# On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Transition to the Ordinary System of Differential Equations Using Quasi-Derivatives

## 3. Construction of Asymptotic Formulas

#### 3.1. Case 1

#### 3.2. Case 2

**Remark**

**1.**

#### 3.3. Case 3

**Theorem**

**1.**

- (1)
- $h\left(x\right),\phantom{\rule{0.277778em}{0ex}}{q}_{2}\left(x\right)\in {L}_{1}[1,\infty ),$
- (2)
- $h\left(x\right),\phantom{\rule{0.277778em}{0ex}}{q}_{3}\left(x\right),\phantom{\rule{0.277778em}{0ex}}\tilde{h}\left(x\right)\in {L}_{1}[1,\infty )$,
- (3)
- ${q}_{3}\left(x\right),\phantom{\rule{0.277778em}{0ex}}{h}_{1}\left(x\right),\phantom{\rule{0.277778em}{0ex}}\tilde{h}\left(x\right)\in {L}_{1}[1,\infty )$.

#### 3.4. Counterexample

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Eastham, M.S.P. The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem; Clarendon Press: Oxford, UK, 1989. [Google Scholar]
- Fedoryuk, M.V. Asymptotic Methods for Linear Ordinary Differential Equations; Nauka: Moscow, Russia, 1983. (In Russian) [Google Scholar]
- Naimark, M.A. Linear Differential Operators; Nauka: Moscow, Russia, 1969. (In Russian) [Google Scholar]
- Nazirova, E.A.; Sultanaev, Y.T.; Valeeva, L.N. On a Method for Studying the Asymptotics of Solutions of Sturm–Liouville Differential Equations with Rapidly Oscillating Coefficients. Math. Notes
**2022**, 112, 1059–1064. [Google Scholar] - Konechnaja, N.N.; Mirzoev, K.A.; Shkalikov, A.A. On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients. Math. Notes
**2023**, 104, 244–252. [Google Scholar] [CrossRef] - Mirzoev, K.A.; Konechnaja, N.N. Asymptotics of solutions to linear differential equations of odd order. Vestnik Moskov. Univ. Ser. 1. Mat. Mekh.
**2020**, 75, 22–26. [Google Scholar] [CrossRef] - Konechnaja, N.N.; Mirzoev, K.A.; Shkalikov, A.A. Asymptotics of Solutions of Two-Term Differential Equations. Math. Notes
**2023**, 113, 228–242. [Google Scholar] [CrossRef] - Nazirova, E.A.; Sultanaev, Y.T.; Valeev, N.F. On a new approach for studying asymptotic behavior of solutions to singular differential equations. Trans. Ufa Math. J.
**2015**, 3, 9–14. [Google Scholar] - Myakinova, O.V.; Sultanaev, Y.T.; Valeev, N.F. On the Asymptotics of Solutions of a Singular nth-Order Differential Equation with Nonregular Coefficients. Math. Notes
**2018**, 104, 606–611. [Google Scholar] - Valeev, N.F.; Nazirova, E.A.; Sultanaev, Y.T. On a Method for Studying the Asymptotics of Solutions of Odd-Order Differential Equations with Oscillating Coefficients. Math. Notes
**2021**, 109, 980–985. [Google Scholar] [CrossRef] - Nazirova, E.A.; Sultanaev, Y.T.; Valeev, N.F. The new asymptotics for solutions of the Sturm-Liouville equation. Proc. Inst. Math. Mech. Acad. Sci. Azerbaijan
**2023**, 49, 253–258. [Google Scholar] - Rossmann, W. Lie Groups—An Introduction Through Linear Groups; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Bellman, R. Stability Theory of Differential Equations; Mc-Graw-Hill: New York, NY, USA; Toronto, ON, Canada; London, UK, 1953. [Google Scholar]
- Everitt, W.N.; Marcus, L. Boundary value problem and symplectic algebra for ordinary differential and quasi-differential operators. AMS. Math. Surv. Monogr.
**1999**, 60, 1–60. [Google Scholar] - Everitt, W.N.; Race, D. Some remarks on linear ordinary quasi-differential expressions. Proc. Lond. Math. Soc.
**1987**, 3, 300–320. [Google Scholar] [CrossRef] - Everitt, W.N.; Zettl, A. Differential operators generated by a countable number of quasi-differential expressions on the real line. Proc. Lond. Math. Soc.
**1992**, 3, 524–544. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Sultanaev, Y.T.; Valeev, N.F.; Nazirova, E.A.
On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations. *Mathematics* **2024**, *12*, 213.
https://doi.org/10.3390/math12020213

**AMA Style**

Sultanaev YT, Valeev NF, Nazirova EA.
On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations. *Mathematics*. 2024; 12(2):213.
https://doi.org/10.3390/math12020213

**Chicago/Turabian Style**

Sultanaev, Yaudat T., Nur F. Valeev, and Elvira A. Nazirova.
2024. "On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations" *Mathematics* 12, no. 2: 213.
https://doi.org/10.3390/math12020213