= Solution
Use $\widehat f(\lambda)=(2\pi)^{-1/2}\int_{\mathbb R}e^{-i\lambda x}f(x)\,dx$. The transform statement requires sufficient decay and regularity for the transform and its <derivatives>; functions in the <Schwartz space> suffice, and all the functions constructed below belong to that space. Integration by parts twice and differentiation under the integral give
$$
\widehat{f''}=-\lambda^2\widehat f,\qquad
\widehat{x^2f}=-\widehat f''.
$$
Transforming the differential equation therefore gives
$$
\boxed{\widehat f''(\lambda)-\lambda^2\widehat f(\lambda)=\mu\widehat f(\lambda).}
$$
The differential equation by itself does not ensure that an ordinary <Fourier transform> exists: $f=e^{x^2/2}$ solves it with $\mu=1$ but grows too rapidly. Thus the usual transformability assumptions are implicit in this part.
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