 Solution

For Laplace's equation, seek a nonzero separated product $f=X(x)Y(y)$. Dividing by $XY$ gives $X^{\prime \prime }/X=-Y^{\prime \prime }/Y$, so both sides are a separation constant. Choosing it as $-k^2$ gives $X^{\prime \prime }+k^{2}X=0$, $Y^{\prime \prime }-k^{2}Y=0$, with trigonometric and hyperbolic functions, respectively. A zero constant gives linear factors; a positive constant interchanges the two types. Boundary conditions determine which modes are allowed, and linearity permits their superposition to match the remaining boundary data.