I can see that you've uploaded a mathematical document with several partial differential equations (PDEs) and boundary value problems (BVPs). Let's go through each one step by step.

Problem 2:

• PDEs to solve using separation of variables: $\text{(a)} \quad u_t + u_{xx} = 0$ $\text{(b)} \quad u_{tt} = c^2 u_{xx}$ Using the method of separation of variables, assume $u(x,t) = \phi(x) g(t)$ and substitute into each PDE. This will lead to ODEs for $\phi(x)$ and $g(t)$, depending on the separation constant you choose.

Problem 3:

• Find the general solution to the BVP: $u_{tt} = a^2 u_{xx}$ $\text{BCs:} \quad u_x(0,t) = 0, \quad u_x(1,t) = 0$ Here, you solve the PDE using separation of variables again and impose the boundary conditions to find the eigenvalues and eigenfunctions for $\phi(x)$.

Problem 4:

• Solve the BVP: $u_{tt} = u_{xx}$ $\text{BCs:} \quad u(0,t) = 0, \quad u(1/2,t) = 0$ Again, the method of separation of variables applies, and you will use the boundary conditions to find the allowed values of the separation constant (eigenvalues).

Problem 5:

• Solve the IBVP: $u_t = u_{xx}$ $\text{BCs:} \quad u(0,t) = 0, \quad u(1,t) = 0$ $\text{IC:} \quad u(x,0) = 3\sin\left( \frac{5\pi x}{2} \right)$ This is an initial boundary value problem where you solve the heat equation with the given boundary conditions and initial conditions. You will likely use Fourier series for the solution.

Problem 6:

• Steady-state temperature for the PDE: $u_t = u_{xx} + 1, \quad 0 < x < L$ $\text{BCs:} \quad u_x(0,t) = 1, \quad u_x(L,t) = \beta$ $\text{IC:} \quad u(x,0) = f(x)$ You need to determine whether a steady-state solution exists, which means finding a time-independent solution $u(x)$ by setting $u_t = 0$. You'll solve the resulting ODE with the given boundary conditions.

Would you like a detailed solution for any of these problems or clarifications? Also, here are some follow-up questions for further exploration:

1. What are the typical forms of solutions for second-order PDEs using separation of variables?
2. How do boundary conditions affect the choice of eigenvalues in a BVP?
3. Can you explain the role of initial conditions in solving IBVPs?
4. What changes when we add a source term like the $+1$ in problem 6 to the heat equation?
5. How can Fourier series be used to solve the IBVP in problem 5?

Tip: When using separation of variables, always check whether the separation constant should be positive, negative, or zero based on the boundary conditions or physical context of the problem.

Let me know how you'd like to proceed!