Find the self-adjoint form of the differential equation -

Question

Question

(a) Consider the eigenvalue problem

4u" + 4u' + (1 + λ)u = 0, u(0) = u(2) = 0.

(i) Find the self-adjoint form of the differential equation.

(ii) Show that if um and un are eigenfunctions corresponding to distinct eigenvalues of the problem, then

(iii) Given that the problem has only positive eigenvalues, find all of the eigenvalues and corresponding cigenfunctions.

(b) Consider the problem for u(x, t) given by

4∂²u/∂x² + 4∂u/∂x -∂u/∂t + u = = (0 < x < 2, t > 0 )

u(0, t) = u(2,t) = 0 (t > 0)

u(x,0) =1  (0 < x < 2).

By applying separation of variables, and using your answers to part. (a) as appropriate, show that the solution to this problem is

u(x,t) = 2Π ^∞∑_n=1 n(1 - (-1)ⁿe)/1 + n²Π²) e-(n²Π²t +x/2 ) sin(1/2nΠx).