= Solution
For the proposed <Gaussian wave packet>, differentiation gives
$$
\partial_t\Psi=(\dot A-A\dot Bx^2)e^{-Bx^2},\qquad \partial_x^2\Psi=(4B^2x^2-2B)\Psi.
$$
Inserting these in the <Time-dependent Schrödinger equation> with <Hamiltonian (quantum mechanics)> $(p^2-x^2)/2$, its right-hand side becomes
$$
\left[\hbar^2B-\left(2\hbar^2B^2+\frac12\right)x^2\right]\Psi.
$$
Equating the constant and quadratic coefficients with $i\hbar\partial_t\Psi$ yields
$$
\boxed{\dot A=-i\hbar AB,\qquad \dot B=-\frac{i}{2\hbar}-2i\hbar B^2}.
$$
Conversely these two equations make the substitution an identity, so they are sufficient for the <Gaussian evolution in an inverted harmonic oscillator>.
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