It's quite a long question and I think you'll need all the info so I apologise in advance.

Consider the problem of a string being held at both ends and acted on by the force of gravity. Take the coordinate x to be along the string with the ends of the string at x=0 and x=L. Take t as time and the variable u(t,x) as the vertical position of the string. The

__equation__governing the string motion is the wave equation:

[note all d's in this example are partial differentials]

d^2 u/d t^2 = c^2(d^2 u/d x^2) + G, x is less than or equal to L and greater than or equal to 0, t>0

Where G is a constant due to gravity and c is the constant wave speed. The ends of the string are held at different heights so that the boundary conditions are:

u(t,x)=0, at x=0, t>0

u(t,x)=H, at x=L, t>0

and the string is initially held completely flat and stationary

u(t,x)=0, at t=0, 0<x<L

du/dt =0, at t=0, 0<x<L

It wants me to find the solution and the steady-state solution as t tends to infinity (It says by setting [d^2 u/d t^2] = 0).

I would usually approach this problem using the ansatz and eigenvalues but the constant G is throwing me off and I don't really know how to approach it. I was given a hint to rewrite the problem in a homogenous form but I haven't had any luck doing that either.

Hope you can help, and thanks in advance.