Differential Equations and The - Braun, Martin 7579 · Problem 1

This is a Laplace equation on a rectangle with three homogeneous boundaries and one non-homogeneous one — a classic setup for separation of variables. The key insight is to assume $u(x,y) = X(x)Y(y)$ and use the three zero boundary conditions to build eigenfunctions in $y$, then solve for the $x$-dependence. The non-homogeneous boundary at $x=0$ becomes a series expansion that determines your coefficients. Think of it as: the homogeneous edges constrain the shape, and the active edge $u(0,y)=f(y)$ tells you the amplitude.

This is a Laplace equation on a rectangle with three homogeneous boundaries and one non-homogeneous one — a classic setup for separation of variables. The key insight is to assume $u(x,y) = X(x)Y(y)$ and use the three zero boundary conditions to build eigenfunctions in $y$, then solve for the $x$-dependence. The non-homogeneous boundary at $x=0$ becomes a series expansion that determines your coefficients. Think of it as: the homogeneous edges constrain the shape, and the active edge $u(0,y)=f(y)$ tells you the amplitude.