Necessary and Sufficient Conditions for Finite Time Blow-Up in Ordinary Differential Equations

This paper gives a theorem which provides necessary and su#cient conditions for the existence of a real finite value of the independent variable at which the general solution of a certain class of ordinary di#erential equations diverges to infinity. This class is a large subset of the set of all autonomous, non-linear polynomial ordinary di#erential equation. These conditions involve the asymptotic form of the local series representation for the general solutions around the singularities and can be checked algorithmically. KEYWORDS: Singularity analysis, Finite time blow-up. RUNNING TITLE: Finite time blow-up for ODEs 1 I. INTRODUCTION The problem of finite time blow-up for partial di#erential equations (PDEs) is a most active domain of research in applied mathematics [1--4]. Literally hundreds of papers have been written on the problem for di#erent sets of equations, and fundamental physical problems, such as the existence of solutions for the Euler equations, rely on the analysis o...