Assume . Then and so the equation becomes . Dividing on both sides of the equation by XT gives . Since the left side depends only on t and the right side only on x, to be equal they must each equal a constant.
That gives two ordinary equations:
You should be able to find the general solution to each of those, depending on of course.
What must be in order that X(0)= 0 and X(L)= 0?
I have a feeling I have repeated what your text book says!