Question

One end of the pipe is closed, which corresponds to the boundary condition
u(0, t) = 0, for t > 0. The other end of the pipe is open, which corresponds
to the boundary condition ux(L, t) = 0, for t > 0.

(a) Suppose that µ < 0, so µ = −k^2 for some k > 0. Find the non-trivial solution X(x) that satisfies equations (3), stating clearly what values k is allowed to take.

(b) Write down the general solution of equation (4) for the case µ = −k^2

(c) You may assume that if µ ≥ 0, then only the trivial solution satisfies
equations (3). Use this assumption to write down the general solution of
the partial differential equation (2) that satisfies the boundary
conditions, by combining your solutions to parts (a) and (b).
(d) Write down the solution that corresponds to the initial conditions
u(x, 0) = 0 and ut(x, 0) = 2 sin (πx/2L)

Answer 1