I want convert a linear differential equation into the Fourier domain and solve the differential equation.

eq = x''[t] + w^2 x[t];
fft = FourierTransform[eq, t, s];
Solve[fft == 0, FourierTransform[x[t], t, s]]

However, I get {} as the output. I tried the same using the LaplaceTransform[fft,t,s] and it works. I know I can get away using the Laplace transform using the following substitution.

s -> i s

where i is (-1)^0.5.

Is there anyway to do this using FourierTransform[fft, t, s]?

Furthermore, if I try LaplaceTransform[x[t] + y[t], t, s] I get LaplaceTransform[x[t], t, s] + LaplaceTransform[y[t], t, s].

However, if I try FourierTransform[x[t] + y[t], t, s] I get FourierTransform[x[t] + y[t], t, s]. Is there a way to get FourierTransform[] to behave like LaplaceTransform[]?