I was wondering if anyone could show me how to solve this ODE using Fourier transforms only, I've tried the problem, it was a past exam question which I'm using to revise. However, we have only used Fourier transforms to solve PDE's in lectures so I'm unsure if what I'm doing is correct
$y'+4y=g(t)$ where $g(t)=0,t<0$ and $g(t)=e^{-3t},t\ge 0$
My solution follows as:
First taking Fourier transforms of both sides gives that $$\mathcal{F}(y'+4y)=\mathcal{F}(g(t))$$ $$\implies i\omega F(\omega)+4F(\omega)=\frac{1}{3+i\omega}$$ $$\implies F(\omega)(i\omega+4)=\frac{1}{3+i\omega}$$ $$\implies F(\omega)=\frac{1}{(3+i\omega)(i\omega+4)}$$
EDIT:
Now using partial fractions gives that $$\frac{1}{(3+i\omega)(i\omega+4)}=\frac{1}{3+i\omega}-\frac{1}{i\omega+4}.$$ Thus $$F(\omega)=\frac{1}{3+i\omega}-\frac{1}{i\omega+4}.$$
Now applying the inverse Fourier transforms gives that $$\mathcal{F}^{-1}(\omega)=e^{-3t}-e^{-4t}$$ $$\implies y(t)=e^{-3t}-e^{-4t}$$ is the solution to the ODE?
Do I need to use convolutions? I'm very confused any help would be great.