Consider the partial differential equation and boundary conditions \frac{\partial^{2} u}{\partial x^{2}}=\frac{1}{D} \frac{\partial u}{\partial t} \quad(00) \frac{\partial u}{\partial x}(0, t)=0, \quad u(\pi, t)=0, \quad(t>0) where D is a non-zero constant.

Use separation of variables, with u(x,t) = X(x) T(t), to show that X^{\prime \prime}(x)=\mu X(x), \quad \text { for some constant } \mu Write down the corresponding differential equation for T(t). Write down the boundary conditions for X(x) implied by the boundary conditions for u(x, t). Show that X(x) = A cos(kx)+B sin(kx), where k > 0, is a solution of the differential equation given in part (a) and clearly state the relationship between k and µ. ) Use the boundary conditions that you found in part (b) to determine the values that k takes for non-trivial solutions. A linear combination of the non-trivial solutions gives the following general solution, which you are not asked to derive: u(x, t)=\sum_{n=1}^{\infty} C_{n} \exp \left(-\left(n-\frac{1}{2}\right)^{2} D t\right) \cos \left(\left(n-\frac{1}{2}\right) x\right) Find the particular solution corresponding to the initial condition u(x, 0)=0.3 \cos \left(\frac{7}{2} x\right), \quad(0 \leq x \leq \pi)