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

With fixed ends at and , and initial velocity and displacement, and respectively.

I am unsure of how to solve this differential equation. I tried assuming a solution of the type and then use seperation of variables. However this method does not work as I cannot seperate the variables due to the constant . Any suggestions on how to start this problem would be greatly appreciated.