Solutions of a class of higher order variable coefficient homogeneous differential equations

Peng E; Tingting Xu; Linhua Deng; Yulin Shan; Miao Wan; Weihong Zhou; Peng E; Tingting Xu; Linhua Deng; Yulin Shan; Miao Wan; Weihong Zhou

doi:10.3934/nhm.2025011

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 213-231. doi: 10.3934/nhm.2025011

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Solutions of a class of higher order variable coefficient homogeneous differential equations

Peng E ¹,
Tingting Xu ^{1
,
,},
Linhua Deng ^{1
,
,},
Yulin Shan ²,
Miao Wan ¹,
Weihong Zhou ^{1,3
,
,}

1.
School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650504, Yunnan, China
2.
School of Mathematics and Computer Science, Chuxiong Normal University, Chuxiong 675000, Yunnan, China
3.
Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming, Yunnan, 650011, China

Received: 03 December 2024 Revised: 18 February 2025 Accepted: 19 February 2025 Published: 27 February 2025

Recently, the variable coefficient homogeneous differential equations (VCHDE) have been widely applied to real-world problems, such as wave propagation and material science. However the exploration and research on higher-order VCHDE is relatively lagging. Given this, this work focuses on the solutions of fourth-order and nth-order VCHDE with polynomial coefficients. By means of the sufficient conditions for the existence of solutions to differential equations, a connection is established between the rank of the variable coefficient matrix and the existence of polynomial particular solutions. The main results show that: (1) the necessary and sufficient conditions for the existence of polynomial particular solutions of fourth-order VCHDE are derived; (2) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of fourth-order VCHDE are proved; (3) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of nth-order VCHDE are proved. These results not only extend the class of solvable differential equations, but also provide a new way of thinking about the existence of solutions to VCHDE.
Citation: Peng E, Tingting Xu, Linhua Deng, Yulin Shan, Miao Wan, Weihong Zhou. Solutions of a class of higher order variable coefficient homogeneous differential equations[J]. Networks and Heterogeneous Media, 2025, 20(1): 213-231. doi: 10.3934/nhm.2025011

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Abstract

Recently, the variable coefficient homogeneous differential equations (VCHDE) have been widely applied to real-world problems, such as wave propagation and material science. However the exploration and research on higher-order VCHDE is relatively lagging. Given this, this work focuses on the solutions of fourth-order and nth-order VCHDE with polynomial coefficients. By means of the sufficient conditions for the existence of solutions to differential equations, a connection is established between the rank of the variable coefficient matrix and the existence of polynomial particular solutions. The main results show that: (1) the necessary and sufficient conditions for the existence of polynomial particular solutions of fourth-order VCHDE are derived; (2) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of fourth-order VCHDE are proved; (3) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of nth-order VCHDE are proved. These results not only extend the class of solvable differential equations, but also provide a new way of thinking about the existence of solutions to VCHDE.

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