I'm to find the vertical displacement of a vibrating string not neglecting gravity, with fixed ends.

The differential equation as given in Schuam's Outline of Advanced Mathematics (Problem 12.57) is

$\displaystyle \[\frac{\partial^2 y}{\partial t^2}=\alpha ^{2}\frac{\partial^2 y}{\partial x^2}-g\]

$

With fixed ends at $\displaystyle x=0$ and $\displaystyle x=l$, and initial velocity and displacement, $\displaystyle f(x)$ and $\displaystyle g(x)$ respectively.

I am unsure of how to solve this differential equation. I tried assuming a solution of the type $\displaystyle y(x,t)=X(x)T(t)$ and then use seperation of variables. However this method does not work as I cannot seperate the variables due to the constant $\displaystyle g$. Any suggestions on how to start this problem would be greatly appreciated.