The most basic method would be to "separate" the variables by looking for a solution of the form U(x,t)= X(x)T(t) and getting two ordinary differential equations for X and T separately. The general solution, for a finite interval, would be a sum of such things and, for an infinite interval, an integral.
Or you could "shortcut" that process by assuming the result- that the solution can be written as a "Fourier Transform" of the form . Then and so that your equation is or . In order for that to be true you must have T"+ isT= 0 for all t. Solve that for T and then use the conditions to determine the constants involved.