§5.3常系数线性微分方程组LinearSystemswithConstantCoefficients5.3.1矩阵指数expA的定义和性质5.3.2基解矩阵的计算公式（可对角化情形）5.3.3基解矩阵的计算公式（不可对角化情形）§5.3LinearSystemswithConstantCoefficients0ccλett()的解，其中常数和向量c是待定的。将(5.43)xAx(5.33)(5.43)的形如我们试图寻求代入(5.33)得cAcλtλteλe§5.3LinearSystemswithConstantCoefficients因为0，上式即tλe0A...
§5.3常系数线性微分方程组LinearSystemswithConstantCoefficients5.3.1矩阵指数expA的定义和性质5.3.2基解矩阵的计算公式（可对角化情形）5.3.3基解矩阵的计算公式（不可对角化情形）§5.3LinearSystemswithConstantCoefficients5.3.1矩阵指数expA的定义和性质无穷矩阵级数121kkkAAAAnnkijnnijnnijaaa)()()(()()()21nnkijijijaaa)(()()()21如果每个nijakkij,,,,()12...
§4.3ReductionofOrderPowerSeriesSolutions4.3.2二阶线性方程的幂级数解法解anxnaxaxay2210为方程的解0(0)y00a例4xydxdy00y()的解。求方程的满足初始条件设anxnaxaxaxy332214.3.1可降阶的方程§4.3高阶方程的降阶和幂级数解法§4.3ReductionofOrderPowerSeriesSolutions弹题1(判断题)任意幂级数的收敛半径都是正实数.答案：错§4.3ReductionofOrderPowerSeriesSolutions...
§4.3ReductionofOrderPowerSeriesSolutions可降低k阶，即可化为n-k阶的齐次线性方程。0111atxdtxatddtxdnnnnn()().)(24已知(4.2)的k个线性无关解，则(4.2)结论0不显含某些变元),,(,,x(n)Ftxx4.3.1可降阶的方程1.方程2.齐次线性方程已知部分线性无关解§4.3ReductionofOrderPowerSeriesSolutions齐次线性方程的右端项.答案：对§4.3ReductionofOrderPowerSeriesSolutionskxxx,,,21xyxkxyxyxkk...
§4.3ReductionofOrderPowerSeriesSolutions可降低k阶，即可化为n-k阶的齐次线性方程。0111atxdtxatddtxdnnnnn()().)(24已知(4.2)的k个线性无关解，则(4.2)结论0不显含某些变元),,(,,x(n)Ftxx4.3.1可降阶的方程1.方程2.齐次线性方程已知部分线性无关解§4.3ReductionofOrderPowerSeriesSolutions弹题1(判断题)齐次线性方程的右端项.答案：对§4.3ReductionofOrderPowerSeriesSolutionskxxx,,,21xyxkxyx...
§4.3ReductionofOrderPowerSeriesSolutions0不显含某些变元),,(,,x(n)Ftxx),(,,1xkxx4.3.1可降阶的方程(1)方程不显含未知函数x及1.方程可降低k阶(2)方程不显含自变量t:0),,,(x(n)Fxx).(594可降低一阶§4.3ReductionofOrderPowerSeriesSolutionsxydtyddtxdx)(方法令dtdxdxdydxydydtdxdxdxydyddxdtydydx)()(2222222dxydydxydydxydydxdyy)())((§4.3Reduction...
§4.3高阶方程的降阶和幂级数解法ReductionofOrderPowerSeriesSolutions§4.3ReductionofOrderPowerSeriesSolutions本节内容/Contents/4.3.1可降阶的方程4.3.2二阶线性方程的幂级数解法§4.3ReductionofOrderPowerSeriesSolutions1.不显含某些变元0),,(,,x(n)Ftxx4.3.1可降阶的方程2.齐次线性方程已知部分线性无关解§4.3ReductionofOrderPowerSeriesSolutions1.(1)方程不显含未知函数x及),(,,1xkxx01),,,(...
第三章一阶微分方程解的存在定理3.8奇解微分方程而言，积分曲线上的每一点处0)(,,yxyF解存在但不唯一的特解在几何学领域有广泛应用。几何学上，曲线族的包络是这样的曲线，它本身并不包含于曲线族，但是过这曲线的每一点，有曲线族中的一条曲线与之相切。例如给定曲线族1)(22ycx1y1y于是，是该曲线族的包络1y弹题给定曲线族的包罗为（）1.Ay1.yB1.xC2.xD1)(22cyxC.答案例如运用第二章...
Chapter4First-OrderOrdinaryDifferentialSystemsManyofthephysicalproblemswhichleadtodifferentialequationsinvolvemorethanoneunknownfunction.Forexample,instudyingthemotionofaparticleinthexyplane,itisnecessarytodeterminethetwocoordinatesxandyoftheparticleasfunctionsoftime.Forthemotionofapointinthree-dimensionalspace,threecoordinatesareneeded.Forthegeneralmotionofarigidbodyinspace,sixcoordinatesarene...
第三章一阶微分方程解的存在定理3.5解的延拓定理1（Cauchy-Piccard）设在的领域,|:|0axxDbyy||0上连续，并满足李氏条件，则00)(,)(yxyfxyy在hxx||0存在唯一解。其中|)(,max|},,min{(,)fxyMMabhDxy(,)fxy),(0y0x局部性弹题（判断题）微分方程10)(2yydxdy而言，是方程的解（）答案：错误yx例1讨论微分方程00)(22yyxdxdy在矩形区域|1|,1|1:|yxD和|2|,2|2:|...
Chapter6IntroductiontoPartialDifferentialEquationsAllofthedifferentialequationswhichhavebeenstudiedinthepreviouschaptershaveonefeatureincommon.Theyalldescribethestateorthemotionofaquantitywhichdependononlyoneindependentvariable.Hencethedifferentialequationsareordinaryones.Inthischapterweshallstudyequationswhichdescribethemotionorchangeofstateofquantitieswhichdependonmorethanoneindependentvariab...
Chapter5*QualitativeAnalysisandStabilityofSolutionsWhentheactualsolutionsofadifferentialequationorsystemcannotbeobtained,weoftenconsidersomemethodofapproximatingsolutions.Althoughthiscandeedhelpusobtainingsomeapproximatesolutions,weoftenfeeltroublesome.Infact,itisnotnecessarytoobtainmaterialformofeachsolution.Whatwhichweneediscorrelativeinformationwiththesolution,namely,theentirehistoryofamotio...
第三章一阶微分方程解的存在定理3.3一阶微分方程的解的唯一性回顾：一阶微分方程解的存在性xxnnxdxfxyxyx0())(,)()(1000如果在闭区域D上连续且存在常数使得(,)xyf,)),(,|,(,||)(,)(,|212121DxyxyyLyfxyfxyL则序列在上连续且一致收敛，且()}{nx],[00hxx()lim()xxnn是积分方程(3.5)的定义于上的连续解。hxxx00问题：连续性和李氏条件下，Cauchy问题的解唯一吗？弹题微分方...
Chapter2First-OrderOrdinaryDifferentialEquationsOurstudiesbeginfromfirst-orderordinarydifferentialequations.First-orderordinarydifferentialequationsaredividedintotwobasiccategories:explicitandimplicitequations.Theequationwhichisintheform:iscalledafirst-orderexplicitequation,andtheequationwhichisintheform:iscalledafirst-orderimplicitequation.Atfirst,weconsiderthreebasictypesoffirst-orderexplicit...
第三章一阶微分方程解的存在定理3.3一阶微分方程的解的唯一性回顾：一阶微分方程解的存在性xxnnxdxfxyxyx0())(,)()(1000如果在闭区域D上连续且存在常数使得(,)xyf,)),(,|,(,||)(,)(,|212121DxyxyyLyfxyfxyL则序列在上连续且一致收敛，且()}{nx],[00hxx()lim()xxnn是积分方程(3.5)的定义于上的连续解。hxxx00问题：连续性和李氏条件下，Cauchy问题的解唯一吗？弹题微分方...
Chapter3Nth-OrderOrdinaryDifferentialEquationsInthischapter,westudythemethodsofsolvingnth-orderdifferentialequationHowever,itwouldbeverydifficulttosolvethisequation.Infact,therearemanyequationswhichcannotbesolvedinasimpleanalyticform.Nevertheless,thereisagoodwayforthelinearequation:Therefore,weshallmainlydiscussthenth-orderlineardifferentialequation.Weshallfirstintroducesomefundamentaltheoriesf...
第三章一阶微分方程解的存在定理3.2皮卡逐步逼近法定理1（Cauchy-Piccard）设在的领域,|:|0axxDbyy||0上连续，并满足李氏条件，则00)(,)(yxyfxyy在hxx||0存在唯一解。其中|)(,max|},,min{(,)fxyMMabhDxy(,)fxy),(x0y0Cauchy问题与积分方程的等价性：命题1在D上连续，则Cauchy问题,),(,)(,)(0000Dyxyxyfxydxdy可以等价地转化成积分方程)5.3(.,)(00xxfxydxyy(,)xyf证明000)(...
Chapter1IntroductiontoDifferentialEquationsInmathematicalanalysis,wehavestudiedmanyfunctionswhichcanreflecttherelationsbetweenquantitiesandquantitiesonmovesofmaterials.However,afewfactsshowthatmanyrelationscannotdirectlybeexpressedinsuchafunctionform.Theyareusuallyexpressedastheequationswhichrelateindependentvariable(s),unknownfunction(s)andthederivatives.Suchanequationiscalleddifferentialequat...
第三章一阶微分方程解的存在定理3.1一阶微分方程的解的存在性回顾：一阶微分方程初等解法(,)fxydxdy()()hxgy分离变量后积分()()qxpxy常数变易法,)(,)(xyNMxy积分因子法而很多现实问题对应的微分方程的形式为Cauchy问题问题1：通过初等解法求通解后再求Cauchy问题可行吗？例考虑微分方程则由刘维尔定理可知，该微分方程不能初等求解。可否考虑近似解？问题2：怎样保证抽象Cauchy问题解的存在性？定理（Peano.G）如果...
AnswersandHintsExercise11)B2)a.Third-order,Non-linearb.First-order,Linearc.Second-order,Lineard.First-order,Non-lineare.First-order,Linearf.Second-order,Non-linear4)5)6)a.b.c.7),8)ShowthatanontrivialsolutionexistsonlyifholdsExercise21)a.b.c.d.e.f.g.h.2)a.b.c.d.e.f.g.h.2813)a.b.c.d.(constant)e.f.g.h.4)a.b.c.d.e.f.g.h.6)7)a.b.c.d.e.;f.g.h.9)10)a.b.None.12)a.Thehalfplaneorthehalfplaneb.Anynotinclu...
