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Rational Solutions of First Order Algebraic Ordinary Differential Equations
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Let f(t,y,y')=Σni=0ai(t,y)y'i=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y'but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)＜deg(f,y')or n max i=0{deg(ai,y)-2(n-i)}＞0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.
Degree boundfirst order AODEheightrational solution
FENG Shuang、SHEN Liyong
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School of Physical and Mathematical Sciences,Nanjing Tech University,Nanjing 211816,China
School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China
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