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So we just apply the Fourier transform to get $\dfrac{{\partial U}}{{\partial t}} + iwU = U(g)$ and $U((x,0))=U(f ),$ so by solving the ODE I get $U(u)(w,t)=ce^{-iwt}+\dfrac{U(g)(w,t)}{iw},$ now I'm quite confusing on placing the initial condition, what's the correct way?