Partial differential equations

The temperature distribution Θ(x, t) along an insulated metal rod of length L is described by the differential equation

The initial temperature distribution in the rod is given by

Θ(x, 0) = 0.4 sin (3πx/2L) (0 ≤ x ≤ L).

(a) Use the method of separation of variables, with Θ(x,t) = X(x)T(t),

to show that the function X(x) satisfies the differential equation X′′ − μX = 0

for some constant μ. Write down the corresponding differential equation that T (t) must satisfy.

1. (b)  Find the boundary conditions that X(x) must satisfy.

2. (c)  Show that if μ = 0, then the only solution of equation (1) that

   satisfies the boundary conditions is the trivial solution X(x) = 0.

3. (d)  Show that if μ = c2 with c > 0, then the only solution of equation (1)

   that satisfies the boundary conditions is the trivial solution X(x) = 0. 

1. (e)  Suppose that μ < 0, so μ = −k2 for some k > 0. Find the non-trivial
   solutions of equation (1) that satisfy the boundary conditions, stating
   clearly what values k is allowed to take. 
   

2. (f)  Solve the differential equation found in part (a) that the function T (t)
   must satisfy. 
   

3. (g)  Use your answers to write down a family of product solutions
   Θn(x, t) = Xn(x) Tn(t) that satisfy the first two boundary conditions. Hence show the general solution of the partial differential equation

(h) Find the particular solution that satisfies the given initial temperature distribution.