I am asked to solve $$u_x+u_y=1$$ If is was homogeneous i.e., $u_x+u_y$ the answer would be $u(x,y)=f(y-x)$ where $f$ is an arbitrary function. I have found the following set of solutions: $$u(x,y)=\lambda x +(1-\lambda)y$$ where $\lambda$ is an arbitrary constant(real or imaginary). I just have no idea what method other then trial and error would have lead me here. Any ideas? Thanks!
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You can observe that the only difference between homogeneous an inhomogeneous equations is $1$. So you can assume that particular solution is linear on both $x$ and $y$, or $u^p = ax + by$. $$ u_x^p + u_y^p = a + b = 1 $$ In your case $a = \lambda$ and $b = 1 - \lambda$. General solution of inhomogeneous PDE given is the sum of general solution of homogeneous PDE and particular solution of inhomogeneous PDE, so $$ u = f(x-y)+\lambda x + (1-\lambda)y $$