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I have to solve this PDE by changing the coordinates: $$ U_x+2U_y+(2x-y)U = (2x-y)(2y+x) $$ So far I've set $x'=x+2y$ and $y'=2x-y$. That has given me: $$ 5U_{x'}+ y'U = y'x' $$ I would have a better idea of how I should work with this one if the $y'x'$ factor did not exist. But I am very confused as to how I should approach this. Are my original new coordinates wrong? Is there a step I need to take that I'm not familiar with?

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Your new equation is a non homogeneous linear ODE in the variable $x'$. It's general solution can be obtained by standard methods, and is given by $$ U(x',y')=x'-\frac{5}{y'}+f(y')\,e^{-\frac{1}{5}x'y'}, \tag{1} $$ where $f$ is an arbitrary differentiable function. Returning to the original variables, we may rewrite $(1)$ as $$ U(x,y)=x+2y-\frac{5}{2x-y}+f(2x-y)\,e^{-\frac{1}{5}(x+2y)(2x-y)}. \tag{2} $$

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