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The problem involves solving the 2D Laplace Partial Differential Equation (PDE) for a specified region, subject to given Dirichlet boundary conditions. Here's a breakdown of how to approach it:

### Part (a)
The equation is:

\[
u_{xx} + u_{yy} = 0
\]

Subject to boundary conditions:

\[
u(x, 0) = 0, \quad u(x, \frac{1}{2}) = 0, \quad u(0, y) = \sin(2\pi y), \quad u(\frac{3}{5}, y) = 0
\]

**Objective**: Use separation of variables to find the analytical solution of this PDE.

#### Step 1: Separation of Variables
Assume the solution \( u(x, y) = F(x)G(y) \). Substituting this into the PDE gives:

\[
F''(x)G(y) + F(x)G''(y) = 0
\]

Dividing by \( F(x)G(y) \) yields:

\[
\frac{F''(x)}{F(x)} = -\frac{G''(y)}{G(y)} = \lambda
\]

This provides two ordinary differential equations (ODEs):

\[
F''(x) - \lambda F(x) = 0, \quad G''(y) + \lambda G(y) = 0
\]

#### Step 2: Solving the ODEs
For \( F(x) \):

\[
F''(x) - \lambda F(x) = 0
\]

The solution depends on \( \lambda \):

- If \( \lambda = 0 \), the general solution is \( F(x) = Ax + B \).
- If \( \lambda > 0 \), the solution is \( F(x) = A\cosh(\sqrt{\lambda}x) + B\sinh(\sqrt{\lambda}x) \).
- If \( \lambda < 0 \), the solution is \( F(x) = A\cos(\sqrt{-\lambda}x) + B\sin(\sqrt{-\lambda}x) \).

For \( G(y) \):

\[
G''(y) + \lambda G(y) = 0
\]

The general solution is:

\[
G(y) = C\cosh(\sqrt{\lambda}y) + D\sinh(\sqrt{\lambda}y)
\]

#### Step 3: Applying Boundary Conditions

The boundary conditions can help you determine the constants. For example:

1. \( u(0, y) = \sin(2\pi y) \) implies that \( F(0) = 1 \) and the Fourier mode will correspond to \( \sin(2\pi y) \) for the y-dependence.
   
2. \( u(\frac{3}{5}, y) = 0 \) leads to further restrictions on \( F(x) \).

You'll also use hyperbolic trigonometric identities like the one given:

\[
\sinh(A - B) = \sinh(A)\cosh(B) - \cosh(A)\sinh(B)
\]

This helps simplify expressions for hyperbolic functions.

### Part (b)
This part asks for a **numerical solution** using the implicit method, with grid spacings \( h = 1/5 \) and \( k = 1/6 \). The task is to calculate the values for different grid points and fill in the table.

#### Implicit Finite Difference Method
For solving numerically, the Laplace equation can be approximated using finite differences. The finite difference approximation for the second derivatives would be:

\[
\frac{u(x+h, y) - 2u(x, y) + u(x-h, y)}{h^2} + \frac{u(x, y+k) - 2u(x, y) + u(x, y-k)}{k^2} = 0
\]

This leads to a system of linear equations that you solve iteratively (e.g., using Gaussian elimination, Jacobi, or Gauss-Seidel methods). The boundary conditions will be used to initialize the grid.

### Next Steps
1. **Solve the system of ODEs** for \( F(x) \) and \( G(y) \) using the boundary conditions.
2. **Set up the finite difference scheme** for Part (b) and compute the solution at the grid points.

Would you like me to help with the detailed separation of variables solution or guide you through the finite difference method?

### Expanding Questions
1. Can you explain how the separation of variables simplifies solving PDEs like this one?
2. How do the boundary conditions restrict the form of the solution?
3. What are hyperbolic functions and where do they appear in the solution?
4. How would you apply the finite difference method to the 2D Laplace equation in this context?
5. What are the advantages of using an implicit method for solving PDEs numerically?

#### Tip
When solving PDEs using separation of variables, always start by carefully analyzing how the boundary conditions influence the choice of solution form for each variable.

Consider the 2D Laplace PDE defined for the region 0 ≤ x ≤ 3/5 and 0 ≤ y ≤ 1/2. Solve using separation of variables and the implicit numerical method.

Explore the solution to a 2D Laplace PDE with given boundary conditions using separation of variables for an analytical approach, and the implicit finite difference method for a numerical solution. This problem is suitable for advanced calculus and differential equations studies.

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