Suppose that a string of length L is held fixed at one end and is being moved up and down, say with a displacement of f(t), at the other end. Assume that the string is initially at rest and has a zero displacement, and similarly, that the forcing of the string, f(t), also has a zero initial velocity and displacement.
c) Solve the equations from part (b) using the method of eigenfunction expansions.
d) Describe what happens when f(t)= 1- cos(cπt/L).
This shows how to write down the system of equations that models a physical problem for homogeneous boundary conditions, and use the method of eigenfunction expansions to solve the equations.