How do you solve Laplace's equation for a rectangular plate with specific boundary conditions as described in the chapter?

To solve Laplace's equation for a rectangular plate with specific boundary conditions, begin by expressing the solution as a product of functions, u(x, y) = X(x)Y(y). Separate the variables and derive ordinary differential equations for X and Y. Apply the boundary conditions to find the coefficients and specific forms of the solutions for X and Y, ultimately leading to the potential distribution over the plate.

The process involves starting with the Laplace equation in the form ∂²u/∂x² + ∂²u/∂y² = 0. By assuming a solution of the form u(x, y) = X(x)Y(y), you separate the variables to obtain two ordinary differential equations. Each equation is solved subject to the given boundary conditions, such as u(0, y) = 0, u(4, y) = 0, and u(x, 0) = f(x). The coefficients in the solutions are determined by matching the boundary conditions, leading to a complete solution that describes the potential distribution across the rectangular plate.