$u_{tt}=u_{xx}+(8-64x^2)e^{-4x^2}$
$u(x,0)=e^{-4x^2},u_t(t,0)=0$
$0<t<\infty,-\infty<x<\infty$
By Fourier Transform
$\frac{d^2u(w,t)}{dt^2}=F[u_{xx}]+F[(8-64x^2)e^{-4x^2}]=-w^2u(w,t)+w^2F[e^{-4x^2}] $
so i can solve the homogenous part of the problem:ODE
$u{''}-w^2u=0\implies u=c_1\cos wt+c_2\sin wt$
$u(w,t)=A(w)\cos wt+B(w)\sin wt\implies B(w)=0,A(w)\cos wt=F(e^{-4x^2})$
i would like to know how to find the particular solution:
$u{''}-w^2u=w^2F(e^{-4x^2})=w^2e^{-w^2/8}$