= Solution
For
$$
\psi=N(t)e^{-F(t)x^2+G(t)x},
$$
direct differentiation gives
$$
\psi_x=(-2Fx+G)\psi,
\qquad
\psi_{xx}=(4F^2x^2-4FGx+G^2-2F)\psi
$$
and
$$
\psi_t=\left(\frac{\dot N}{N}-\dot F x^2+\dot Gx\right)\psi.
$$
Substitution into the Schrodinger equation and equality of the coefficients of $x^2,x,1$ give the necessary and sufficient <Riccati equation> system
$$
\boxed{
\begin{aligned}
i\hbar\dot F&=\frac{2\hbar^2}{m}F^2-\frac12m\omega^2,\\
i\hbar\dot G&=\frac{2\hbar^2}{m}FG,\\
i\hbar\frac{\dot N}{N}
&=\frac{\hbar^2}{m}F-\frac{\hbar^2}{2m}G^2.
\end{aligned}}
$$
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