  We present analytic solutions to the spatially homogeneous axion field
equation, using a model potential which strongly resembles the standard
anharmonic $(1-\cos N\theta)$ potential, but contains only a piece-wise second
order term. Our exactly soluble model for $\theta(t)$ spans the entire range
$[-\pi/N,\pi/N]$. In particular, we are able to confirm (i) Turner's numeric
correction factors \cite{Turner} to the adiabatic and harmonic analytic
treatments of homogeneous axion oscillations, and (ii) Lyth's estimate
\cite{Lyth} valid near the metastable misalignment angle $\pi/N$ at the peak of
the potential. We compute the enhancement of axion density fluctuations that
occurs when the axion mass becomes significant at $T\sim 1$ GeV. We find that
the anharmonicity amplifies density \mbox{f}luctuations, but only significantly
for relatively large initial misalignment angles. The enhancement factor is
$\sim$ (2,3,4,13) for $\theta_{\rm in}\sim (0.85,0.90,0.95,0.99)\times\pi$.
