I am currently stuck on this question. I have done a and b, but cannot do c or d or e.

Any help appreciated. Thank you.

The temperature distribution $\Theta(x, t)$ along an insulated metal rod of length $L$ is described by the differential equation $$\frac{\partial^2\Theta}{\partial x^2} = \frac{1}{D} \frac{\partial\Theta}{\partial t} \quad(0 < x < L, \,t > 0),$$ where $D \neq 0$ is a constant. The rod is held at a fixed temperature of 0° C at one end and is insulated at the other end, which gives rise to the boundary conditions $\Theta(0, t) = 0$ and $\Theta_x(L, t) = 0$, for $t > 0$.

The initial temperature distribution in the rod is given by $$\Theta(x, 0) = 0.5 \sin\left(\frac{7\pi x}{2L}\right) \quad(0 ≤ x ≤ L).$$(a) Use the method of separation of variables, with $\Theta(x, t) = X(x) T(t)$, to show that the function $X(x)$ satisfies the differential equation $$X'' − \mu X = 0 \qquad(1)$$ for some constant $\mu$. Write down the corresponding differential equation that $T(t)$ must satisfy.

(b) Write down the boundary conditions that $X(x)$ must satisfy.

Consider the functions $$X_n(x) = \sin(k_nx), \text{ where } k_n = \frac{(2n − 1)\pi}{2L} \text{ and }n = 1,\, 2,\, 3,\, .\, .\, .\, .$$ Show that function $X_n(x)$ satisfies the boundary conditions that you found. Show that $X_n(x)$ satisfies differential equation (1) for some constant $\mu$ (which you should specify).

(c) Solve the differential equation found in part (a) that the function $T(t)$ must satisfy. [3]

(d) Use your answers to write down a family of product solutions $\Theta_n(x, t) = X_n(x) T_n(t)$ that satisfy the first two boundary conditions. Hence show that the general solution of the partial differential

equation may be expressed as $$\Theta(x,t) = \sum_{n=1}^\infty C_n\exp\left(-\frac{D(2n-1)^2\pi^2t}{4L^2}\right)\sin\left(\frac{(2n-1)\pi x}{2L}\right). [2]$$ (e) Find the particular solution that satisfies the given initial temperature distribution.