= Solution
Insert the expansion with amplitudes $c_1,c_2$ and the two unperturbed <dynamical phases> into $i\hbar\dot\Psi=(H_0+V)\Psi$. Differentiation of the explicit <phases> cancels $H_0$, leaving
$$
i\dot c_1=\gamma e^{-i(\omega-\omega_0)t}c_2,\qquad
i\dot c_2=\gamma e^{i(\omega-\omega_0)t}c_1,
\qquad\omega_0=\frac{2\mu_BB_0}{\hbar}.
$$
Equivalently, with $\Omega=(\omega-\omega_0)/2$,
$$
\boxed{\dot c_1=-i\gamma e^{-2i\Omega t}c_2,\qquad\dot c_2=-i\gamma e^{2i\Omega t}c_1.}
$$
These interaction-picture equations conserve $|c_1|^2+|c_2|^2$. They are exact for the <circularly rotating magnetic drive>; a <rotating-wave approximation> has not been used.
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