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2024 | OriginalPaper | Chapter

Graphical Portraits of the Solutions of Binary First Order Nonlinear Ordinary Differential Equation Near Their Singular Point

Authors : Petar Popivanov, Angela Slavova

Published in: New Trends in the Applications of Differential Equations in Sciences

Publisher: Springer Nature Switzerland

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Abstract

This paper deals with the appropriate normal homeomorphic forms of a nonlinear binary first order ordinary differential equation (ODE) with smooth coefficients near the critical (singular) point (0, 0) . The normal forms (A. Davydov, then T. Fukui) depend on a real parameter \( \lambda \) and can be semicubic parabola, folded saddle point, folded node and folded focus. The corresponding graphical portraits in the plane are proposed in Theorem 2 and are illustrated geometrically by several figures.

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previous chapter 2-D Minimal Surface Flow with the Oblique Condition and Translators
next chapter Physics Informed Cellular Neural Networks for Solving Partial Differential Equations
Literature
1.
Arnold, V.I. Additional chapters on the theory of ordinary differential equations, Nauka, Moscow, 1978.
2.
Davydov, A. The normal forms of a differential equation that is not solved with respect to the derivative in a neighbourhood of its singular point, Funct. Analysis i prilozhenia, v.19, 1985, 1–10.
3.
Fukui, T. Local differential geometry of cuspidal edge and swallow tail, Osaka J. of Math., v.57:4, 2020, 961–992.
4.
Gradstein, M., Rijik, I. Table of integrals, series and products, Academic Press, NY, 1980.
5.
Petrovskii I. Lectures on the theory of ordinary differential equations, Nauka, Moscow, 1964.
6.
Popivanov, P. Geometrical methods for solving of fully nonlinear partial differential equations, Math. and its Appl. Ser., vol.2, Union of Bulgarian Mathematicians ed., 2006.
7.
Kamka E. Differentialgleichungen Lösunogsmethoden und Lösungen, I Gewöhnliche differentialgleichungen, Leipzig, 1959.
8.
Thom, R. Un lemme sur les applications differentiables, Bol. Soc. Mat. Mexicana, 1956, 59–71.
Metadata
Title
Graphical Portraits of the Solutions of Binary First Order Nonlinear Ordinary Differential Equation Near Their Singular Point
Authors
Petar Popivanov
Angela Slavova
Copyright Year
2024
Book
New Trends in the Applications of Differential Equations in Sciences

Print ISBN: 978-3-031-53211-5

Electronic ISBN: 978-3-031-53212-2

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-53212-2_2

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