  We find all non-rational meromorphic solutions of the equation
$ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and
$\gamma$ are rational functions of $z$. In so doing we answer a question of
Hayman by showing that all such solutions have finite order. Apart from special
choices of the coefficient functions, the general solution is not meromorphic
and contains movable branch points. For some choices for the coefficient
functions the equation admits a one-parameter family of non-rational
meromorphic solutions. Nevanlinna theory is used to show that all such
solutions have been found and allows us to avoid issues that can arise from the
fact that resonances can occur at arbitrarily high orders. We actually solve
the more general problem of finding all meromorphic solutions that are
admissible in the sense of Nevanlinna theory, where the coefficients $\alpha$,
$\beta$ and $\gamma$ are meromorphic functions.
