# Partial differential equation

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.

a) Write down the system of equations that models this physical problem.

b) Think of some change of dependent variables that produces another wave equation problem with homogeneous boundary conditions. (The other equations may become inhomogeneous as a result.)

c) Solve the equations from part (b) using the method of eigenfunction expansions.

d) Describe what happens when f(t)= 1- cos(cπt/L).

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#### Solution Summary

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.