To solve the given boundary value problem:

Problem Breakdown:

• PDE: $u_{xx} + u_{yy} = 0$, for $0 < x < 1$ and $0 < y < \frac{1}{2}$.
• Boundary Conditions:
  1. $u_x(0, y) = 0$, $u_x(1, y) = 7 \cos(3\pi y)$, for $0 \leq y \leq \frac{1}{2}$.
  2. $u_y(x, 0) = -2 \cos(4\pi x) + x^2(x-1)^2$, $u(x, \frac{1}{2}) = 0$, for $0 \leq x \leq 1$.

Solution Procedure:

This is a two-dimensional Laplace equation problem, solved by separation of variables.

1. Assume a solution: $u(x, y) = X(x)Y(y)$ Substituting into the PDE gives: $\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0$ Let: $\frac{X''(x)}{X(x)} = -\lambda, \quad \frac{Y''(y)}{Y(y)} = \lambda$

2. Solve $X(x)$ and $Y(y)$:

   • For $X(x)$: $X''(x) + \lambda X(x) = 0$.
   • For $Y(y)$: $Y''(y) - \lambda Y(y) = 0$.

3. Apply the boundary conditions to determine the eigenvalues ($\lambda$) and construct the general solution: $u(x, y) = \sum_{n=1}^\infty \left( A_n \cos(\sqrt{\lambda_n}x) + B_n \sin(\sqrt{\lambda_n}x) \right) \left( C_n e^{\sqrt{\lambda_n}y} + D_n e^{-\sqrt{\lambda_n}y} \right)$

4. Boundary Condition Application:

   • From $u_x(0, y) = 0$, we determine that $A_n = 0$ for all $n$, as the derivative at $x = 0$ must vanish.
   • The boundary conditions $u_x(1, y) = 7\cos(3\pi y)$ and $u_y(x, 0) = -2\cos(4\pi x) + x^2(x-1)^2$ will dictate the coefficients $B_n, C_n, D_n$.

5. Use orthogonal expansion:

   • Expand the given boundary functions in Fourier series to match the form of the eigenfunctions.

6. Solve for the coefficients $B_n, C_n, D_n$ by comparing terms.

Would you like me to perform detailed computations for specific steps or expand on any part?