I would feel better if you would at least show that you know what "separation of variables" is!

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:

or

and

or

.

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!