Using Fourier Transform to solve an ODE 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.

You can combine


*

*Linearity of the Fourier transform:
$$\mathcal{F}\left[\sum_{\forall k} g_k\right] = \sum_{\forall k} \mathcal{F}[g_k]$$

*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.
