常微分方程课件_第11讲(53)

1、Lecture 1112 25.2 General Theory of Linear Systems of Differential Equations5.2.1 Homogeneous Linear Systems of Differential Equations (齐次线性微分方程组齐次线性微分方程组)5.2.2 NonHomogeneous Linear Systems of Differential Equations (非齐次线性微分方程组非齐次线性微分方程组)3 3(5.15) ,)(xAxtDefinition C/P206(5.15). of)( a be tosaid ar

2、e then they(5.15), of solutionst independenlinearly are )(,),(),( that Suppose 21基基本本解解组组 solutions of system lfundamentaxxxntttnExample Cf E/P349 of solutionst independenlinearly 2 are 3)(,23)(t Verify tha 552221tttteeteetxx.76,34212211xxxxxx4 4(5.15) ,)(xAxtDefinition C/P206(5.15). of)( a be tosai

3、d are then they(5.15), of solutionst independenlinearly are )(,),(),( that Suppose 21基基本本解解组组 solutions of system lfundamentaxxxntttnProposition Cf C/P125/Theorem 6ncombinatiolinear Then the (5.15). of solutions of system lfundamenta a are they is, that (5.15), ofsolutionst independenlinearly be )(,

4、),(),(Let 21ntttnxxx. )()()()( 2211tctctctnnxxxxconstants.arbitrary are , where(5.15), ofsolution general theis21nccc? unique (5.15) of solutions of systems lfundamenta theIs:Question 5 5DEs. of systemlinear shomogeneou theof that and DE shomogeneoulinear order -higher theof problem valuedinitial eb

5、etween th iprelationsh that the207-C/PP206 seecan We6 6(5.15) ,)(xAxtDefinition C/P207(Solution matrix and fundamental solution matrix解矩阵和基解矩阵解矩阵和基解矩阵)matrix Then the (5.15). of solutions be )(,),(),(Let 21nnntttnxxx)(,),(),()(21ttttnxxx(5.15). system for the )( a called is tors,column vec its as ec

6、torssolution v thesehaving解解矩矩阵阵matrix solution. )( a called is solutions ofmatrix Then the (5.15). of solutions of system lfundamenta a are they is, that (5.15), of solutionst independenlinearly are )(,),(),( if e,Furthermor21基解矩阵基解矩阵matrix solution lfundamentaxxxntttn7 7(5.15) ,)(xAxtTheorem 1* C/

7、P208 ).(matrix solution lfundamenta a havemust (5.15) t then (5.15), ofsolution is )( Ift ,)( )(ctttor.column vecconstant ldimensionacertain a is wherenc8 8(5.15) ,)(xAxtmatrix?solution lfundamenta a is ofsolution matrix a whether determine can we How Theorem 2* C/P208 ).(matrix oft determinan thede

8、notes )(det where),( 0)(det ifonly and if , interval on thematrix solution lfundamenta a is (5.15) of )( ofmatrix solution A tnntbtatbat!, allfor 0)(detthen , 0)(detsuch that , a is thereif e,Furthermor 00batttbat9 9Example 5.2.1 C/P208systemlinear theofmatrix solution lfundamenta a is 0)(t Verify t

9、ha ttteteet. 1011 xxproblem valueinitial thesolve try toe,Furthermor .11)0(,1011xxx10Corollary 1* C/P209(5.15) ,)(xAxt.,on (5.15) ofmatrix solution lfundamenta a also is )(matrix then , 0det isthat matrix,constant singular -non an is ,on (5.15) ofmatrix solution lfundamenta a is )( If batnnnnbatCCCQ

10、uestionmatrix?solution lfundamenta a also is )( whether is,That ?)(matrix about the What ttnnCC? unique (5.15) of solutions of systems lfundamenta theIs:QuestionProof:1111Corollary 2* C/P210(5.15) ,)(xAxt. )()(such that matrix constant singular -non an exists re then the,on (5.15) of matricessolutio

11、n lfundamenta twoare )( and )( If CCttnnbattProof:1212125.2 General Theory of Linear Systems of Differential Equations5.2.1 Homogeneous Linear Systems of Differential Equations (齐次线性微分方程组齐次线性微分方程组)5.2.2 NonHomogeneous Linear Systems of Differential Equations (非齐次线性微分方程组非齐次线性微分方程组)13135.2 General The

12、ory of Linear Systems of Differential Equations5.2.1 Homogeneous Linear Systems of Differential Equations (齐次线性微分方程组齐次线性微分方程组)5.2.2 NonHomogeneous Linear Systems of Differential Equations (非齐次线性微分方程组非齐次线性微分方程组)14145.1.2 NonHomogeneous Linear Systems of Differential Equations (非齐次线性微分方程组非齐次线性微分方程组)(5

13、.14) ),()(ttfxAxis systemlinear shomogeneou associated The . ,on continuous are )( and )( that SupposebattfA(5.15) ,)(xAxtProperty 1C/P211; Cf C/P127/Property 1(5.14). ofsolution a is )()(then (5.15), ofsolution a is (t) (5.14), ofsolution a is )( If tttProperty 2C/P211; Cf C/P127/Property 2(5.15).

14、ofsolution a is )()( then (5.14), of solutions twoare )( and )( If tttt1515Theorem 7C/P211(Structure of solution(解的结构解的结构) as expressed becan (5.14) of )(solution any then (5.14), ofsolution a is )( , (5.15) ofmatrix solution lfundamenta a is )( If ttt(5.14) ),()(ttfxAx(5.15) ,)(xAxt ),()()( tttctor

15、.column vecconstant ldimensionacertain a is wherencNote: This theorem tells us, in order to find any solution of (5.14), we need only to find a particular solution of (5.14) and a fundamental solution matrix of (5.15)! C/P211;E/P353+3541616(5.14) ),()(ttfxAx(5.15) ,)(xAxt In fact, as done in C/PP127

16、-129 for the higher-order linear nonhomogenerous DE, if we have known a fundamental solution matrix (or a fundamental system of solutions) of (5.14), we may also use the method of variation of constants to find a particular solution of (5.14)!1717(5.14) ),()(ttfxAx(5.15) ,)(xAxt (5.14). ofnot (5.15)

17、, ofsolution a is )()( then , (5.15) ofmatrix solution lfundamenta a is )( Suppose cttt funnction. valued- vectoredundeterminan is )( here (5.14), ofsolution a is(5.24) )()()( Let ttttcc ).()()()()( )()()( ttttttttfcAcc(5.25) ).()( )(tttfc ).()()( 1tttfc, ,)()()(010battdssstttfc! 0)( that take wewhe

18、re0tc1818(5.14) ),()(ttfxAx(5.15) ,)(xAxt (5.14). ofnot (5.15), ofsolution a is )()(then , (5.15) ofmatrix solution l fundamenta a is )( Suppose cttt funnction. valued- vectored undetermin a is )( here (5.14), ofsolution a is(5.24) )()()( Let ttttcc, ,)()()(010battdssstttfc(5.24)(5.26) , ,)()()()(01

19、0battdsssttttf(5.26). formula by the determined isit then (5.24), form theof (t)solution a has (5.14) if Therefore, 1919(5.14) ),()(ttfxAx(5.15) ,)(xAxt (5.14). ofnot (5.15), ofsolution a is )()(then , (5.15) ofmatrix solution l fundamenta a is )( Suppose cttt funnction. valued- vectored undetermin

20、a is )( here (5.14), ofsolution a is(5.24) )()()( Let ttttcc(5.26) , ,)()()()(010battdsssttttf(5.26). formula by the determined isit then (5.24), form theof (t)solution a has (5.14) if Therefore, (5.14). ofsolution a bemust (5.26)formular by the determined )(function valued- vector the,Conversely t2

21、020Theorem 8C/P212function valued-vector then , (5.15) ofmatrix solution l fundamenta a is )( If t(5.14) ),()(ttfxAx(5.15) ,)(xAxt.)(condition initial thesatisfies and (5.14), ofsolution a is00tRemarkC/P212is )(condition initial thesatisfying (5.14) ofsolution The 0t(5.27) ,)()()()()()(0101ttdsssttt

22、tf.)(condition initial satisfying (5.15) ofsolution a is )()()( here001tttthh(5.26) , ,)()()()(010battdsssttttf2121(5.14) ),()(ttfxAx(5.15) ,)(xAxt(5.27) ,)()()()()()(0101ttdsssttttf(5.26) , ,)()()()(010battdsssttttf.)(condition initial thesatisfies and (5.14), ofsolution a is00t.)(condition initial

23、 satisfing (5.15) ofsolution a is )()()( here001tttthh.DEs of (5.14) systemlinear eousnonhomogen for the )( be tosaidboth are (5.27) and (5.26) Formulas 常数变易公式常数变易公式constants of variation of formulas the (5.27) and (5.26) Formulas in the 0 keusually ta wegeneral,In 0t2222Example 5.2.2 C/P213problem

24、valueinitial theofsolution particular thefind Try to . 11)0( ,01011 xxxte8DEs.C/P20 of systemlinear shomogeneou associated theofmatrixsolution lfundamenta a is 0)( known that isIt ttteteetSolution:(5.27) ,)()()()()()(0101ttdsssttttf2323Corollary 3 C/P214(the formula of variation of constants for nth

25、 order nonhomogeneous linear DE cf. C/PP127-129 (n n阶非齐次线性微分方程的常数变易公式阶非齐次线性微分方程的常数变易公式)(5.28) )()()()(1)1(1)(tfxtaxtaxtaxnnnn(5.21) 0)()()(1)1(1)(xtaxtaxtaxnnnn.,on continuous all are )( and )(),(,),( that Suppose11batftatatann(5.21), of solutions of system lfundamenta a is )(,),(),( If21txtxtxn:con

26、ditons initial thesatisfying (5.28) ofsolution particular then the, , 0)(, 0)( , 0)(00)1(00battttnisnkttnnkkdssfsxsxWsxsxWtxt1110(5.29) .)()(,),()(,),()()();(,),( of Wronkian theis )(,),( Here11sxsxsxsxWnn.1 , 0 , 0 , 0by dsubstitude is )(,),(icolumn th thefrom derivedt determinan theis )(,),(11Tnnk

27、sxsxWksxsxW2424Example 5.2.3 C/P215;E/P155equation theofsolution particular a find Try to .tan txxSolutionnkttnnkkdssfsxsxWsxsxWtxt1110(5.29) .)()(,),()(,),()()( hence and ,sin)( and cos)( is 0 DE shomogeneouassociated theof solutions of systems lfundamenta a Obviously, 21ttxttxxx. 1cossinsincos)(),

28、(21tttttxtxWis DEgiven theofsolution particular desired a Therefore,tdssfsxsxWsxtxsxtxt0212112)()(),()()()()()(2525Example 5.2.3 C/P215;E/P155equation theofsolution particular a find Try to .tan txxSolutionnkttnnkkdssfsxsxWsxsxWtxt1110(5.29) .)()(,),()(,),()()(tdssfsxsxWsxtxsxtxt0212112)()(),()()()(

29、)()(tsdsstst0tan1sincoscossinttsdsstdsssst00tansincoscossincossintttttttanseclnsincoscos1sintttttanseclncossinDE, associated theofsolution a is sin AstDE.given theofsolution a also is tanseclncos)(tttt262626Question:From the relationship between the high-order DE and the system of Des in C/P191, do

30、you understand the equations (4.17) in C/PP128-129 which we took? Example 5.2.3 C/P215;E/P155equation theofsolution particular a find Try to .tan txxnkttnnkkdssfsxsxWsxsxWtxt1110(5.29) .)()(,),()(,),()()(Linear Systems of Differential Equations(线性微分方程组线性微分方程组)Chapter V (第五章第五章)5.1 Existence and Uniq

31、ueness Theorem (存在唯一性定理存在唯一性定理)5.2 General Theory of Linear Systems of Differential Equations (线性微分方程组的一般理论线性微分方程组的一般理论)5.3 Linear Systems of Differential Equations with constant coefficients(常系数线性微分方程组常系数线性微分方程组)27Linear Systems of Differential Equations(线性微分方程组线性微分方程组)Chapter V (第五章第五章)5.1 Existen

32、ce and Uniqueness Theorem (存在唯一性定理存在唯一性定理)5.2 General Theory of Linear Systems of Differential Equations (线性微分方程组的一般理论线性微分方程组的一般理论)5.3 Linear Systems of Differential Equations with constant coefficients(常系数线性微分方程组常系数线性微分方程组)285.3.1 Definition and properties of exponential matrix exp A (矩阵指数矩阵指数 exp

33、A 的定义和性质的定义和性质)5.3.2 Computational formula of fundamental solution matrix (基解矩阵的计算公式基解矩阵的计算公式)5.3 Linear Systems of Differential Equations with constant coefficients (常系数线性微分方程组常系数线性微分方程组)295.3.1 Definition and properties of exponential matrix exp A (矩阵指数矩阵指数 exp A 的定义和性质的定义和性质)5.3.2 Computational f

34、ormula of fundamental solution matrix (基解矩阵的计算公式基解矩阵的计算公式)5.3 Linear Systems of Differential Equations with constant coefficients (常系数线性微分方程组常系数线性微分方程组)30 series lexponentia theof meansby defined bemay number complex theof lexponentia The zez.! 3! 21exp32nzzzzzenz series by the definedmatrix theis e

35、xp: )( then thematrix, an is if Similarly,nnennAmatrix lexponentiaAA矩矩阵阵指指数数(5.34) ! 3! 2exp:032kkmkmeAAAAAEAA. 1 if y,inductivel and , matrix,identity theis where10mnnmmAAAEAEE/P402 C/P219; exp 1.Amatrix lexponentia of Definition5.3.1 Definition and properties of exponential matrix exp A (矩阵指数矩阵指数

36、exp A 的定义和性质的定义和性质)这这个个定定义义合合理理吗吗? ? ? reasonable difinition thisIs:Question31(5.34) ! 3! 2exp032kkmkmeAAAAAEAAby given is (5.34)in right on the series infinite theof meaning The(*) !lim!00mkkmkkkkAA.exp clearly, And, .matrix squareevery for (5.34)by welldefined is expmatrix lexponentia theis,That E

37、/P403.220; matrixC/P square every for exists (*)in limit theshown that becan It E0AA0enn32Example 5.3.1E/P403matrix diagonal 22for matrix lexponentia theCalculate Ae.00baA.00exp :AnswerbaeeA33Solution:E/P403C/P222;matrix diagonal of lexponentiamatrix theGenerally,nn.,diag:00000002121nnaaaaaaDmatrix

38、diagonal theisnn.000000021naaaeeeeD34Exercise Cf E/P404/Example 3matrix triangularupper theof lexponentiamatrix thecalculate Try toAe.000100010A.100110! 2/111exp :AnswerA35Solution:Exercise Cf E/P404/Example 3matrix triangularupper theof lexponentiamatrix thecalculate Try toAe.000100010ARemarkE/P404

39、 series lexponentia then the,integer positive somefor 0 If nnA(5.34) ! 3! 2exp032kkmkmeAAAAAEAA . )( be tosaid ispower vanishingawith matrix aSuch exercise.in as calculatedreadily is matrix lexponentia theso terms,ofnumber finite aafter terminates幂零的nilpotentAe36Exercise Cf E/P404/Example 3matrix tr

40、iangularupper theof lexponentiamatrix thecalculate Try toAe.000100010AQuestionmatrixngular upper tria theof lexponentiamatrix thecalculate can we HowAe? 001001A37Amatrix lexponentia the of properties basic Someexp 2. ;exp (1)E0 then , i.e., e,commutativ are and matrices twoif (2)BAABBAnn ;expexpexpB

41、ABA, 0011kllkllkkkkkkkkkkCCCCBABBAABA0!1exp kkkBABA 00;!kkllkllklBA !expexp00mmmmmmBABA00!1kkllkllkCkBA 00;!kkllkllklBA ;expexpexpBABA383939ExerciseE/P405 matrixCf triangularupper theof lexponentiamatrix thecalculate Try toAe.001001A.000! 2/exp :AnswereeeeeeAAmatrix lexponentia the of properties bas

42、ic Someexp 2. ;exp (1)E0 then , i.e., e,commutativ are and matrices twoif (2)BAABBAnn ;expexpexpBABA404040 ?)exp( then ,0001000010001 SupposeAAkkAmatrix lexponentia the of properties basic Someexp 2. ;exp (1)E0 then , i.e., e,commutativ are and matrices twoif (2)BAABBAnn ;expexpexpBABAand ,exp ofmat

43、rix inverse theexists always there,matrix any for (3)AAnn ;expexp1AA then),0det (i.e.,matrix r nonsingula an is ,matrix an is that suppose (4)TTAnnnn ;expexpTATATT1141Amatrix lexponentia the of properties basic Someexp 2. ;exp (1)E0 then , i.e., e,commutativ are and matrices twoif (2)BAABBAnn ;expex

44、pexpBABAi.e., ,matrix diagonal-quasi an is that Suppose (5)nnA,diag:00000002121ssAAAAAAAthensAAAAexp0000exp000expexp21.exp,exp,expdiag:21sAAA42 solutions lexponentiaMatrix )( 3.矩矩阵阵指指数数解解(5.34) ! 3! 2exp032kkmkmeAAAAAEAAgives (5.34)in ofon substituti then iable,scalar var a is If ttA(5.35) ! ! ! 3!

45、2exp:0032kkkkkmtktktmttttteAAAAAAEAA.by ofelement each gmultiplyinby simply obtained is wherettAA./P220numbers)C real ofset (thein contained interval finiteevery on convergentuniformly is exp shown that becan isIt RAt43(5.33) Axx Theorem 9C/P221.)0( and (5.33), of a is exp)( function valued-matrix T

46、he Ematrix solution lfundamentaAtt44Proof:45(5.33) Axx Theorem 9C/P221.)0( and (5.33), of a is exp)( function valued-matrix The Ematrix solution lfundamentaAtt. (5.33) of C/P217/5 )( or )( thebe to)exp(: call We解矩阵解矩阵标准基标准基矩阵指数解矩阵指数解matrix solution lfundamenta standardSolution lExponentiaMatrix AAtetmatrix.r nonsingul