Utt = 4Uxx + Sin(wt)Cos(x)

What do you want to do with this?

 

If you are looking for a specific function, U, and frequency w, that satisfies this, then by setting w = sqrt(5) and

U(t,x) = -sin(wt)cos(x) you get:

 

U_tt = 5sin(wt)cos(x)  and U_xx = sin(wt)cos(x)  so that  U_tt = 4U_xx + sin(wt)cos(x)

 

However, you might have had something entirely different in mind!