Hi,

I was working on my hwk but would like to double check my answer. Could you please solve it?

PROBLEM 2: Forced wave equation Consider the one-dimensional inhomogeneous wave equation in the interval [0, L]: at2 (2) where c > 0 and w > 0 is given constants. The boundary conditions on the function u(x, t) are u(0, t) = 0, u( L, t) = 0, (3) while the initial conditions are u(x, 0) = 0, at (2, 0) = 0. (4) (a) We first look for a particular solution of the form up(x, t) = f(x)e-it. Show that, in order for up(x, t) to satisfy equation (2), the function f(x) must satisfy an ODE of the form d' f + dx2 + k f = g (x ) . (5) Determine k and g(x). Assuming that up satisfies the same boundary conditions as u, what are the boundary conditions on f(x)? (b) Solve equation (5) for the function f(x) and determine all integration constants. (c) We define a new variable v(x, t) = u(x, t) - Up(x, t). Determine the PDE, boundary conditions, and initial conditions satisfied by the function v. (d) Obtain the general solution for v(x, t) by separation of variables and determine the coefficients in the series. (e) Write down the solution for u(x, t)