MHB Solution given by sum of functions on a PDE

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Consider $u_t+u_x=g(x),\,x\in\mathbb R,\,t>0$ and $u(x,0)=f(x).$ Given $f,g\in C^1,$ then show that $u(x,t)$ has the form $u(x,t)=f(x-t)+\sqrt{2\pi}(g*h)(x)$ where $h(x)=\chi_{[0,t]}(x).$

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?
 
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I'm quite interested on this one, I want to know how to use the initial condition and plug it into $U(u)(w,t),$ please! :D
 
Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
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