本文選題：反射積分 + 逆積分因子　； 參考：《揚(yáng)州大學(xué)》2017年碩士論文

【摘要】：研究微分系統(tǒng)x'= X(t,x)的解的性態(tài),不僅推動(dòng)著微分方程理論的發(fā)展,同時(shí)對(duì)研究客觀世界中物體的運(yùn)動(dòng)規(guī)律也具有很大的實(shí)際應(yīng)用價(jià)值.當(dāng)微分系統(tǒng)為自治系統(tǒng),對(duì)于它的解的性態(tài)的研究成果已有很多.而對(duì)于非自治系統(tǒng)的研究成果就相對(duì)有限了.我們知道,對(duì)于周期時(shí)變系統(tǒng)的研究,可以借助于Poincare映射和Lyapunov變換[1-4],但有時(shí)尋找這些變換是很困難的.上世紀(jì)八十年代,Mironenko[5]創(chuàng)建了反射函數(shù)理論.利用反射函數(shù),我們可以建立周期時(shí)變系統(tǒng)（？）的Poincare映射,借助它能研究該系統(tǒng)的解的定性性態(tài).我們稱具有相同反射函數(shù)的兩個(gè)微分系統(tǒng)類是等價(jià)的,而等價(jià)的周期系統(tǒng)的周期解的性態(tài)是相同的.所以當(dāng)研究一類復(fù)雜的非自治微分系統(tǒng)解的性態(tài)時(shí),只需研究與該系統(tǒng)等價(jià)的簡(jiǎn)單系統(tǒng)或自治系統(tǒng)解的性態(tài)即可.Mironenko在[5-6]中研究了微分系統(tǒng)（？）與x' = Y(t,x)(2)的等價(jià)性,得出(2)等價(jià)于(1),當(dāng)且僅當(dāng)(2)可表示為（？）.(3)這里F(t,x)為(1)的反射函數(shù).但是對(duì)于一般微分系統(tǒng)要求出其反射函數(shù)是相當(dāng)困難的.那如何在反射函數(shù)未知的情況下,判定(1)與(2)等價(jià)?于是Mironenko在[7]中給出,若△(t,x)滿足（？）時(shí),（？）與(1)等價(jià),這里α(t)為t的奇的純量函數(shù),由此并推出x' = X(t,x)+ ∑αi(t)△,(t,x)(6)也與(1)等價(jià),這里αi(t)為奇的純量函數(shù),△i.(t,x)為(4)的解.由此可看出,求出(4)的解△(t,x)即反射積分,對(duì)判定兩個(gè)微分系統(tǒng)的等價(jià)性尤為重要.Belskii 在[26]中給出 Riccati 方程（？）和 Abel 方程（？）及一般多項(xiàng)式方程x'=∑i-0nai(t)xi的反射積分的結(jié)構(gòu)形式,及這些方程具有這些反射積分的充分條件.Veresovich[19],Varenikova[25]研究了一個(gè)平面多項(xiàng)式微分系統(tǒng)與其線性部分等價(jià)的判定準(zhǔn)則.在本文中,本人主要研究了幾類一階非自治有理分式型微分方程的反射積分及逆積分因子.通過(guò)它們建立了與這些方程等價(jià)的一階微分方程類,利用逆積分因子研究了這些方程的可積性及其解的定性性態(tài).其次還研究了兩個(gè)非自治線性方程組的等價(jià)性,并給出了若干判定的準(zhǔn)則.在這篇文章的第三章中,本人研究了一次有理分式方程具有各種類型的反射積分的充分條件,建立了與(7)等價(jià)的微分方程類.并利用這些反射積分討論了微分系統(tǒng)的逆積分因子、首次積分及其解的定性性態(tài).其次研究了二次有理分式方程(?)具有二次有理分式形式的反射積分的充分條件.建立了與(9)等價(jià)的微分方程類,并利用反射積分研究了微分系統(tǒng)(?)的逆積分因子及可積性問(wèn)題及其解的定性性態(tài).在第四章中,研究了兩個(gè)非自治線性微分系統(tǒng)（？）等價(jià)性,并給出當(dāng)它們等價(jià)時(shí),其系數(shù)矩陣A(t),B(t)所滿足的必要條件,以及它們等價(jià)的若干判定準(zhǔn)則.特別地,還討論了（？）等價(jià)時(shí),(這里φ(t)為純量函數(shù)C為常數(shù)矩陣),φ(t),C所具有的特征性質(zhì).
[Abstract]:The study of the behavior of the solution of the differential system xn = X t X) not only promotes the development of the theory of differential equations, but also has great practical application value for the study of the law of motion of objects in the objective world. When the differential system is an autonomous system, many researches have been done on the behavior of its solution. However, the research results of non-autonomous systems are relatively limited. We know that the study of periodic time-varying systems can be done by means of Poincare maps and Lyapunov transformations [1-4], but sometimes it is difficult to find these transformations. Mironenko [5] founded the theory of reflection function in the 1980s. Using the reflection function, we can establish the periodic time-varying system. By means of Poincare mapping, the qualitative behavior of the solution of the system can be studied. We say that two classes of differential systems with the same reflection function are equivalent, and the behavior of the periodic solutions of the equivalent periodic systems is the same. Therefore, when we study the behavior of solutions of a class of complex nonautonomous differential systems, we only need to study the properties of solutions of simple systems or autonomous systems equivalent to the system. Mironenko has studied the differential systems in [5-6]. The equivalence with x'= Y ~ (t) ~ (X ~ (+) ~ (2) is obtained, which is equivalent to ~ (1), if and only if ~ (2) can be expressed as the reflection function of ~ (1), where F _ (t ~ (X) is a ~ (1). However, it is very difficult for the general differential system to require its reflection function. How can we determine the equivalence of 1) and 2) when the reflection function is unknown? So Mironenko gives in [7], if TX) satisfies the ) It is equivalent to 1), where 偽 t) is an odd scalar function of t, from which we derive the solution of x'= XTX) 鈭，

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