Meromorphic solutions of second order differential equations which are not solutions of first order algebraic differential equations (Q1059757)

Let \(L_{\alpha}\) denote the field of all meromorphic functions of order less than or equal to \(\alpha\). For any meromorphic function f denote by \(\lambda (f)= \limsup_{r\to \infty}\log^+n(r,0)/\log r,\) \(\mu (f)= \limsup_{r\to \infty}\log^+n(r,\infty)/\log r;\) and by \({\bar \lambda}\)(f), \({\bar \mu}\)(f) these expressions when n(r,0) and n(r,\(\infty)\) are replaced by \(\bar n(\)r,0) (the number of distinct zeros of f in \(| z| \leq r)\) and \(\bar n(\)r,\(\infty)\). Theorem. Suppose all solutions of (i) \(W''+PW'+QW=0\) are meromorphic and P, \(Q\in L_{\alpha}\). If every nontrivial solution W is such that max(\({\bar \lambda}\)(W),\({\bar \mu}\)(W))\(>\alpha\) then no nontrivial solution satisfies a first order ADE (algebraic differential equation) with coefficients in \(L_{\alpha}\). Two corollaries follow. Corollary 2. Suppose Q(z) is an entire function with \(\lambda (Q)<\sigma (Q)<\infty\), (\(\sigma\) (Q) order of Q). At least one nontrivial solution of the equation (ii) \(W''-Q(z)W=0\) satisfies a first order ADE with coefficients in \(L_{\sigma (Q)}\), if and only if there is a solution W of (ii) for which \(\lambda (W)=\sigma (Q)\). Let \(L_ R\) denote the field of rational functions. Theorem. If all the nontrivial solutions of (iii) \(W''+P(z)W'+Q(z)W=0\) are meromorphic functions with infinite number of zeros, then no nontrivial solution of (iii) satisfies a first order ADE with coefficients in \(L_{\alpha}\) for any \(\alpha <1/2\). Finally we discuss the DE \(W''+aW'+bW=0\) where a, b belong to the field of complex numbers and point out when a solution of the DE will not satisfy a first order ADE with coefficients in K, the field of meromorphic functions of order less than one together with functions of order equal to one and of minimal type.