Computational Methods and Function Theory 9 (2009), No. 2, 551--563
Copyright Heldermann Verlag 2009
Galina Filipuk
g.filipuk@impan.gov.pl , Loughborough University, Department of Mathematical Sciences, Loughborough, LEICS LE11 3TU, U.K.; current address: Polish Academy of Sciences, Institute of Mathematics, 8 Sniadeckich Street, Warsaw, 00-956 Poland.
Rodney G. Halburd
r.halburd@ucl.ac.uk , University College London, Department of Mathematics, Gower Street, London CC1E 6BT, U.K.
The movable singularities of solutions of equations of the form y''=F(z,y)y'+G(z,y) are studied, where F and G are polynomials in y. It is shown that if degy G≤ degy F+1 and a certain resonance condition is satisfied, then any movable singularity of y that can be reached by analytic continuation along a finite length curve is algebraic. The case in which degy G≤ degy F-1 and the only explicit dependence on z in the equation is in the y-independent term of G(z,y) was considered by R. Smith. The movable algebraic and non-algebraic singularities in a particular class of equations of Liénard type satisfying the "maximum balance" condition degy G=2degy F+1 are also analyzed.
Keywords: Algebraic singularities, movable singularities, Liénard equations.
MSC 2000: Primary 34M35; Secondary 34A34, 34A12, 34M55.
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