= Solution
Substitution of these expressions into the <cosmological perfect-fluid continuity equation> gives
$$
\dot\phi\left(\ddot\phi+3H\dot\phi+V'(\phi)\right)=0.
$$
Continuity through a turning point therefore yields the homogeneous <Klein-Gordon equation>
$$
\boxed{\ddot\phi+3H\dot\phi=-V'(\phi)}.
$$
During <slow-roll inflation>, the acceleration and kinetic-energy terms are negligible. In terms of the <reduced Planck mass>, the approximate evolution is
$$
\boxed{3H\dot\phi\simeq-V'(\phi)},
\qquad
\boxed{H^2\simeq\frac{V(\phi)}{3M_{\rm Pl}^2}}.
$$
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