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1、航空工業(yè)管理學(xué)院畢業(yè)論文（設(shè)計(jì)）2015屆數(shù)學(xué)與應(yīng)用數(shù)學(xué)專(zhuān)業(yè)1111062班級(jí)題目二階常微分方程的降階解法姓名賈靜靜學(xué)號(hào)111106213指導(dǎo)教師程春蕊職稱講師2015年4月5號(hào)二階常微分方程的降階解法摘要常微分方程是數(shù)學(xué)領(lǐng)域的一個(gè)非常重要的課題，并在實(shí)踐中廣泛于解決問(wèn)題，分析模型。常微分方程在微分理論中占據(jù)首要位置,普遍應(yīng)用在工程應(yīng)用、科學(xué)研究以及物理學(xué)方面,不少應(yīng)用例都?xì)w結(jié)為二階線性常微分方程的求解問(wèn)題。而正常情況下，常系數(shù)微分方程依據(jù)線性常微分方程的日常理論是可以求解的.不過(guò)對(duì)于變系數(shù)二階線性常微分方程的求解卻有一定程度的困難,迄今為止還沒(méi)有一個(gè)行之有效的普遍方法
2、。本文主要考慮了二階常系數(shù)線性微分方程的降階法。關(guān)于二階常系數(shù)線性微分方程的求解問(wèn)題,首先,我們給出二階齊次常系數(shù)線性微分方程的特征方程，并求解出特征方程的兩個(gè)特征根；其次，利用積分因子乘以微分方程和導(dǎo)數(shù)的運(yùn)算，將二階常系數(shù)線性微分方程化為一階微分形式；最后，將一階微分形式兩邊同時(shí)積分，求解一階線性微分方程，可求得二階常系數(shù)線性微分方程的一個(gè)特解或通解。關(guān)于二階變系數(shù)齊次線性微分方程的求解問(wèn)題，化為恰當(dāng)方程通過(guò)降階法求解二階齊次變系數(shù)微分方程的通解。對(duì)于非齊次線性微分方程，只需再運(yùn)用常數(shù)變易法求出它的一個(gè)特解，問(wèn)題也就相應(yīng)地解決了。關(guān)鍵詞二階常微分方程；降階法；特征根；
3、常數(shù)變易法；一階微分形式OrderreductionmethodofsecondorderordinarydifferentialequationsJingjingJiaChunruiCheng111106213AbstractOrdinarydifferentialequationisaveryimportanttopicinthefieldofmathematics,ithasbeenwidelyusedinsolvingtheproblemandanalyzingmodelinpractice.Ordinarydifferentialequationsinthet
4、heoryofdifferentialoccupiedfirstplace,ithasbeenwidelyusedinengineeringapplicationandscientificresearchaswellasphysics,manyapplicationexamplesareattributedtosecondorderlinearordinarydifferentialequationsolvingproblem.Andundernormalcircumstances,ordinarycoefficientdifferentialequationonthe
5、basisofthelinearoftendailytheoryofdifferentialequationsiscanbesolved.Butforthesolutionforvariablecoefficientsecondorderlinearordinarydifferentialequationshaveacertaindegreeofdifficulty,sofarwehaven'tawell-establishedgeneralmethod.Thispapermainlyintroducesthemethodofreductionofordertwoord
6、erlineardifferentialequationwithconstantcoefficients.Ontheproblemofsolvingthelineardifferentialequationwithtwoorderconstantcoefficients,first,wegivehomogeneousordinarycoefficientlineardifferentialequationofthecharacteristicequationandsolvethetwocharacteristicrootsofcharacteristicequation
7、;secondly,weshouldusetheintegralfactortimesdifferentialequationandderivativeoperationandturntwoorderconstantcoefficientlineardifferentialequationintothefirstorderdifferentialequation.Finally,Wefirstorderdifferentialandintegralformonbothsides,solvethefirstorderlineardiffer
