I'm new here, hope it is the right place to ask the question.
The PDE question is ∂2u/∂x2+∂2u/∂y2=0
and u(x,0)=f(x), u(x,1)=0, u(0,y)=0, u(1,y)=0.
I use the method of separate the variables, with is let u(x,y)=X(x)Y(y) and get X''/X+Y''/Y=0.
Then let X''/X=-Y''/Y=-λ, i.e. X''+λX=0...