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I am very lost regarding the method of separation of variables for my PDE course. I have to solve the Laplace equation over a right triangular region, mainly: $$u_{xx}+u_{yy}=0$$ $$u(1,y)=u(x,x)=0$$ $$u(x,0)=f(x)$$, where $0<x<1$ and $0<y<x$. I have been reading some forums online (Solving Laplace's Equation for 2D isosceles right triangle) in which the technique they apply has to do with considering a square region, solving the problem there, and then adding the separate solutions. Even though I know the mechanical process of solving a Laplace equation over a square, I do not understand the reasoning behind it or how to add the solutions at the end. Can someone please give me some insight?

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    $\begingroup$ I'm pretty sure what you need here is a change of variables to a nicer domain that you do know how to solve. $\endgroup$ – DaveNine Dec 15 '17 at 3:02

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