A note on deriving linearizing transformations for a class of second order nonlinear ordinary differential equations
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TL;DR: In this article, a general form of a nonlinear ODE that admits Bernoulli equation as its first integral is constructed, and conditions for this integral to yield three different linearizing transformations, namely point, Sundman and generalized linearizing transformation.
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Abstract: We present a method of deriving linearizing transformations for a class of second order nonlinear ordinary differential equations. We construct a general form of a nonlinear ordinary differential equation that admits Bernoulli equation as its first integral. We extract conditions for this integral to yield three different linearizing transformations, namely point, Sundman and generalized linearizing transformations. The explicit forms of these linearizing transformations are given. The exact forms and the general solution of the nonlinear ODE for these three linearizables cases are also enumerated. We illustrate the procedure with three different examples.
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Citations
On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations
TL;DR: In this article , the authors present a (n−r)th-order ODE that admits a three-dimensional solvable Lie algebra and provide four illustrative examples of the integration algorithm.
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A. D. Poli︠a︡nin,V. F. Zaĭt︠s︡ev +1 more
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Abstract: Annotation Foreward Some Remarks and Notation First Order Differential Equations Simplest Equations with Arbitrary Functions Integrable in a Closed Form Riccati Equations: g(y)y'x = f2(x)y2 + f1(x)y + f0(x) Abel Equations of the Second Kind Equations Containing Polynomial Functions of y Nonlinear Equations of the Form f(x,y)y'x = g(x,y) Containing Arbitrary Parameters Equations Not Solved for Derivative Equations of the Form F(x,y)y'x = G(x,y) Containing Arbitrary Functions Equations of the Form F(x,y,y'x) = 0 Not Solved for the Derivative and Containing Arbitrary Functions Second Order Differential Equations Linear Equations Autonomous Equations y"xx = F(y,y'x) Emden-Fowler Equation y"xx = Axnym Equations of the Form y"xx = A1xn1ym1 + A2xn2ym2 Generalized Emden-Fowler Equation y"xx = Axnym(y'x)l Equations of the Form y"xx = A1xn1ym1(y'x)l1 + A2xn2ym2(y'x)l2 Equations of the Form y"xx = f(x)g(y)h(y'x) Some Nonlinear Equations with Arbitrary Parameters Equations Containing Arbitrary Functions Third Order Differential Equations Linear Equations Equations of the Form y'"xxx = Axayss(y'x)g(y"xx)d Equations of the Form y'"xxx = f(y)g(y'x)h(y"xx) Some Nonlinear Equations with Arbitrary Parameters Nonlinear Equations Containing Arbitrary Functions Fourth Order Differential Equations Linear Equations Nonlinear Equations Higher Order Differential Equations Linear Equations Nonlinear Equations Supplement 1. Some Elementary Functions and Their Properties Trigonometric Functions Hyperbolic Functions Inverse Trigonometric Functions Inverse Hyperbolic Functions Some Conventional Symbols Supplement 2. Some Special Functions Gamma-Function Bessel Functions Jn and Yn Modified Bessel Functions In and Kn Degenerate Hypergeometric Functions Legendre Functions The Weierstrass Function References Index
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Elementary Lie Group Analysis and Ordinary Differential Equations
N. Kh. Ibragimov
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TL;DR: In this paper, the authors present a Lie Group Analysis of Ordinary Differential Equations (ODE) for the first order and second order differential equations, respectively, and integrate them into Third Order Equations.
923
Chapter 5 Integrability of polynomial differential systems
Jaume Llibre
- 01 Jan 2004
TL;DR: In this paper, the planar polynomial differential systems were introduced and the notion of first integral integral was introduced; and the definition of integrating factor was discussed. And the concept of an exponential factor due to Christopher was introduced.
110
Lagrangians for dissipative nonlinear oscillators: the method of jacobi last multiplier
TL;DR: In this paper, the authors presented a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier.
87
Invariant linearization criteria for systems of cubically nonlinear second-order ordinary differential equations
Fazal M. Mahomed,Asghar Qadir +1 more
TL;DR: In this article, it is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs and its reduction to the simplest system.