【正文】 , 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 之間的對應(yīng)關(guān)系 ........................... 30 實例分析 ....................................................................................................... 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 求解實例 ................................................................................. 38 給出一個現(xiàn)實問題通過 MATLAB 求解 ......................................................... 41 II 總結(jié) ....................................................................................................... 45 參考文獻 ............................................................................................... 46 致謝 ....................................................................................................... 47 畢業(yè)設(shè)計（論文）知識產(chǎn)權(quán)聲明 ........................................................ 48 畢業(yè)設(shè)計 （ 論文） 獨創(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。 2 帶 *項可根據(jù)學科特點選填。 （含起始時間、設(shè)計地點）： 第 1周 — 第 2周： 查閱資料，了解常微分方程有關(guān)主理論與計算方法； 第 3周 — 第 4周： 查閱資料，掌握高 階微分方程與一階微分組的邏輯關(guān)系； 第 5周 — 第 13周： 著手寫論文，師生討論相關(guān)問題，第一稿完成； 第 14周 — 第 16周：修改論文； 第 17周 — 第 18周：定稿，準備答辯； 以上所有工作均在學校完成。 本科 畢業(yè)設(shè)計（論文） 題目： 高階線性微分方程與線性微分方程組之間關(guān)系的研究 院（系） 專 業(yè) 班 級 姓 名 學 號 導 師 xxxx 年 x 月 畢業(yè)設(shè)計（論文）任務(wù)書 院（系） 專業(yè) 數(shù)學與應(yīng)用數(shù)學 班級 姓名 學號 （ 論文）題目： 高階線性微分方程與線性微分方程組之間關(guān)系的研究 ： 常微分方程是數(shù)學分析或基礎(chǔ)數(shù)學的一個組成部分，在整個數(shù)學大廈中占據(jù)著非常重要的位置，在反映客觀現(xiàn)實世界運動過程的量與量之間的關(guān)系中，大量存在滿足常微分方程關(guān)系式的數(shù)學模型，需要我們通過求解常微分方程來了解位置函數(shù)的性質(zhì)。 (論文 )的主要內(nèi)容（理工科含技術(shù)指標）： （ 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）給出幾個算例。 指導教師簽名： 年 月 日 學生簽名：