【正文】 斷的適應(yīng)新的要求提出新的解決方法。 MATLAB 起源于矩陣的運(yùn)算，并且已經(jīng)發(fā)展成為一種高度集成的計(jì)算機(jī)語(yǔ)言。應(yīng)該說(shuō)，應(yīng)用常微分方程理論已經(jīng)取得了很大的成就，但是，它現(xiàn)有的理論也遠(yuǎn)遠(yuǎn)不能滿足我們的需求，還有待于進(jìn)一步的發(fā)展，從而使這門學(xué)科的理論更加完善。而這些問(wèn)題都是可以轉(zhuǎn)化成為求解常微分方程的解，或者化為研究解的性質(zhì)的相關(guān)問(wèn)題。 本文主要解決的問(wèn)題及所用方法 （ 1）高階線性微分方程解得結(jié)構(gòu)及初等解法的綜述； （ 2）一階線性微分組方程解結(jié)構(gòu)解得的性質(zhì)； （ 3）討論高階線性微分方程與線性微分方程組之間關(guān)系； （ 4）給出幾個(gè)算例； （ 5）使用 MATLAB 求解相關(guān)問(wèn)題。雖然常微分方程的理論發(fā)展已經(jīng)歷幾百年，但目前仍在發(fā)展中。因此，常微分方程的理論和方法不僅廣泛應(yīng)用于自然科學(xué)，而且越來(lái)越多的應(yīng)用于社會(huì)科學(xué)的各個(gè)領(lǐng)域。 常微分方程的理論在逐步完善的時(shí)候，利用它可以精確地表述事物變化的基本規(guī)律，只要列出相應(yīng)的微分方程，有了解方程的方法，常微分方程也就成為了最有生命力的數(shù)學(xué)分支之一?？萍己蛿?shù)學(xué)界的重大發(fā)現(xiàn)是混沌、孤立子和分形，其中混沌、孤立子直接與微分方程有關(guān)。 19 世紀(jì)末，天體力學(xué)中的太陽(yáng)系穩(wěn)定性問(wèn)題需要研究常微分方程解的大范圍性態(tài)，從而使常微分方程的研究從“求定解問(wèn)題”轉(zhuǎn)向“求所有解”的新時(shí)代。加上柯西初值問(wèn)題的提出，常微分方程從“求通解”轉(zhuǎn)向“求定解”時(shí)代。常微分方程是解決實(shí)際問(wèn)題的重要工具。 常微分方程是數(shù)學(xué)分析或基礎(chǔ)數(shù) 學(xué)的一個(gè)組成部分，在整個(gè)數(shù)學(xué)大廈中占據(jù)非常重要的位置。同樣，如果知道自變量、未知函數(shù)及函數(shù)的導(dǎo)數(shù)（或微分）組成的關(guān)系式，得到的便是微分方程，通過(guò)求解微分方程求出未知函數(shù)。直觀的體現(xiàn)了數(shù)學(xué)軟件在求解 高階線性 微分方程 時(shí)，采用的也是把高階線性微分方程轉(zhuǎn)化為線性微分方程組來(lái)求解的解題思路 . 關(guān)鍵詞： 高階 線性 微分方程； 線性 微分方程組； MATLAB II Research on the relationship between higher order linear differential equations and linear differential equations Abstract In this thesis, relevant theories and structures of highorder differential equation and differential equations are mainly introduced, and the calculation methods of relevant equations of the two and therefore further researches and analyzes the mutual transformation relationship between the two are discussed. In the introduction part, the research background, current development situation and research significance of ordinary differential equations are mainly introduced in the thesis and the major research contents of this thesis will be given. In the second chapter, the basic theory of linear highorder ordinary differential equations and the nature of the solutions to homogeneous linear differential equations and nonhomogeneous linear differential equations are researched, and several solutions to linear highorder differential equations are solved. Combining with the relevant theories in the second chapter, the structure and solution forms of linear ordinary differential equations and utilizes the method of elimination are deeply researched in the third chapter , applying first integral method and the method of variation of constant to analyze and solve related examples. Through solving and analyzing related examples and practical problems,in the fourth chapter, the mutual transformation relationship between linear highorder differential equations and linear differential equations is explored. Differential equations can be solved by being transformed into linear highorder differential equation through the method of elimination. Likewise, highorder differential equation can be solved by being transformed into differential equations by the form of function substitution and the variation of constants formula. At last, through the understanding of MATLAB correlation function, the transformation from highorder linear differential equations to differential equations and the solution are realized, which intuitively embodies the thought of solving highorder differential equations of mathematical software of transforming higher order differential equations to differential equations. Keywords:Linear highorder ordinary differential,Linear differential equations,MATLAB I 目錄 中文 摘要 ............................................ I 英文摘要 ........................................... II .............................................. 1 常微分方程的背景和發(fā)展現(xiàn)狀 .................................. 1 本文主要解決的問(wèn)題及所用方法 ................................ 2 課題成果及意義 .............................................. 2 ............................ 3 高階齊次線性微分方程 ........................................ 6 特征根是單根 ............................................ 7 特征根是重根 ............................................ 8 高階非齊次線性微分方程 ...................................... 9 常數(shù)變易法 ............................................. 10 比較系數(shù)法 ............................................. 12 拉普拉斯變換法 ......................................... 13 幾種可降階的微分方程的解法 ................................. 16 ........................................................ 20 齊次線性微分方程組的解的相關(guān)定理 ............................ 23 非齊次線性微分方程組的解的相關(guān)定理 .......................... 25 ....... 30 高階線性微分方程與線性微分方程組 之間的對(duì)應(yīng)關(guān)系 ........................... 30 實(shí)例分析 ....................................................................................................... 34 MATLAB 中高階微分方程到微分方程組的轉(zhuǎn)化及求解 .............................. 37 解微分方程的 MATLAB 命令 ................................................................ 38 MATLAB 求解實(shí)例 ................................................................................. 38 給出一個(gè)現(xiàn)實(shí)問(wèn)題通過(guò) MATLAB 求解 ......................................................... 41 II 總結(jié) ....................................................................................................... 45 參考文獻(xiàn) ............................................................................................... 46 致謝 ....................................................................................................... 47 畢業(yè)設(shè)計(jì)（論文）知識(shí)產(chǎn)權(quán)聲明 ........................................................ 48 畢業(yè)設(shè)計(jì) （ 論文） 獨(dú)創(chuàng)性聲明 ............................................................ 49 附錄 A 外文翻譯原文 ........................................................................... 50 附錄 B 外文翻譯譯文 ........................................................................... 57 1 緒論 1 常微分方程的背景和發(fā)展現(xiàn)狀 數(shù)學(xué)分析（微積分）中研究了變量的各種函數(shù)及函數(shù)的微分與積分。 第二章，對(duì)常微分高階線性微分方程的基礎(chǔ)理論以及齊次和非齊次線性 高階微分方程的解的性質(zhì)進(jìn)行了深入的分析與研究，并結(jié)合例題 歸納 了幾種高階線性微分的解法。 2 帶 *項(xiàng)可根據(jù)學(xué)科特點(diǎn)選填。 ① 實(shí)驗(yàn)（時(shí)數(shù)） *或?qū)嵙?xí)（天數(shù)）： 1— 18周 ② 圖紙（幅面和張數(shù)） *： ③ 其他要求： 查閱相關(guān)參考文獻(xiàn)至少 18篇，其中至少 2篇