How do I fourier transform the differential equation $$tf'' + 3f' + tf = 0$$? I.e I want the equation in terms of $\hat{f}$.
I think I'm able to use the scaling rule and differential rule on the middle term, but I can't figure out how to handle the terms $tf''$ and $tf$.
Hint: $$\mathscr{F}\{t^nf(t)\}(\omega)=i^n\frac{\mathrm{d}^n}{\mathrm{d}\omega^n}\bigg(\mathscr{F}\{f(t)\}(\omega)\bigg)$$