Admissible solutions of higher order differential equations
Let \(\Omega\) [w] be a differential polynomial, i.e. a finite sum of differential monomials \(M[w]=a(z)w^{n_ 0}(w')^{n_ 1}...(w^{(m)})^{n_ m}.\) The coefficient a(z)\(\neq 0\) is a meromorphic function in the plane and the degree and weight-\(\mu\) of \(\Omega\) and M are defined by \(\gamma_{\Omega}=\max \gamma_ M,\Gamma^{\mu}_{\Omega}=\max \Gamma_ M^{\mu}\gamma_ M=n_ 0+n_ 1+...+n_ m\) and \(\Gamma^{\mu}_ M=\mu n_ 0+(\mu +1)n_ 1+...+(\mu +m)n_ m,\) respectively. A meromorphic function f is called admissible with respect to \(\Omega\) if \(T(r,a)=S(r,f)\) for all coefficients occuring in \(\Omega\). The following is proved: Theorem 1. Let H,P,Q be differential polynomials (H and P nontrivial) and let w be an admissible solution of the ODE \(H[w]^ n=w^ mP[w]+Q[w].\) If \(n>\max \{(\Gamma_ p+m)/\gamma_ H\), \((\gamma_ p+1)(\gamma_ p+m)\}\) and \(m\geq \Gamma_ Q+2\), then Q[w]\(\equiv 0\), provided H satisfies the so- called condition (GL): Given \(\mu\geq 1\) there is an index \(j=j(\mu)\) such that \(\Gamma^{\mu}_{\mu_ j}>\Gamma^{\mu}_{\mu_ i}\) for \(i\neq j\).
scientific article; zbMATH DE number 4165337