with solving the above. I've gone up to the general solution which wasn't too tricky: $$U(x,y)=e^{-x}(\frac{R_1e^{4x}}{4}+f(y-x))$$ However, it gets a bit strange for me when I'...