Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 79, pp. 1-25.

Title: Solutions approaching polynomials at infinity to nonlinear 
       ordinary differential equations

Author: Christos G. Philos (Univ. of Ioannina, Greece)
        Panagiotis Ch. Tsamatos  (Univ. of Ioannina, Greece)

Abstract: 
 This paper concerns the solutions approaching polynomials at
 $\infty $ to $n$-th order ($n>1$) nonlinear ordinary differential
 equations, in which the nonlinear term depends  on  time $t$ and
 on $x,x',\dots ,x^{(N)}$, where $x$ is the unknown function and
 $N$ is an integer with $0\leq N\leq n-1$. For each given integer
 $m$ with $\max \{1,N\}\leq m\leq n-1$, conditions are given which
 guarantee that, for any real polynomial of degree at most $m$,
 there exists a solution that is asymptotic at $\infty $ to this
 polynomial. Sufficient conditions are also presented for every
 solution to be asymptotic at $\infty $ to a real polynomial of
 degree at most $n-1$. The results obtained extend those by the
 authors and by Purnaras [25] concerning the particular case
 $N=0$.
  
Submitted February 13, 2005. Published July 11, 2005.
Math Subject Classifications: 34E05, 34E10, 34D05
Key Words: Nonlinear differential equation; asymptotic properties;
    asymptotic expansions; asymptotic to polynomials solutions.