常微分方程第8章.ppt

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1、上页下页铃结束返回首页 8 systems of linear first-order differential equations1 preliminary theory4 Matrix exponential3 variation of parameters 2 homogeneous linear systems with constant coefficients上页下页铃结束返回首页上页下页铃结束返回首页8.1 preliminary theoryIn this chapter we are going to concentrate on only systems of linear

2、 first-order equations,and develop a general theory for these kinds of systems and,in the case of systems with constantcoefficients,a method of solution that utilizes some basic concepts from the algebra of matrices.上页下页铃结束返回首页We study systems of first-order differential equations that have the norm

3、al form上页下页铃结束返回首页A system such as(1)of n first-order equationsis called a first-order system.上页下页铃结束返回首页Linear systems When each of the functionsin(1)is linear in the dependent variableswe get the normal form of a first-order systemof linear equation:上页下页铃结束返回首页We refer to a system of form(2)simply

4、 as alinear system（线性方程组）.上页下页铃结束返回首页We assume that the coefficientsand thefunctionsare continuous on a commoninterval I.When,thelinear system is said to be homogeneous;otherwise it is nonhomogeneous.上页下页铃结束返回首页Matrix form of a linear systemIf X,A(t),and F(t)denote the respective matrices上页下页铃结束返回首页

5、Then the system of linear first-order differentialequations(2)can be written asor simplyIf the system ishomogeneous,its matrix form is then上页下页铃结束返回首页For example:(a)Ifthen(b)Ifthen上页下页铃结束返回首页Solution vectorA solution vector on an interval I is any columnmatrix上页下页铃结束返回首页Example 1Verify that on the i

6、ntervalare solutions ofWhose entries are differentiable functions satisfying the systemon the interval上页下页铃结束返回首页Solution:We see that:And上页下页铃结束返回首页Much of the theory of systems of n linearfirst-order differential equations is similar tothat of linear nth-order differential equations.Initial-value p

7、roblemletdenote a point on an interval I and上页下页铃结束返回首页Where theare given constants.Then the problemis an Initial-value problem on the interval.上页下页铃结束返回首页Theorem 8.1Existence of a Unique solutionLet the entries of the matrices A(t)and F(t)befunctions continuous on a common interval Ithat contains t

8、he point.Then there exists aunique solution of the initial-value problem(3)on the interval.上页下页铃结束返回首页Theorem 8.2 Superposition Principle Letbe a set of solution vectors of thehomogeneous systemon an interval I.Then the linear combinationWhere theare arbitrary constants,is also a solution on the int

9、erval.上页下页铃结束返回首页You should practice by verifying thata)the two vectorsare solutions of the systemAnd satisfy the initial-value上页下页铃结束返回首页b)is the solution ofsystem ofand satisfy the initial-valueWhereare arbitrary constants.上页下页铃结束返回首页Definition 8.2Linear Dependence/IndependenceLetbe a set of solut

10、ion vectors ofthe homogeneous systemon an intervalI.We say that the set is linearly dependenton the interval if there exist constantsnot all zero,such thatfor every t in the interval.If the set of vectors isnot linearly dependent on the interval,it issaid to be linearly independent.上页下页铃结束返回首页THEORE

11、M 8.3Criterion for Independent SolutionsLetbe n solution vectors of homogeneous systemon an interval I.Then the set of solutionvectors is linearly independent on I上页下页铃结束返回首页if and only if the Wronskianfor every t in the interval.上页下页铃结束返回首页For exampleWe saw that(in example 1)are solutions of system

12、上页下页铃结束返回首页1 neither vector is a constant multiple of the other.2 we haveFor all real values of t.For 1 or 2,we haveare linearlyindependent.上页下页铃结束返回首页Definition 8.3fundamental set of solutionsAny setof n linearly independentsolution vectors of the homogeneous systemon an interval I is said to be af

13、undamental set of solutions on the interval.上页下页铃结束返回首页Existence of a fundamental setThere exists a fundamental set of solutions forthe homogeneous systemon the interval I.上页下页铃结束返回首页Definition 8.5General Solution-Homogeneous SystemsLetbe a fundamental set of solutionsof homogeneous systemson aninte

14、rval I.Then the general solution of thesystem on the interval iswhere theare arbitrary constants上页下页铃结束返回首页Example 3The vectorsare solutions of the system上页下页铃结束返回首页Nowfor all real values of t.we conclude thatform a fundamental set ofsolutions on上页下页铃结束返回首页Thus the general solution of the system on

15、the interval is the linear combination THEOREM 8.6General Solution-Nonhomogeneous SystemsLetbe a given solution of the nonhomogeneoussystemon an interval I,and let 上页下页铃结束返回首页denote the general solution on the same intervalof the associated homogeneous systemThen the general solution of the nonhomog

16、eneoussystem on the interval isThe general solutionof the homogeneoussystemis called the complementaryfunction of the nonhomogeneous system上页下页铃结束返回首页For exampleThe nonhomogeneous systemWe saw that the general solution of the associatedhomogeneous system isAnd we can verify that the vector上页下页铃结束返回首

17、页Is a particular solution ofHence,Is the general solution of上页下页铃结束返回首页preview1 Matrix form of a linear systemFor example2 Linear independence and Wronskian3 Fundamental set of solutions上页下页铃结束返回首页8.2 homogeneous linear systems with constant coefficientsWe have known thatIs the general solution of t

18、he homogeneous system上页下页铃结束返回首页We guess that a solution of the formFor the general homogeneous linear first-ordersystemWhere A is anmatrix of constants.上页下页铃结束返回首页Eigenvalues（特征值）and Eigenvectors（特征向量）Ifis a solution vector of the linear system,then we have上页下页铃结束返回首页The matrix equation(3)is equiva

19、lent to thesimultaneous algebraic equationsThus to find a nontrival solution X of(2)we mustfind a nontrival solution of the foregoing system.上页下页铃结束返回首页Find a nontrival vectorthat satisfies(3),we must haveThis polynomial equation inis called thecharacteristic equation of the matrix A;itssolutions ar

20、e the eigenvalues of A.A solutionof(3)corresponding to aneigenvalueis called an eigenvector of A.a solution of the homogeneous system(2)is then上页下页铃结束返回首页In the discussion that follows,We examinethree cases:1,real and distinct eigenvalues2,repeated eigenvalues3,complex eigenvalues上页下页铃结束返回首页8.2.1 DI

21、STINCT REAL EIGENVALUESWhen the nxn matrix A possesses n distinct realeigenvaluesthen a set of n linearlyindependent eigenvectorscan alwaysbe found andis a fundamental set of solutions of(2)on上页下页铃结束返回首页上页下页铃结束返回首页THEOREM 8.7 General solution-homogeneous systemsLetbe n distinct real eigenvalues of t

22、he coefficient matrix A of the homogeneoussystem(2),and letbe thecorresponding eigenvectors.Then the generalsolution of(2)on the intervalis given by上页下页铃结束返回首页EXAMPLE 1 Distinct EigenvaluesSolve上页下页铃结束返回首页Solution we findand so the eigenvalues are上页下页铃结束返回首页ForGauss-Jordan elimination givesTherefore

23、The choicegives an eigenvector an corresponding solution vector上页下页铃结束返回首页Similarly,forImpliesChoosingwe get a second eigenvector and solution vector上页下页铃结束返回首页Finally whenthe augmented matricesyield上页下页铃结束返回首页The general solution of(6)is a linear combinationof the solution vectors in(7),(8),and(9):

24、上页下页铃结束返回首页8.2.2 REPEATED EIGENVALUESThe character equation of the coefficient matrixin the systemIs shown to beIs a root of multiplicity two.上页下页铃结束返回首页For this value we find the single eigenvectoris one solution of(10),but the second solution?In general:If m is a positive integer andis a factorof

25、the characteristic equation whileis not a factor,thenis said to be an eigenvalueof multiplicity m.The next three examples 上页下页铃结束返回首页illustrate the following cases:For some nxn matrices A it may be possibleto find m linearly independent eigenvectorscorresponding to an eiqenvalueof multiplicityIn thi

26、s case the general solution of of the systemcontains the linear combination上页下页铃结束返回首页If there is only one eigenvector correspondingto the eigenvalueof multiplicity m linearlyindependent solutions of the form上页下页铃结束返回首页EXAMPLE 3Repeated EigenvaluesSolve上页下页铃结束返回首页Solution Expanding the determinant i

27、n the characteristic equationYields上页下页铃结束返回首页ForGauss-Jordan eliminationimmediately givesThe first row of the last matrix meansThe choicesyield,上页下页铃结束返回首页in turn,Thus two eigenvectorscorresponding toareSince neither eigenvector is a constant multiple ofthe other,we have found two linearly independ

28、entsolutions,上页下页铃结束返回首页corresponding to the same eigenvalue.Last,forthe reductionImpliesPricking gives and thus a third eigenvector is上页下页铃结束返回首页We conclude that the general solution of the system is上页下页铃结束返回首页The matrix of coefficients A in Example 3 is aspecial kind of matrix known as a symmetric

29、 matrix.If its transposeis same as A(),anmatrix A is said to be symmetric（对称）.If the matrix A in the systemis symmetric and has real entries,then we canalways find n linearly independent eigenvectors上页下页铃结束返回首页Second SolutionNow suppose thatis an eigenvalue of multiplicitytwo and that there is only

30、one eigenvectorassociated with this value.A second solution canbe found of the formWhere上页下页铃结束返回首页To see this we substitute(12)into the systemand simplify:Since this last equation is to hold for all valued oft,we must have上页下页铃结束返回首页Equation(13)simply states that K must be aneigenvector of A associ

31、ated withBy solving(13),we find one solutionTo find the second solutionwe need only solvethe additional system(14)for the vector P.上页下页铃结束返回首页EXAMPLE 4 Repeated EigenvaluesFind general solution of the system given in(10).SolutionFrom(11)we know thatand that one solution isIdentifying上页下页铃结束返回首页we fi

32、nd from(14)that we must now solveWe can chooseso thatHence,Thus,The general solution is上页下页铃结束返回首页Eigenvalue of multiplicity threeWhen the coefficient matrix A has only oneeigenvector associated with an eigenvalueof multiplicity three,we can find a second solutionof the form(12)and a third solution

33、of the form上页下页铃结束返回首页By substituting(15)into the systemwe find that the column vectors K,P,Q must satisfy上页下页铃结束返回首页Example 5 SolveSolution the characteristic equationshows thatis an eigenvector of multiplicity three.By solvingWe find the single eigenvector上页下页铃结束返回首页We next solve the systemsIn suc

34、cession and find thatWe see that the general solution of the system is上页下页铃结束返回首页8.2.3 complex eigenvaluesThe character equation of the systemis上页下页铃结束返回首页From the quadratic formula,haveFor,we solveSince,the choicethen corresponding solution vector:上页下页铃结束返回首页In like manner,for,we haveWe can verify

35、by means of the Wronskian thatthese solution vectors are linearly independent,the general solution is上页下页铃结束返回首页Note:上页下页铃结束返回首页THEOREM 8.8 Solutions Corresponding to a Complex EigenvalueLet A be the coefficient matrix having real entriesof the homogeneous system(2),an letbe aneigenvector correspond

36、ing to the complexeigenvaluereal.Thenare solutions of(2).上页下页铃结束返回首页It is desirable to rewrite a solution in terms ofreal functions.Use Eulers formula to writeAfter multiply complex numbers,collect terms,(20)becomes上页下页铃结束返回首页WhereAndThe two vectors and are themselves linearlyindependent real soluti

37、ons of the original system,hence the linear combination(21)is an alternativegeneral solution.上页下页铃结束返回首页Letbe an eigenvector of the coefficient matrixA corresponding to the complex eigenvalue,then上页下页铃结束返回首页By superposition principle,the following vectors are also solutions:上页下页铃结束返回首页Bothare real n

38、umbers forany complex numberTherefore,the entriesin the column vectors are real numbers.By definingwe are led to the following theorem.上页下页铃结束返回首页THEOREM 8.9 Real Solutions Corresponding to a Complex EigenvalueLetbe a complex eigenvalue of the coefficientmatrix A in the homogeneous system(2),and let

39、denote the column vectors defined in(22).Thenare linearly independent solutions of(2)on上页下页铃结束返回首页The matricesare often denoted byFor example上页下页铃结束返回首页EXAMPLE 6 Complex EigenvaluesSolve the initial-value problemSolution First we obtain the eigenvalues from上页下页铃结束返回首页The eigenvalues areForthe system

40、givesBy choosingwe get上页下页铃结束返回首页Now from(24)we formSinceit follows from(23)that the general solution of the system is上页下页铃结束返回首页8.3 VARIATION OF PARAMETERSBefore developing a matrix version of parameters fornonhomogeneous linear systemwe need to examine a special matrix that is formedout of the sol

41、ution vectors of the correspondinghomogeneous system上页下页铃结束返回首页A Fundamental matrix Ifis a fundamental set of solutionsof the homogeneous systemthen its general solution on the interval ison an interval I,上页下页铃结束返回首页or上页下页铃结束返回首页Where C iscolumn vector of arbitrary constantsand thematrix,whose colum

42、ns consistof the entries of the solution vectors of the systemis called a fundamental matrix（基本矩阵）（基本矩阵）of the system on the interval.上页下页铃结束返回首页Two properties of a fundamental matrix:A fundamental matrixis nonsingular(不可约).The linear independence of the columns ofon the interval I.is nonsingularThe

43、 multiplicative inverseexists for every t in the interval.上页下页铃结束返回首页 If is a fundamental matrix of the system thenVariation of parameters Replace the matrix of constants C in(1)by a column matrix of functions上页下页铃结束返回首页is a particular solution of the nonhomogeneous systemBy the product rule,haveSub

44、stituting(4)and(6)into(5)gives 上页下页铃结束返回首页We have上页下页铃结束返回首页Example 1 Find the general solution of the nonhomogeneous systemOn the interval .Solution we first solve the homogeneous system 上页下页铃结束返回首页The characteristic equation of the coefficient matrix isThe eigenvalues corresponding to and are,resp

45、ectively,上页下页铃结束返回首页The solution vectors of the system are then hence上页下页铃结束返回首页From the formula we obtain 上页下页铃结束返回首页Hence,the general solution is上页下页铃结束返回首页Initial-value problemThe general solution of(5)can be written asThe form is useful in solving(5)subject to an initial condition .上页下页铃结束返回首页8.

46、4 matrix exponential The simple linear first-order differential equation ,where is a constant,has the general solution The matrix exponential is a solution of the system上页下页铃结束返回首页Homogeneous systems We define a matrix exponential so that is solution of the homogeneous system Here A is an matrix of

47、constants,and C is an column matrix of arbitrary constants.上页下页铃结束返回首页The power series(幂级数)representation of the scalar exponential functionThe series in(2)converges for all t.上页下页铃结束返回首页DEFINITION 8.4 Matrix Exponential For any matrix A,The series given in(3)converges to an matrix for value of t.and 上页下页铃结束返回首页Derivative of We have known and we can justify 上页下页铃结束返回首页We can getHence,is a solution of for every vector C of constants.