Let's solve the given Laplace equation step by step as requested. --- **Step 1: Write the Laplace equation and assume a separable solution.** Equation: \[ u_{xx} + u_{yy} = 0 \] Boundary conditions: \[ \begin{cases} u(x,0) = 0 \\ u(0,y) = 0 \\ u(x,b) = 0 \\ u(a,y) = f(y) \end{cases} \] Domain: \(0 < x < a\), \(0 < y < b\) Assume a separable solution: \[ u(x, y) = F(x) \cdot G(y) \] **Explanation:** We assume the solution can be written as a product of functions, each depending only on one variable. This is a standard method for solving PDEs with homogeneous boundary conditions. --- **Step 2: Plug the separable solution into the Laplace equation and separate variables.** Plug \(u(x, y) = F(x)G(y)\) into the PDE: \[ u_{xx} + u_{yy} = 0 \implies F''(x)G(y) + F(x)G''(y) = 0 \] Divide both sides by \(F(x)G(y)\): \[ \frac{F''(x)}{F(x)} + \frac{G''(y)}{G(y)} = 0 \] \[ \frac{F''(x)}{F(x)} = -\frac{G''(y)}{G(y)} = -\lambda \] where \(\lambda\) is a separation constant. So, we get two ODEs: \[ F''(x) + \lambda F(x) = 0 \] \[ G''(y) - \lambda G(y) = 0 \] **Explanation:** By substituting the separable form into the PDE and dividing by \(F(x)G(y)\), we separate the equation into two ODEs—one for each variable. The separation constant \(\lambda\) appears with opposite signs in the two equations. --- **Final Answer:** **Step 1:** Assume \(u(x, y) = F(x) G(y)\). **Step 2:** Substitute into the Laplace equation to get: \[ F''(x) + \lambda F(x) = 0, \quad G''(y) - \lambda G(y) = 0 \] These ODEs, along with the boundary conditions, can now be solved to find the specific form of \(u(x, y)\).