November  2022, 15(11): 3413-3427. doi: 10.3934/dcdss.2022149

New approach to the singular solution of implicit ordinary differential equations

1. 

School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250358, China

2. 

School of Mathematical Sciences, Soochow University, Suzhou 215006, China

*Corresponding author: Shaoheng Zhang

Received  January 2022 Revised  June 2022 Published  November 2022 Early access  August 2022

In this paper, for a class of nonlinear implicit ordinary differential equation with variable coefficients, a new approach to get the analytical singular solutions are given both for first order and higher order equations. Sufficient conditions on singular solutions are strengthened into necessary and sufficient conditions in specific case. Finally, the new results are verified by practical examples and MATLAB illustrations compared with traditional methods.

Citation: Shasha Zheng, Shaoheng Zhang. New approach to the singular solution of implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (11) : 3413-3427. doi: 10.3934/dcdss.2022149
References:
[1]

V. I. Arnold, Ordinary Differential Equations Third Edition, Springer-Verlag, Berlin, 2006.

[2]

M. Bartusek and K. Fujimoto, Singular solutions of nonlinear differential equations with $p(t)$-Laplacian, J. Differential Equations, 269 (2020), 11646-11666.  doi: 10.1016/j.jde.2020.08.046.

[3]

M. Bartusek and E. Pekarkova, On existence of proper solutions of quasilinear second order differential equations, Electron. J. Qual. Theory Differ. Equ., (2007), No. 5, 14 pp. doi: 10.14232/ejqtde.2007.1.5.

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N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102.  doi: 10.1016/j.na.2012.04.025.

[5] L. M. B. C. Campos, Higher-Order Differential Equations and Elasticity, CRC Press, Boca Raton, 2020. 
[6]

J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discret. Cont. Dynam. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[7]

C. V. Coffman and J. S. W. Wong, Oscillation and nonoscillation theorems for second order ordinary differential equations, Funkcial. Ekvac., 15 (1972), 119-130. 

[8]

M. A. del Pino and R. F. Manasevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277.  doi: 10.1006/jdeq.1993.1050.

[9]

T. Ding, Approaches to the Qualitative Theory of Ordinary Differential Equations, Peking University Series in Mathematics, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.

[10]

A. Garcia and J. Llibre, Existence of periodic solutions for a class of second order ordinary differential equations, Acta Applicandae Mathematicae, 169 (2020), 193-197.  doi: 10.1007/s10440-019-00295-9.

[11]

L. Ge, Discrimination of singular solutions of a class of first order differential equations(in Chinese), Scientific and Technology Information, 11 (2011).

[12]

A. Ghosh and S. Maitra, The first integral method and some nonlinear models, Comput. Appl. Math., 40 (2021), Paper No. 79, 16 pp. doi: 10.1007/s40314-021-01470-1.

[13]

E. L. Ince, Ordinary Differential Equations, Illustrated, Unabridged, Courier Corporation, Chicago, 2008.

[14] M. Kline, Mathematical Thought from Ancient to Modern Times: Volume 2, Oxford University Press, London, 1990. 
[15]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[16]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913. 

[17]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Transactions on Automatic Control, 64 (2019), 4024-4034. 

[18]

X. LiX. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031.

[19]

G. M. Murphy, Ordinary Differential Equations and Their Solutions, Courier Corporation, Chicago, 2011.

[20]

M. Raisinghania, Ordinary and Partial Differential Equations, S. Chand Publishing, New Delhi, 2013.

[21]

M. van NoortM. A. PorterY. Yi and S. N. Chow, Quasiperiodic dynamics in Bose CEinstein condensates in periodic lattices and superlattices, J. Nonlinear Sci., 17 (2007), 59-83.  doi: 10.1007/s00332-005-0723-4.

[22]

Y. Wu and D. Qian, Periodic solutions of singular second order equations at resonance, Acta Mathematica Sinica, English Series, 31 (2015), 1599-1610.  doi: 10.1007/s10114-015-4596-7.

[23]

Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Applicandae Mathematicae, 110 (2010), 871-883.  doi: 10.1007/s10440-009-9482-9.

show all references

References:
[1]

V. I. Arnold, Ordinary Differential Equations Third Edition, Springer-Verlag, Berlin, 2006.

[2]

M. Bartusek and K. Fujimoto, Singular solutions of nonlinear differential equations with $p(t)$-Laplacian, J. Differential Equations, 269 (2020), 11646-11666.  doi: 10.1016/j.jde.2020.08.046.

[3]

M. Bartusek and E. Pekarkova, On existence of proper solutions of quasilinear second order differential equations, Electron. J. Qual. Theory Differ. Equ., (2007), No. 5, 14 pp. doi: 10.14232/ejqtde.2007.1.5.

[4]

N. D. Brubaker and J. A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102.  doi: 10.1016/j.na.2012.04.025.

[5] L. M. B. C. Campos, Higher-Order Differential Equations and Elasticity, CRC Press, Boca Raton, 2020. 
[6]

J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discret. Cont. Dynam. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[7]

C. V. Coffman and J. S. W. Wong, Oscillation and nonoscillation theorems for second order ordinary differential equations, Funkcial. Ekvac., 15 (1972), 119-130. 

[8]

M. A. del Pino and R. F. Manasevich, Infinitely many $T$-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277.  doi: 10.1006/jdeq.1993.1050.

[9]

T. Ding, Approaches to the Qualitative Theory of Ordinary Differential Equations, Peking University Series in Mathematics, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.

[10]

A. Garcia and J. Llibre, Existence of periodic solutions for a class of second order ordinary differential equations, Acta Applicandae Mathematicae, 169 (2020), 193-197.  doi: 10.1007/s10440-019-00295-9.

[11]

L. Ge, Discrimination of singular solutions of a class of first order differential equations(in Chinese), Scientific and Technology Information, 11 (2011).

[12]

A. Ghosh and S. Maitra, The first integral method and some nonlinear models, Comput. Appl. Math., 40 (2021), Paper No. 79, 16 pp. doi: 10.1007/s40314-021-01470-1.

[13]

E. L. Ince, Ordinary Differential Equations, Illustrated, Unabridged, Courier Corporation, Chicago, 2008.

[14] M. Kline, Mathematical Thought from Ancient to Modern Times: Volume 2, Oxford University Press, London, 1990. 
[15]

X. LiD. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024.

[16]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Transactions on Automatic Control, 65 (2020), 4908-4913. 

[17]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Transactions on Automatic Control, 64 (2019), 4024-4034. 

[18]

X. LiX. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031.

[19]

G. M. Murphy, Ordinary Differential Equations and Their Solutions, Courier Corporation, Chicago, 2011.

[20]

M. Raisinghania, Ordinary and Partial Differential Equations, S. Chand Publishing, New Delhi, 2013.

[21]

M. van NoortM. A. PorterY. Yi and S. N. Chow, Quasiperiodic dynamics in Bose CEinstein condensates in periodic lattices and superlattices, J. Nonlinear Sci., 17 (2007), 59-83.  doi: 10.1007/s00332-005-0723-4.

[22]

Y. Wu and D. Qian, Periodic solutions of singular second order equations at resonance, Acta Mathematica Sinica, English Series, 31 (2015), 1599-1610.  doi: 10.1007/s10114-015-4596-7.

[23]

Q. Yao, Periodic positive solution to a class of singular second-order ordinary differential equations, Acta Applicandae Mathematicae, 110 (2010), 871-883.  doi: 10.1007/s10440-009-9482-9.

Figure 1.  Singular solution and general solutions of Clairaut Equation($ \gamma = 1 $)
Figure 2.  Singular solution general solutions of (9)
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