I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$.
I have tried solving it by using superposition, to no avail. The superposition yielded an infinite series whereas the solution is supposedly expected to be finite!
I'd appreciate any advice.
I'm assuming that instead of $ 3 x$ twice, one of them should be $ 3 y $. In this case, you can use superposition to solve separately the boundary value problem with the linear terms, and then the trigonometric terms, and then add together.. For example, $ \psi = 3 x y $ solves laplaces equations with the boundary conditions $ 3 x $ and $ 3 y $. For the other part of the boundary conditions, I think you should be able to get it :) .