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ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 97 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 24 September 2014, pp. 391-403
- Print publication:
- December 2014
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- Article
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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 With this in hand, we also prove that if
$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$
$f$ 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].
