i used this property
because X(x) won't vary with t
i applied it and still didnt get the needed expresion
http://i50.tinypic.com/67uyc7.jpg
Your equation is originally,
If you let , you have and (since X and T are functions of a single variable you should not have " ") so your equation becomes
which is what you have. Now divide both sides by XT:
the equation you want to get to.
The importance of this, by the way, is the fact that the right side now depends only of x while the left side depends only on t. In order to be equal for all x and t, each must be a constant! That allows you to separate the partial differential equation into two ordinary differential equations,
and
which are the same as
and
.
Of course, you have to determine possible values for and you usually do that by looking at boundary conditions on X. Notice that is an "eigenvalue" for both differential operators.