scholarly journals Algorithmic Reduction and Rational General Solutions of First Order Algebraic Differential Equations

2006 ◽  
pp. 201-212 ◽  
Author(s):  
Guoting Chen ◽  
Yujie Ma
Keyword(s):  
Differential Equations ◽  
General Solutions ◽  
Algebraic Differential Equations ◽  
First Order ◽  
Order Algebraic

scholarly journals The motions of algebraic differential equations

1984 ◽  
Vol 25 (1) ◽  
pp. 93-96
Author(s):  
Lee A. Rubel
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Algebraic Differential Equations ◽  
First Order ◽  
Order Algebraic ◽  
Independent Variable

We confine ourselves, for simplicity, to first-order algebraic differential equations (ADE's), although analogous considerations may be made for higher-order ADE's:P(x, y(x), y'(x)) = 0. (*)A motion of (*) is a change of independent variable that takes solutions to solutions, that is, a suitable map <p of the underlying interval I into itself so that if y is a solution of (*) then y ° φ is a solution of (*), i.e.

scholarly journals On the Growth of Solutions of Algebraic Differential Equations Whose Coefficients are Arbitrary Entire Functions

1970 ◽  
Vol 39 ◽  
pp. 107-117 ◽  
Author(s):  
Steven Bank
Keyword(s):  
Differential Equations ◽  
Entire Functions ◽  
Algebraic Differential Equations ◽  
First Order ◽  
Rate Of Growth ◽  
Growth Of Entire Functions ◽  
Order Algebraic ◽  
Growth Of Solutions

In this paper we treat the problem of determining the rate of growth of entire functions which are solutions of first order algebraic differential equations whose coefficients are arbitrary entire functions (i.e. equations of the form Ω(z, y, dy/dz) = 0, where Ω(z, y, dy/dz) = is a polynomial in y and dy/dz, whose coefficients fkJ(z) are entire functions).

scholarly journals General solutions depending algebraically on arbitrary constants

1989 ◽  
Vol 113 ◽  
pp. 1-6 ◽  
Author(s):  
Keiji Nishioka
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
General Solutions ◽  
Algebraic Differential Equations ◽  
First Order ◽  
Order Case ◽  
Rational Dependence ◽  
Strong Normality ◽  
Solutions Of Equations ◽  
The Given

In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

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