What is the inverse Fourier transform of e^3iωt for solving ut+3ux=0?

[Conservation]

Homework Statement

Solve u_t+3u_x=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).

Homework Equations

Fourier Transform where (U=fourier transform of u)
Convolution Theorem

The Attempt at a Solution

I've used Fourier transform to get that U_t-3iwU=0 and that U=F(w)e^3iwt. However, I'm completely stuck trying to find the inverse Fourier transform of e^3iwt in order to use the convolution theorem; I suspect that it's a dirac delta function of t-3t, but the table that I'm using doesn't have this general form and contains a 1/2pi constant at the front for the function to be transformed.

Thanks.

 

[blue_leaf77]

I suspect that it's a dirac delta function of t-3t

Yes, look at this. But the argument of the delta function will be 3t+x.