Elastic transformation method for solving ordinary differential equations with variable coefficients

Pengshe Zheng; Jing Luo; Shunchu Li; Xiaoxu Dong; Pengshe Zheng; Jing Luo; Shunchu Li; Xiaoxu Dong

doi:10.3934/math.2022077

AIMS Mathematics

2022, Volume 7, Issue 1: 1307-1320. doi: 10.3934/math.2022077

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Research article

Elastic transformation method for solving ordinary differential equations with variable coefficients

Institute of Applied Mathematics, Xihua University, Chengdu 610039, Sichuan, China

Received: 24 July 2021 Accepted: 11 October 2021 Published: 22 October 2021
MSC : 34A34, 34A05, 33C45

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.
- ordinary differential equation,
- variable coefficient,
- Laguerre equation,
- general solution,
- elastic transformation method
Citation: Pengshe Zheng, Jing Luo, Shunchu Li, Xiaoxu Dong. Elastic transformation method for solving ordinary differential equations with variable coefficients[J]. AIMS Mathematics, 2022, 7(1): 1307-1320. doi: 10.3934/math.2022077

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Abstract

Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

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