We consider solutions to the algebraic differential equation
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$
, where
$Q_d(z,f)$
is a differential polynomial in
$f$
of degree
$d$
with rational function coefficients,
$u$
is a nonzero rational function and
$v$
is a nonconstant polynomial. In this paper, we prove that if
$n\ge d+1$
and if it admits a meromorphic solution
$f$
with finitely many poles, then
$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$
With this in hand, we also prove that if
$f$
is a transcendental entire function, then
$f'p_k(f)+q_m(f)$
assumes every complex number
$\alpha $
, with one possible exception, infinitely many times, where
$p_k(f), q_m(f)$
are polynomials in
$f$
with degrees
$k$
and
$m$
with
$k\ge m+1$
. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’,
Ann. of Math. (2)70(2) (1959), 9–42].