= Solution
In <neutral meson mixing>, the <mass eigenstates> and <flavor eigenstates> differ. A state prepared as $D^0=c\bar u$ develops a $\bar D^0=u\bar c$ component because the effective <Hamiltonian operator> has off-diagonal <mass> and <decay width> entries. In the <CP symmetry> limit implicit in the source, the propagating states are equal superpositions, with average <mass> $M=(M_1+M_2)/2$ and average <decay width> $\Gamma=(\Gamma_1+\Gamma_2)/2$.
To retain the source's \b[positive] $i\sin$ convention for $g_-$, take $M_2-M_1=\Delta M$ and $\Gamma_2-\Gamma_1=\Delta\Gamma$, with $g_\pm=(e_1\pm e_2)/2$, where $e_j=\exp(-iM_jt-\Gamma_jt/2)$. Then $x=\Delta M/\Gamma$ and $y=\Delta\Gamma/(2\Gamma)$ reproduce the given functions. Reversing an eigenstate label changes signs in intermediate formulae; it must also change the phase conventions consistently.
For equal widths, the mixed probability is proportional to $e^{-\Gamma t}\sin^2(\Delta M t/2)$. Thus the angular <angular frequency> is $|\Delta M|$ in natural units, or $|\Delta M|/\hbar$ with units restored; the period is $2\pi\hbar/|\Delta M|$. A nonzero difference of <decay widths> adds hyperbolic time dependence.
The ordinary neutral flavour-carrying <mesons> that mix are $K^0$, $D^0$, $B_d^0$ and $B_s^0$ with their <antiparticles>. A self-conjugate $\pi^0$ has no distinct antiparticle flavour to oscillate into. There are no long-lived top-flavoured <mesons>, since the <top quark> decays before <hadronization>.
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