I'm trying to solve $f'=f$ using fourier transform. I know that $$\hat f'(\alpha )=2i\pi \alpha \hat f(\alpha )$$
therefore
$$f'=f\iff 2i\pi \alpha \hat f(\alpha )=\hat f(\alpha )\iff \hat f(\alpha )=0\iff f(x)=0,$$

what wrong ? Because the solution should be $f(x)=e^{x}$.