= Solution
Use the <Fourier transform> $\widetilde f(k)=\int_{\mathbb R}f(x)e^{-ikx}\,dx$. For a rapidly decaying smooth function, <integration by parts> gives $\widetilde{f'}=ik\widetilde f$, and differentiation under the integral gives $\widetilde{xf}=i\,d\widetilde f/dk$. Both are justified here by Gaussian decay. Since $f'=-2a^2xf$,
$$
ik\widetilde f=-2ia^2\widetilde f',\qquad
\widetilde f'(k)=-\frac{k}{2a^2}\widetilde f(k).
$$
The <Gaussian integral> supplied in the hint gives $\widetilde f(0)=\sqrt\pi/a$. Solving this first-order equation yields
$$
\boxed{\widetilde f(k)=\frac{\sqrt\pi}{a}e^{-k^2/(4a^2)}.}
$$
The calculation states the transform normalization and the two differentiation properties explicitly.
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