I have the following equation:

$$m \frac{d^2 x}{dt^2} = -Kx - \alpha \frac{dx}{dt} + f(t) $$

Which I introduced into Mathematica as:

eqn = m D[x[t], {t, 2}] == -K x + f[t] - a D[x[t], t]

I want to take the Fourier transform of this equation to get (after simplyfing terms):

$$ x(\omega) [ - m \omega^2 + i \omega \alpha + K] = f(\omega)$$

(There might be a prefactor missing because I defined the Fourier transform as the inverse Fourier transform of the one that Mathematica uses but that is not important).

In order to do that I tried:

FourierTransform[f[t], t, \[Omega]]

But the output that I get is just:

FourierTransform[ m x''[t] == -K x + f[t] -   a x'[t], t, \[Omega]]

It seems that Mathematica doesn't know how to do the Fourier transform of the functions $x(t)$ and $f(t)$ without their explicit form. Is there any way to do this directly with the built in functions of Mathematica?


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