A uniform string with fixed endpoints at x= 0 and L is driven at midpoint with an external force = F₀ d(x-L/2) cos wt.

At t=0 the string has initial conditions u(x,0) = 0 and u(x,0)/t = 0.

Find the complete solution u(x,t) for all times by computing the Fourier coefficients of the homogeneous solution.

(the homogeneous solution at t=0 is simply equal to the negative of the special inhomogeneous solution obtained from the Green function)