Forced damped simple harmonic oscillator

Question

Consider the equation of motion of the following forced dampded simple harmonic oscillator $$\ddot x + \gamma \dot x + \Omega^2x = f (t)$$ where f (t) = e^(iω0t) if |t| ≤ T 0 otherwise (3) with ω0 a known angular frequency. a) Find fb(ω) the Fourier transform of f(t). b) By taking the Fourier transform of Eq.(1) show that one obtains xb(ω) = Gb (ω) fb(ω) (4) and find the explicit expression of Gb(ω). What can you say about the inverse transform G(t) of Gb(ω)? c) Consider now a function f(x) and its Fourier transform fb(k). Introduce a new function g(x) = f(x − x0). Find its Fourier transform gb(k) in terms of fb(k). d) Do the same for a function g(x) = f(λx) with λ ∈ R.