On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

Abstract The method of blowing up points indeterminacy certain systems two ordinary differential equations is applied to obtain information about the singularity structure solutions corresponding non-linear equations. We first deal with so-called Painlevé example, which passes test, but have more complicated singularities. Resolving base in equivalent system we can explain singularities original equation. Smith example has a solution non-isolated singularity, an accumulation point algebraic Smith’s equation be written as ways. show that sequence blow-ups for both infinite. Another consider Painlevé-Ince When usual analysis applied, it possesses positive and negative resonances. three there infinite another one terminates, further gives Laurent expansion around movable pole. Moreover, even possible general after blow-ups.