Using Fourier Transform to solve an ODE

Question

Consider the differential equation

$$f^{iv}+3f^{''}-f=g$$

I have read that taking the Fourier Transform of both sides gives $$\left(i\lambda\right)^{4}F\left(\lambda\right)+3\left(i\lambda\right)^2F\left(\lambda\right)=G\left(\lambda\right)-F\left(\lambda\right)=G\left(\lambda\right)$$

Im not sure how they have done this. I think it involves using $\mathcal{F}\left[f^{'}\right]=i\lambda\mathcal{F}\left[f\right]$ but I don't know how.

Answer 1

You can combine

1. Linearity of the Fourier transform: $$\mathcal{F}\left[\sum_{\forall k} g_k\right] = \sum_{\forall k} \mathcal{F}[g_k]$$
2. Differentiation becomes multiplication with the frequency: $$\mathcal{F}[g'] = i\lambda\mathcal{F}[g]$$

First use 1) to separate each term. Then use 2) as many times you need to get down to $F[f]$ for each individual term.