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Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order
Author(s) -
Sobolevsky S.
Publication year - 2004
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.0022-2526.2004.01472.x
Subject(s) - gravitational singularity , mathematics , order (exchange) , nonlinear system , class (philosophy) , ordinary differential equation , mathematical analysis , differential equation , combinatorics , pure mathematics , physics , finance , quantum mechanics , artificial intelligence , computer science , economics
In this paper we consider nonlinear ordinary differential equations y ( n ) = F ( y ′, y , x ) of arbitrary order n ≥ 3 , where F is algebraic in y , y ′ and locally analytic in x . We prove that for n > 3 these equations always admit movable branch points. In the case n = 3 these equations admit movable branch points unless they are of the known class y ′′′= a ( x )( y ′) 2 + ( b 2 ( x ) y 2 + b 1 ( x ) y + b 0 ( x )) y ′+ ( c 4 ( x ) y 4 + c 3 ( x ) y 3 + c 2 ( x ) y 2 + c 1 ( x ) y + c 0 ( x )) , where a , b j , c j are locally analytic in x .