Nagle R.K., Saff E.B., Snider A.D. Fundamentals of differential ...

or in matrix notation

!Dx1x2x3x4

T œ

! D 0 1 0 0"3 0 1 0

0 0 0 12 0 "2 0

T Dx1x2x3x4

T .502 Chapter 9 Matrix Methods for Linear Systems

9.1 EXERCISES

In Problems 1–6, express the given system of differentialequations in matrix notation.

1.

2.

3.

4.

5.

6.

x¿3 ! x1 " x2

x ¿2 ! Asin 2tBx2 , x ¿1 ! Acos 2tBx1 , y¿ ! Acos tBx # Aa # bt3By x¿ ! Asin tBx # ety , x¿4 ! 0 x ¿3 ! 1px1 " x3 , x ¿2 ! x1 # x4 , x ¿1 ! x1 " x2 # x3 " x4 , z¿ ! 4y y¿ ! 2z " x , x¿ ! x # y # z , y¿ ! "x x¿ ! y , y¿ ! 3x " 2y x¿ ! 7x # 2y ,

In Problems 7–10, express the given higher-order differ-ential equation as a matrix system in normal form.

7. The damped mass–spring oscillator equation

8. Legendre’s equation 9. The Airy equation

10. Bessel’s equation

In Problems 11–13, express the given system of higher-order differential equations as a matrix system in normalform.11.

12.

13.y‡ # y– " tx¿ # y¿ # etx ! 0x– " 3x¿ # t2y " Acos tBx ! 0 ,

y– # x¿ # 3y¿ # y ! 0x– # 3x¿ " y¿ # 2y ! 0 ,

y– " 2x ! 0x– # 3x # 2y ! 0 ,

y– #1t

y¿ # a1 "n2

t2b y ! 0

y– " ty ! 0

A1 " t2By– " 2ty¿ # 2y ! 0my– # by¿ # ky ! 0

9.2 REVIEW 1: LINEAR ALGEBRAIC EQUATIONSHere and in the next section we review some basic facts concerning linear algebraic systemsand matrix algebra that will be useful in solving linear systems of differential equations innormal form. Readers competent in these areas may proceed to Section 9.4.

A set of equations of the form

o

(where the ’s and ’s are given constants) is called a linear system of n algebraic equationsin the n unknowns The procedure for solving the system using eliminationmethods is well known. Herein we describe a particularly convenient implementation of the

x1, x2, . . . , xn.biaij

an1x1 # an2x2 # p # annxn ! bn

a21x1 # a22x2 # p # a2nxn ! b2 ,

a11x1 # a12x2 # p # a1nxn ! b1 ,

There are no constraints on either or ; thus we take them to be free variables and charac-terize the solutions by

!

In closing, we note that if the execution of the Gauss-Jordan algorithm results in a displayof the form 0 ! 1 (or where ), the original system has no solutions; it isinconsistent. This is explored in Problem 12.

k " 00 ! k,

x1 ! 2 # s $ 2t , x2 ! s , x3 ! t , x4 ! $1, $q 6 s, t 6 q .

x3x2

506 Chapter 9 Matrix Methods for Linear Systems

In Problems 1–11, find all solutions to the system usingthe Gauss–Jordan elimination algorithm.

1.

2.

3.

4.

5.

6.

7.

8.

9.

$x1$ A1# iB x2 ! 0

A1$ iBx1 # 2x2 ! 0 ,

x1 # 3x2 $ 2x3 ! 3 2x1 # 4x2 $ x3 ! 0 ,

x1 # 2x2 # x3 ! $3 ,

$3x1 # 9x2 ! 0 $x1 # 3x2 ! 0 ,

4x1 $ 4x2 # 2x3 ! 0 x1 $ 3x2 # x3 ! 0 ,

$2x1 # 2x2 $ x3 ! 0 ,

2x1 # 3x2 ! 0 $x1 # 2x2 ! 0 ,

2x1 $ x2 # x3 # x4 ! 0 2x1 $ x2 # x3 # 2x4 ! 0 ,

x1 # x2 # x3 # x4 ! 1 , x3 # x4 ! 0 ,

x1 # x2 $ x3 ! 0 $x1 $ x2 # x3 ! 0 ,

x1 # x2 $ x3 ! 0 ,

x1 # 2x2 $ x3 # x4 ! 0 2x1 # 2x2 $ x3 # x4 ! 0 ,

x1 # # x4 ! 0 , x1 # x2 # x3 # x4 ! 1 ,

x1 # x2 # 3x3 ! 6 2x1 # x2 # x3 ! 6 ,

x1 # 2x2 # 2x3 ! 6 ,

10.

11.

12. Use the Gauss–Jordan elimination algorithm toshow that the following systems of equations areinconsistent. That is, demonstrate that the exis-tence of a solution would imply a mathematicalcontradiction.(a)

(b)

13. Use the Gauss–Jordan elimination algorithm toshow that the following system of equations has aunique solution for r ! 2, but an infinite number ofsolutions for r ! 1.

14. Use the Gauss–Jordan elimination algorithm toshow that the following system of equations has aunique solution for but an infinite numberof solutions for r ! 2.

4x1 $ 4x2 # 5x3 ! rx3

x1 # x3 ! rx2 , x1 # 2x2 $ x3 ! rx1 ,

r ! $1,

x1 $ 2x2 ! rx2

2x1 $ 3x2 ! rx1 ,

$x1 # x2 # 5x3 ! 1 $3x1 # x2 # 4x3 ! 1 ,

2x1 # x3 ! $1 ,

$6x1 # 3x2 ! 4 2x1 $ x2 ! 2 ,

$x1 # x2 # 5x3 ! 0 $3x1 # x2 # 4x3 ! 1 ,

2x1 # x3 ! $1 , x1 # 2x2 # x3 ! i

2x1 # 3x2 $ ix3 ! 0 , x1 # x2 # x3 ! i ,

9.2 EXERCISES

Show that is a solution of the matrix differential equation , where

We simply verify that and are the same vector function:

!

The basic properties of differentiation are valid for matrix functions.

x¿ AtB ! c"v sin vtv cos vt

d ; Ax ! c 0 "vv 0

d c cos vtsin vt

d ! c"v sin vtv cos vt

d .Ax AtBx¿ AtB

A ! c 0 "vv 0

d .x¿ ! Axx AtB ! c cos vt

sin vtd

Section 9.3 Review 2: Matrices and Vectors 515

Example 3

Solution

Differentiation Formulas for Matrix Functions

ddt

AABB ! A

dBdt

#dAdt

B .

ddt

AA # BB !dAdt

#dBdt

.

ddt

ACAB ! C

dAdt

(C a constant matrix) .

In the last formula, the order in which the matrices are written is very important because,as we have emphasized, matrix multiplication does not always commute.

9.3 EXERCISES

1. Let and

Find: (a) (b)

2. Let and

Find: (a) (b)

3. Let and

Find: (a) (b) (c)

4. Let and

Find: (a) (b) BA .AB .

B J c1 1 "10 3 1

d .A J £ 2 10 4

"1 3§

B2 ! BB .A2 ! AA .AB .

B J c"1 35 2

d .A J c 2 41 1

d 7A " 4B .A # B .

B J c1 "1 20 3 "2

d .A J c 2 0 52 1 1

d 3A " B .A # B .

B J c"1 02 "3

d .A J c 2 13 5

d 5. Let and

.

Find: (a) (b) (c)

6. Let and

Find: (a) (b) (c)7. (a) Show that if u and v are each column vec-

tors, then the matrix product is the same asthe dot product u # v.

uTvn $ 1

AA # BBC .AABBC .AB .

C J c1 "41 1

d .

A J c 1 21 1

d , B J c 0 31 2

d , A AB # CB .AC .AB .

C J c"1 12 1

dA J c 1 "2

2 "3d , B J c 1 0

1 1d ,

(b) Let v be a column vector withShow that, for A as given in

Example 1,(c) Does hold for every

matrix A and vector (d) Does hold for every pair of

matrices A, B such that both matrix products aredefined? Justify your answer.

8. Let and

Verify that

In Problems 9–14, compute the inverse of the givenmatrix, if it exists.

9. 10.

11. 12.

13. 14.

15. Prove that if satisfies then every solu-tion to the nonhomogeneous system Ax ! b is of theform , where xh is a solution to thecorresponding homogeneous system Ax ! 0.

16. Let

(a) Show that A is singular.

(b) Show that has no solutions.

(c) Show that has infinitely many

solutions.

In Problems 17–20, find the matrix function whose value at t is the inverse of the given matrix

17.

18. X AtB ! c sin 2t cos 2t2 cos 2t "2 sin 2t

dX AtB ! c et e4t

et 4e4t dX AtB.X"1 AtB

Ax ! £ 303§

Ax ! £ 313§

A ! £ 2 "1 1"1 2 1

1 1 2§ .

x ! xp # xh

Axp ! b,xp

£ 1 1 11 "1 21 1 4

§£"2 "1 12 1 03 1 "1

§£ 1 1 1

1 2 30 1 1

§£ 1 1 11 2 12 3 2

§c 4 15 9

dc 2 1"1 4

dAB $ BA.

B J c1 23 2

d .A J c 2 "1"3 4

dAABBT ! BTAT

v?n % 1m % nAAvBT ! vTAT

AAvBT ! vTAT.vT ! 3 2 3 5 4 .3 % 1


19.

20.

In Problems 21–26, evaluate the given determinant.

21. 22.

23. 24.

25. 26.

In Problems 27–29, determine the values of r for whichdet

27. 28.

29.

30. Illustrate the equivalence of the assertions (a)–(d) inTheorem 1 (page 513) for the matrix

as follows.(a) Show that the row-reduction procedure applied

to fails to produce the inverse of A.(b) Calculate det A.(c) Determine a nontrivial solution x to Ax ! 0.(d) Find scalars and not all zero, so that

where and are the columns of A.

In Problems 31 and 32, find for the given vectorfunctions.

31. 32. x AtB ! £ e"t sin 3t0

"e"t sin 3t§x AtB ! £ e3t

2e3t

"e3t§

dx /dt

a3a1, a2,c1a1 # c2a2 # c3a3 ! 0,c3,c1, c2,

3A!I 4A ! £ 4 "2 2

"2 4 22 2 4

§

A ! £ 0 0 00 1 01 0 1

§A ! c 3 3

2 4dA ! c 1 1

"2 4dAA " rIB ! 0.

† 1 4 43 0 "31 6 2

†† 1 4 3"1 "1 2

4 5 2†

† 1 0 20 3 "1

"1 2 1†† 1 0 0

3 1 21 5 "2

†` 12 8

3 2`` 4 3

"1 2`

X AtB ! £ e3t 1 t3e3t 0 19e3t 0 0

§X AtB ! £ et e"t e2t

et "e"t 2e2t

et e"t 4e2t§

In Problems 33 and 34, find for the given matrixfunctions.

33.

34.

In Problems 35 and 36, verify that the given vectorfunction satisfies the given system.

35.

36.

In Problems 37 and 38, verify that the given matrixfunction satisfies the given matrix differential equation.

37.

38.

X AtB ! £ et 0 00 et e5t

0 et "e5t§

X¿ ! £ 1 0 00 3 "20 "2 3

§ X ,

X¿ ! c 1 "12 4

d X , X AtB ! c e2t e3t

"e2t "2e3t d

x¿ ! £ 0 0 00 1 01 0 1

§ x , x AtB ! £ 0et

"3et§

x¿ ! c 1 1"2 4

d x , x AtB ! c e3t

2e3t d

X AtB ! £ sin 2t cos 2t e"2t

"sin 2t 2 cos 2t 3e"2t

3 sin 2t cos 2t e"2t§

X AtB ! c e5t 3e2t

"2e5t "e2t ddX /dt

Section 9.4 Linear Systems in Normal Form 517

In Problems 39 and 40, the matrices aregiven. Find

(a) (b) (c)

39.

40.

41. An matrix A is called symmetric if that is, if for all Show thatif A is an matrix, then is a symmetricmatrix.

42. Let A be an matrix. Show that is a sym-metric matrix and is a symmetric

matrix (see Problem 41).43. The inner product of two vectors is a generalization

of the dot product, for vectors with complex entries.It is defined by

where

, y ! col Ay1, y2, . . . , ynBx ! col Ax1, x2, . . . , xnBAx, yB J an

i!1 xiyi ,

m # mAATn # n

ATAm # n

A $ ATn # ni, j ! 1, . . . , n.aij ! aji,

AT ! A;n # n

A AtB ! c 1 e"2t

3 e"2t d , B AtB ! c e"t e"t

"e"t 3e"t dB AtB ! c cos t "sin t

sin t cos tdA AtB ! c t et

1 et d ,ddt

3A AtBB AtB 4 .!1

0 B AtB dt .! A AtB dt .

A AtB and B AtB

9.4 LINEAR SYSTEMS IN NORMAL FORMIn keeping with the introduction presented in Section 9.1, we say that a system of n lineardifferential equations is in normal form if it is expressed as

(1)

where col , col and is anmatrix. As with a scalar linear differential equation, a system is called homogeneous

when otherwise, it is called nonhomogeneous. When the elements of A are allf AtB " 0;n # n

A AtB ! 3aij AtB 4A f1 AtB, . . . , fn AtBB ,f AtB !Ax1 AtB, . . . , xn AtBBx AtB !

x¿ AtB ! A AtBx AtB $ f AtB ,

are complex vectors and the overbar denotescomplex conjugation.

(a) Show that , wherecol

(b) Prove that for any vectors x, y, z and anycomplex number we have

Ax, lyB ! l Ax, yB .Alx, yB ! l Ax, y) , Ax, y ! zB ! Ax, yB $ Ax, zB ,Ax, y) ! Ay, xB ,

l,n # 1

Ay1, y2, . . . , ynB .y !Ax, yB ! xTy

The proof of this theorem is almost identical to the proofs of Theorem 4 in Section 4.5 andTheorem 4 in Section 6.1. We leave the proof as an exercise.

The linear combination of in (10) written with arbitrary constants is called a general solution of (9). This general solution can also be expressed as

where X is a fundamental matrix for the homogeneous system and c is anarbitrary constant vector.

We now summarize the results of this section as they apply to the problem of finding ageneral solution to a system of n linear first-order differential equations in normal form.

x ! xp " Xc,

c1, . . . , cnxp, x1, . . . , xn

Section 9.4 Linear Systems in Normal Form 523

We devote the rest of this chapter to methods for finding fundamental solution sets for homo-geneous systems and particular solutions for nonhomogeneous systems.

In Problems 1–4, write the given system in the matrixform

1.

2.

3. 4.

dzdt

! x " 5zdzdt

! tx # y " 3z # et

dydt

! 2x # y " 3z ,dydt

! etz " 5 ,

dxdt

! x " y " z ,dxdt

! t2x # y # z " t ,

u¿ AtB ! r AtB # u AtB " 1

r¿ AtB ! 2r AtB " sin t ,

y¿ AtB ! #x AtB " 2y AtB " et

x¿ AtB ! 3x AtB # y AtB " t2 ,

x¿ ! Ax " f.

Approach to Solving Normal Systems1. To determine a general solution to the homogeneous system :

(a) Find a fundamental solution set that consists of n linearlyindependent solutions to the homogeneous system.

(b) Form the linear combination

where col is any constant vector and is thefundamental matrix, to obtain a general solution.

2. To determine a general solution to the nonhomogeneous system :

(a) Find a particular solution to the nonhomogeneous system.(b) Form the sum of the particular solution and the general solution

to the corresponding homogeneous system in part 1,

to obtain a general solution to the given system.

x ! xp " Xc ! xp " c1x1 " . . . " cnxn ,

c1x1 " . . . " cnxn

Xc !xp

x¿ ! Ax " f

X ! 3x1 . . . xn 4Ac1, . . . , cnBc !

x ! Xc ! c1x1 " . . . " cnxn ,

Ex1, . . . , xnF x¿ ! Axn $ n

9.4 EXERCISES

In Problems 5–8, rewrite the given scalar equation as afirst-order system in normal form. Express the system inthe matrix form

5.

6. 7.

8.

In Problems 9–12, write the given system as a set ofscalar equations.

9. x¿ ! c 5 0#2 4

d x " e#2t c 2#3d

d3y

dt3 #dydt

" y ! cos t

d4wdt4 " w ! t2x– AtB " x AtB ! t2

y– AtB # 3y¿ AtB # 10y AtB ! sin tx¿ ! Ax " f.

10.

11.

12.

In Problems 13–19, determine whether the given vectorfunctions are linearly dependent or linearly inde-pendent on the interval

13. 14.

15. 16.

17.

18.

19.

20. Let

.

(a) Compute the Wronskian.(b) Are these vector functions linearly independent

on ?(c) Is there a homogeneous linear system for which

these functions are solutions?

In Problems 21–24, the given vector functions are solu-tions to the system Determine whetherthey form a fundamental solution set. If they do, find afundamental matrix for the system and give a generalsolution.

21. x1 ! e2t c 1"2d , x2 ! e2t c"2

4d

x¿ AtB ! Ax AtB.

("q, q)

x1 ! £ cos t00

§ , x2 ! £sin tcos tcos t

§ , x3 ! £ cos tsin tcos t

§

£ 101§ , £ t

0t§ , £ t2

0t2§

c sin tcos t

d , c sin tsin td , c cos t

cos td

e2t £ 105§ , e2t £ 1

1"1§ , e3t £ 01

0§

c sin tcos t

d , c sin 2tcos 2t

det c 15d , et c "3

"15d

c te"t

e"t d , c e"t

e"t dc t3d , c 4

1d A"q, q B.ALIB ALDB

x¿ ! £ 0 1 00 0 1

"1 1 2§ x # t £ 1

"12§ # £ 31

0§

x¿ ! £ 1 0 1"1 2 5

0 5 1§ x # et £ 10

0§ # t £ 01

0§

x¿ ! c 2 1"1 3

d x # et c t1d


22.

23.

24.

25. Verify that the vector functions

and

are solutions to the homogeneous system

on and that

is a particular solution to the nonhomogeneoussystem where colFind a general solution to

26. Verify that the vector functions

are solutions to the homogeneous system

on and that

is a particular solution to wherecol Find a general solution to

x¿ ! Ax # f AtB.A"9t, 0, "18tB.f AtB !x¿ ! Ax # f AtB,

xp ! £ 5t # 12t

4t # 2§

A"q, q B,x¿ ! Ax ! £ 1 "2 2

"2 1 22 2 1

§ xx3 ! £"e"3t

"e"3t

e"3t§x1 ! £ e3t

0e3t§ , x2 ! £"e3t

e3t

0

§ ,

x¿ ! Ax # f AtB. Aet, tB.f AtB !x¿ ! Ax # f AtB,xp !

32

c tet

tet d "14

c et

3et d # c t2td " c 0

1dA"q, q B,

x¿ ! Ax ! c 2 "13 "2

d x ,

x2 ! c e"t

3e"t dx1 ! c et

et dx3 ! £"cos t

sin tcos t

§x1 ! £ et

et

et§ , x2 ! £ sin t

cos t"sin t

§ ,

x3 ! £ e3t

"e3t

2e3t§x1 ! £ e"t

2e"t

e"t§ , x2 ! £ et

0et§ ,

x1 ! e"t c 32d , x2 ! e4t c 1

"1d

27. Prove that the operator defined by J where is an matrix function and x is an

differentiable vector function, is a linear oper-ator.

28. Let be a fundamental matrix for the systemShow that is the

solution to the initial value problem ,

In Problems 29–30, verify that is a fundamentalmatrix for the given system and compute Use theresult of Problem 28 to find the solution to the giveninitial value problem.

29.

30.

31. Show that

on but that the two vector functions

are linearly independent on 32. Abel’s Formula. If are any n solutions

to the system then Abel’sformula gives a representation for the Wronskian

Namely,

where are the main diagonal ele-ments of Prove this formula in the special casewhen n ! 3. [Hint: Follow the outline in Problem 30of Exercises 6.1.]

33. Using Abel’s formula (Problem 32), prove that theWronskian of n solutions to on the intervalI is either identically zero on I or never zero on I.

34. Prove that a fundamental solution set for the homo-geneous system always exists on anx¿ AtB ! A AtBx AtB

x¿ ! Ax

A AsB.a11 AsB, . . . , ann AsBW AtB ! W At0Bexp a! t

t0 Ea11 AsB " p " ann AsB F dsb ,

W AtB J W 3 x1, . . . , xn 4 AtB.x¿ AtB ! A AtBx AtB,n " nx1, . . . , xn

A#q, q B.c t2

2td , c t 0 t 0

2 0 t 0 dA#q, q B,` t2 t 0 t 0

2t 2 0 t 0 ` " 0

X AtB ! c e#t e5t

#e#t e5t dx¿ ! c 2 3

3 2d x , x A0B ! c 3

#1d ;

X AtB ! £ 6e#t #3e#2t 2e3t

#e#t e#2t e3t

#5e#t e#2t e3t§

x¿ ! £ 0 6 01 0 11 1 0

§ x , x A0B ! £#101§ ;

X#1 AtB.X AtBx At0B ! x0.x¿ ! Ax

x AtB ! X AtBX#1 At0Bx0x¿ ! Ax.X AtB

n " 1n " nA

x¿ # Ax,L 3 x 4Section 9.4 Linear Systems in Normal Form 525

interval I, provided is continuous on I. Hint:Use the existence and uniqueness theorem (Theorem 2)and make judicious choices for

35. Prove Theorem 3 on the representation of solutionsof the homogeneous system.

36. Prove Theorem 4 on the representation of solutionsof the nonhomogeneous system.

37. To illustrate the connection between a higher-orderequation and the equivalent first-order system, con-sider the equation

(11)

(a) Show that is a fundamental solutionset for (11).

(b) Using the definition of Section 6.1, compute theWronskian of

(c) Setting show thatequation (11) is equivalent to the first-ordersystem

(12)

where

(d) The substitution used in part (c) suggests that

is a fundamental solution set for system (12).Verify that this is the case.

(e) Compute the Wronskian of S. How does itcompare with the Wronskian computed in part (b)?

38. Define x1 , x2 , and x3 , for , by

(a) Show that for the three scalar functions in eachindividual row there are nontrivial linear combi-nations that sum to zero for all t.

(b) Show that, nonetheless, the three vector functionsare linearly independent. (No single nontrivialcombination works for each row, for all t.)

(c) Calculate the Wronskian .(d) Is there a linear third-order homogeneous differen-

tial equation system having as solutions?

x1 AtB, x2 AtB, and x3 AtBW 3x1, x2, x3 4 AtB

x1 AtB ! £sin tsin t0§ , x2 AtB ! £sin t

0sin t§ , x3 AtB ! £0sin t

sin t§ .

#q 6 t 6 qAtBAtBAtB

S J # £ et

et

et§ , £ e2t

2e2t

4e2t§ , £ e3t

3e3t

9e3t§ $

A J £ 0 1 00 0 16 #11 6

§ .x# ! Ax ,

x1 ! y, x2 ! y¿, x3 ! y–,Eet, e2t, e3tF.

Eet, e2t, e3tFy$ AtB % 6y& AtB " 11y# AtB % 6y AtB ! 0 .

x0. 4 3A AtB

or AU ! UD, where U is the matrix whose column vectors are eigenvectors and D is a diago-nal matrix whose diagonal entries are the eigenvalues. Since U’s columns are independent, U isinvertible and we can write

(24) A ! UDU"1 or D ! U"1AU ,

and we say that A is diagonalizable. [In this context equation (24) expresses a similarity trans-formation.] Because the argument that leads from (2) to (23) to (24) can be reversed, we have anew characterization: An matrix has n linearly independent eigenvectors if, and only if,it is diagonalizable.

n # n


9.5 EXERCISES

In Problems 1–8, find the eigenvalues and eigenvectorsof the given matrix.

1. 2.

3. 4.

5. 6.

7. 8.

In Problems 9 and 10, some of the eigenvalues of thegiven matrix are complex. Find all the eigenvalues andeigenvectors.

9. 10.

In Problems 11–16, find a general solution of the systemfor the given matrix A.

11. 12.

13. A ! £ 1 2 22 0 32 3 0

§A ! c 1 3

12 1dA ! £"1 3

4

"5 3§

x¿ AtB ! Ax AtB£ 1 2 "1

0 1 10 "1 1

§c 0 "11 0

d

£"3 1 00 "3 14 "8 2

§£ 1 0 02 3 10 2 4

§£ 0 1 1

1 0 11 1 0

§£ 1 0 00 0 20 2 0

§c 1 51 "3

dc 1 "12 4

dc 6 "32 1

dc"4 22 "1

d 14.

15. 16.

17. Consider the system with

(a) Show that the matrix A has eigenvalues and with corresponding eigenvectors

col and col(b) Sketch the trajectory of the solution having

initial vector (c) Sketch the trajectory of the solution having

initial vector (d) Sketch the trajectory of the solution having

initial vector

18. Consider the system with

(a) Show that the matrix A has eigenvalues and with corresponding eigenvectors

col and col(b) Sketch the trajectory of the solution having

initial vector x A0B ! u1.

A1, "1B.u2 !A1, 1Bu1 !r2 ! "3

r1 ! "1

A ! c"2 11 "2

d .x¿ AtB ! Ax AtB, t $ 0,

x A0B ! u2 " u1.

x A0B ! u2.

x A0B ! "u1.

A1, "23 B .u2 !A23, 1Bu1 !

r2 ! "2r1 ! 2

A ! C 1 2323 "1S .

x¿ AtB ! Ax AtB, t $ 0,

A ! £"7 0 60 5 06 0 2

§A ! £ 1 2 30 1 02 1 2

§A ! £"1 1 0

1 2 10 3 "1

§

(c) Sketch the trajectory of the solution havinginitial vector

(d) Sketch the trajectory of the solution havinginitial vector

In Problems 19–24, find a fundamental matrix for thesystem for the given matrix A.

19. 20.

21.

22.

23.

24.

25. Using matrix algebra techniques, find a general solu-tion of the system

26. Using matrix algebra techniques, find a general solu-tion of the system

In Problems 27–30, use a linear algebra software pack-age such as MATLAB, MAPLE, or MATHEMATICA tocompute the required eigenvalues and eigenvectors andthen give a fundamental matrix for the system

for the given matrix A.

27. 28.

29. A ! D0 1 0 00 0 1 00 0 0 12 "6 3 3

T A ! £ 2 1 1"1 1 0

3 3 3§A ! £ 0 1.1 0

0 0 1.30.9 1.1 "6.9

§x¿ AtB ! Ax AtB

y¿ ! 4x " 7y .x¿ ! 3x " 4y ,

z¿ ! 4x " 4y # 5z .y¿ ! x # z ,x¿ ! x # 2y " z ,

A ! D4 "1 0 00 0 0 00 0 2 "30 0 1 "2

TA ! D2 1 1 "1

0 "1 0 10 0 3 10 0 0 7

TA ! £ 3 1 "11 3 "13 3 "1

§A ! £ 0 1 0

0 0 18 "14 7

§A ! c 5 4

"1 0dA ! c"1 1

8 1dx¿ AtB ! Ax AtB

x A0B ! u1 " u2.

x A0B ! "u2.

Section 9.5 Homogeneous Linear Systems with Constant Coefficients 535

30.

In Problems 31–34, solve the given initial value problem.

31.

32.

33.

34.

35. (a) Show that the matrix

has the repeated eigenvalue and that all the eigenvectors are of the form s col

(b) Use the result of part (a) to obtain a nontrivialsolution to the system

(c) To obtain a second linearly independent solutionto try Hint:Substitute into the system andderive the relations

Since must be an eigenvector, set col and solve for

(d) What is ? (In Section 9.8, will beidentified as a generalized eigenvector.)

36. Use the method discussed in Problem 35 to find ageneral solution to the system


A ! £ 2 1 60 2 50 0 2

§x¿ AtB ! c 5 "3

3 "1d x AtB .

u2AA # IB2u2

u2. 4A1, 2B u1 !u1

AA # IBu1 ! 0 , AA # IBu2 ! u1 .

x¿ ! Axx2

3x2 AtB ! te"tu1 # e"tu2.x¿ ! Ax,

x¿ ! Ax.x1 AtBA1, 2B. u !

r ! "1

A ! c 1 "14 "3

dx¿ AtB ! £ 0 1 1

1 0 11 1 0

§ x AtB , x A0B ! £"140§

x A0B ! £"2"3

2§

x¿ AtB ! £ 1 "2 2"2 1 "2

2 "2 1§ x AtB ,

x¿ AtB ! c 6 "32 1

d x AtB , x A0B ! c"10"6d

x¿ AtB ! c 1 33 1

d x AtB , x A0B ! c 31d

A ! D0 1 0 01 "1 0 00 0 0 10 0 "2 4

T

has the repeated eigenvalue r ! 2 with multi-plicity 3 and that all the eigenvectors of A are ofthe form u ! s col(1, 0, 0).

(b) Use the result of part (a) to obtain a solution tothe system of the form

(c) To obtain a second linearly independent solutionto try Hint:Show that and must satisfy

(d) To obtain a third linearly independent solutionto try

Hint: Show that and must satisfy

(e) Show that



has the repeated eigenvalue r ! 1 of multiplicity3 and that all the eigenvectors of A are of theform col col

(b) Use the result of part (a) to obtain two linearlyindependent solutions to the system ofthe form

and(c) To obtain a third linearly independent solution

to try Hint:Show that and must satisfy

Choose an eigenvector of A, so that you cansolve for

(d) What is ?


x¿ AtB ! £ 1 3 "20 7 "40 9 "5

§ x AtB .AA " IB2u4

u4. 4u3,

AA " IBu3 ! 0 , AA " IBu4 ! u3 .u4u3

3x3 AtB ! tetu3 # etu4.x¿ ! Ax,

x2 AtB ! etu2.x1 AtB ! etu1

x¿ ! Ax

A"1, 0, 1B.A"1, 1, 0B # yu ! s

A ! £ 2 1 11 2 1

"2 "2 "1§

x¿ AtB ! £ 3 "2 12 "1 1

"4 4 1§ x AtB.

AA " 2IB2u2 ! AA " 2IB3u3 ! 0.

AA " 2IBu3 ! u2. 4AA " 2IBu1 ! 0 , AA " 2IBu2 ! u1 ,

u3u1, u2,3x3 AtB !t2

2 e2tu1 # te2tu2 # e2tu3 .

x¿ ! Ax,

AA " 2IBu1 ! 0 , AA " 2IBu2 ! u1. 4u2u1

3x2 AtB ! te2tu1 # e2tu2.x¿ ! Ax,

x1 AtB ! e2tu1.x¿ ! Ax


41. Use the substitution to convert thelinear equation where and c are constants, into a normal system. Show thatthe characteristic equation for this system is the sameas the auxiliary equation for the original equation.

42. (a) Show that the Cauchy–Euler equationcan be written as a

Cauchy–Euler system

(25)

with a constant coefficient matrix A, by settingx1 ! y/t and x2 ! .

(b) Show that for t $ 0 any system of the form (25)with A an n % n constant matrix has nontrivialsolutions of the form if and only if ris an eigenvalue of A and u is a correspondingeigenvector.

In Problems 43 and 44, use the result of Problem 42 tofind a general solution of the given system.

43.

44.

45. Mixing Between Interconnected Tanks. Twotanks, each holding 50 L of liquid, are intercon-nected by pipes with liquid flowing from tank A intotank B at a rate of 4 L/min and from tank B into tankA at 1 L/min (see Figure 9.2). The liquid inside eachtank is kept well stirred. Pure water flows into tank Aat a rate of 3 L/min, and the solution flows out oftank B at 3 L/min. If, initially, tank A contains 2.5 kg of salt and tank B contains no salt (onlywater), determine the mass of salt in each tank attime Graph on the same axes the two quanti-ties and where is the mass of salt intank A and is the mass in tank B.x2 AtB x1 AtBx2 AtB,x1 AtBt & 0.

tx¿ AtB ! c"4 22 "1

d x AtB , t 7 0

tx¿ AtB ! c 1 3"1 5

d x AtB , t 7 0

x AtB ! tru

y¿

t x¿ ! Ax

at2y– # bty¿ # cy ! 0

a, b,ay– # by¿ # cy ! 0,x1 ! y, x2 ! y¿

3 L/min

Pure waterx 1 (t)

50 L

x 1 (0) = 2.5 kg

A 4 L/min

x 2 (t)

50 L

x 2 (0) = 0 kg

B

3 L/min

1 L/min

Figure 9.2 Mixing problem for interconnected tanks

46. Mixing with a Common Drain. Two tanks, eachholding 1 L of liquid, are connected by a pipethrough which liquid flows from tank A into tank Bat a rate of The liquidinside each tank is kept well stirred. Pure waterflows into tank A at a rate of 3 L/min. Solution flowsout of tank A at L/min and out of tank B at L/min. If, initially, tank B contains no salt (onlywater) and tank A contains 0.1 kg of salt, determinethe mass of salt in each tank at time How doesthe mass of salt in tank A depend on the choice of ?What is the maximum mass of salt in tank B? (SeeFigure 9.3.)

at ! 0.

3 " aa

3 " a L/min A0 6 a 6 3B.

Section 9.5 Homogeneous Linear Systems with Constant Coefficients 537

3 L/min

Pure waterx 1 (t)

1 L

x 1 (0) = 0.1 kg

A L/min

x 2 (t)

1 L

x 2 (0) = 0 kg

B

3 L/min

L/min ) L/min

Figure 9.3 Mixing problem for a common drain, 0 6 a 6 3

48. To complete the proof of Theorem 6, assume theinduction hypothesis that arelinearly independent.(a) Show that if

then

(b) Use the result of part (a) and the induction hypoth-esis to conclude that are linearlyindependent. The theorem follows by induction.

49. Stability. A homogeneous system withconstant coefficients is stable if it has a fundamentalmatrix whose entries all remain bounded as

(It will follow from Lemma 1 in Section9.8 that if one fundamental matrix of the system hasthis property, then all fundamental matrices for thesystem do.) Otherwise, the system is unstable. Astable system is asymptotically stable if all solu-tions approach the zero solution as Stabil-ity is discussed in more detail in Chapter 12.†

(a) Show that if A has all distinct real eigenvalues,then is stable if and only if alleigenvalues are nonpositive.

(b) Show that if A has all distinct real eigenvalues,then is asymptotically stable ifand only if all eigenvalues are negative.

(c) Argue that in parts (a) and (b), we can replace“has distinct real eigenvalues” by “is symmet-ric” and the statements are still true.

50. In an ice tray, the water level in any particular icecube cell will change at a rate proportional to the dif-ference between that cell’s water level and the levelin the adjacent cells.(a) Argue that a reasonable differentiable equation

model for the water levels x, y, and z in the sim-plified three-cell tray depicted in Figure 9.4 isgiven by

(b) Use eigenvectors to solve this system for the ini-tial conditions , y A0B # z A0B # 0.x A0B # 3

z¿ # y " z .x¿ # y " x , y¿ # x $ z " 2y ,

x¿ AtB # Ax AtBx¿ AtB # Ax AtB

t S $q.

t S $q.

x¿ # Ax

u1, . . . , uk$1

c1 Ar1 " rk$1Bu1 $ p $ ck Ark " rk$1Buk # 0 .

c1u1 $ p $ ckuk $ ck$1uk$1 # 0 ,

u1, . . . , uk, 2 % k,

47. To find a general solution to the system

proceed as follows:(a) Use a numerical root-finding procedure to

approximate the eigenvalues.(b) If r is an eigenvalue, then let col

be an eigenvector associated with r. To solve foru, assume (If not then either or may be chosen to be 1. Why?) Now solve thesystem

for and Use this procedure to find approx-imations for three linearly independent eigen-vectors for A.

(c) Use these approximations to give a general solu-tion to the system.

u3.u2

AA " rIB £ 1u2u3

§ # £ 000§

u3u2u1,u1 # 1.

Au1, u2, u3Bu #

x¿ # Ax # £ 1 3 "13 0 1

"1 1 2§ x ,

x y z

Figure 9.4 Ice tray

†All references to Chapters 11–13 refer to the expanded text Fundamentals of Differential Equations and BoundaryValue Problems, 6th ed.

From the quadratic formula we find so the four eigenvalues of A are

and Hence, the two normal frequencies for this system are

and cycles per second. !34 ! 25

2p! 0.397

34 " 252p

! 0.211

#i34 ! 25.#i34 " 25

r2 $ "4 # 25,

Section 9.6 Complex Eigenvalues 541

9.6 EXERCISES

In Problems 1–4, find a general solution of the systemfor the given matrix A.

1. 2.

3.

4.

In Problems 5–8, find a fundamental matrix for the sys-tem for the given matrix A.

5. 6.

7.

8.

In Problems 9–12, use a linear algebra software packageto compute the required eigenvalues and eigenvectors forthe given matrix A and then give a fundamental matrixfor the system

9. A $ £ 0 1 1"1 0 1"1 "1 0

§x¿ AtB $ Ax AtB.

A $ D0 1 0 01 0 0 00 0 0 10 0 "13 4

TA $ £ 0 0 10 0 "10 1 0

§A $ c"2 "2

4 2dA $ c"1 "2

8 "1dx¿ AtB $ Ax AtB

A $ £ 5 "5 "5"1 4 2

3 "5 "3§

A $ £ 1 2 "10 1 10 "1 1

§A $ c"2 "5

1 2dA $ c 2 "4

2 "2dx¿ AtB $ Ax AtB

10.

11.

12.

In Problems 13 and 14, find the solution to the given sys-tem that satisfies the given initial condition.

13.

(a) (b)

(c) (d)

14.

(a) (b) x A"pB $ £ 011§x A0B $ £"2

2"1§

x¿ AtB $ £ 1 0 "10 2 01 0 1

§ x AtB ,x Ap/2B $ c 0

1dx A"2pB $ c 2

1d

x ApB $ c 1"1dx A0B $ c"1

0d

x¿ AtB $ c"3 "12 "1

d x AtB ,

A $ E1 0 0 0 00 0 1 0 00 1 0 0 00 0 0 0 10 0 0 "29 "4

UA $ D 0 1 0 0

0 0 1 00 0 0 1

"2 2 "3 2

TA $ D 0 1 0 0

0 0 1 00 0 0 1

13 "4 "12 4

T

15. Show that and given by equations (4) and(5) are linearly independent on provided

and a and b are not both the zero vector.16. Show that and given by equations (4) and

(5) can be obtained as linear combinations of and given by equations (2) and (3). [Hint:Show that

In Problems 17 and 18, use the results of Problem 42 inExercises 9.5 to find a general solution to the givenCauchy–Euler system for

17.

18.

19. For the coupled mass–spring system governed by sys-tem (10), assume kg, N/m,and N/m. Determine the normal frequenciesfor this coupled mass–spring system.

20. For the coupled mass–spring system governed bysystem (10), assume kg, !

N/m, and assume initially that m,m/sec, m, and m/sec.

Using matrix algebra techniques, solve this initialvalue problem.

21. RLC Network. The currents in the RLC networkgiven by the schematic diagram in Figure 9.6 aregoverned by the following equations:

I1 AtB ! I2 AtB " I3 AtB ,13I3 AtB " 52q1 AtB ! 10 ,4I¿2 AtB " 52q1 AtB ! 10 ,

x¿2 A0B ! 0x2 A0B ! 2x¿1 A0B ! 0x1 A0B ! 0k3 ! 1

k1 ! k2m1 ! m2 ! 1

k3 ! 3k1 ! k2 ! 2m1 ! m2 ! 1

tx¿ AtB ! c#1 #19 #1

d x AtBtx¿ AtB ! £#1 #1 0

2 #1 10 1 #1

§ x AtBt 7 0.

x2 AtB !w1 AtB # w2 AtB

2i. dx1 AtB !

w1 AtB " w2 AtB2

,

w2 AtB w1 AtBx2 AtBx1 AtBb $ 0A#q, q B,x2 AtBx1 AtB


where is the charge on the capacitor,and initially coulombs and

amps. Solve for the currents and Hint: Differentiate the first two equations, elimi-

nate and form a normal system with and

22. RLC Network. The currents in the RLC networkgiven by the schematic diagram in Figure 9.7 aregoverned by the following equations:

where is the charge on the capacitor,and initially coulombs

and amps. Solve for the currents andHint: Differentiate the first two equations, use

the third equation to eliminate and form a normalsystem with and x3 ! I2. 4x1 ! I1, x2 ! I¿1,

I3,3I3.

I1, I2,I3 A0B ! 0q3 A0B ! 0.5I3 AtB ! q¿3 AtB,q3 AtBI1 AtB ! I2 AtB " I3 AtB ,50I¿1 AtB " 800q3 AtB ! 160 ,

50I¿1 AtB " 80I2 AtB ! 160 ,

x3 ! I3. 4x2 ! I¿2,x1 ! I2,I1,

3 I3.I1, I2,I1 A0B ! 0q1 A0B ! 0I1 AtB ! q¿1 AtB,q1 AtB

10 volts

I 1

I 1

I 1

I 2

farads

4 henries 13 ohms

I 3

1 —– 52

Figure 9.6 RLC network for Problem 21

800

160 volts

I 1

I 1

I 2

1

I 3

50 henries

80 ohms farads

Figure 9.7 RLC network for Problem 22

23. Stability. In Problem 49 of Exercises 9.5, wediscussed the notion of stability and asymptotic sta-bility for a linear system of the form !Assume that A has all distinct eigenvalues (real orcomplex).(a) Show that the system is stable if and only if all

the eigenvalues of A have nonpositive real part.(b) Show that the system is asymptotically stable if

and only if all the eigenvalues of A have nega-tive real part.

24. (a) For Example 1, page 539, verify that

is another general solution to equation (8).(b) How can the general solution of part (a) be directly

obtained from the general solution derived in(9) on page 539?

" c2 c#e#2t sin t # e#2t

cos te#2t

sin td

x AtB ! c1 c#e#2t cos t " e#2t

sin te#2t

cos td

Ax AtB.x¿ AtB

Substituting into formula (13), we obtain the solution

! ! £"92 et " 5

6 e"t # 4

3 e2t # 3

" 32 et " 5

6 e"t # 1

3 e2t # 2§ .

! £"

32 et # 1

2 e"t

" 12 et # 1

2 e"t§ # c3et e"t

et e"t d £ 12 et # 1

2 e"t " 1

32 et " 1

6 e3t " 4

3

§ ! £"

32 et # 1

2 e"t

" 12 et # 1

2 e"t§ # c3et e"t

et e"t d ! t

0

£ 12 es " 1

2 e"s

" 12 e3s # 3

2 es§ ds

! # c3et e"t

et e"t d ! t

0

£"

12 e"s "

12 e"s

" 12 es "

32

es§ c e2s

1d ds

x AtB ! c3et e"t

et e"t d £ 12 "

12

" 12

32

§ c"10d

Section 9.7 Nonhomogeneous Linear Systems 547

9.7 EXERCISES

In Problems 1–6, use the method of undetermined coeffi-cients to find a general solution to the system

are given.

1.

2.

3.

4.

5.

6. A ! c 1 10 2

d , f AtB ! e"2t c t3d

A ! £ 0 "1 0"1 0 0

0 0 1§ , f AtB ! £ e2 t

sin tt§

A ! c 2 22 2

d , f AtB ! c"4 cos t"sin t

dA ! £ 1 "2 2

"2 1 22 2 1

§ , f AtB ! £ 2et

4et

"2et§

A ! c 1 14 1

d , f AtB ! c "t " 1"4t " 2

dA ! c 6 1

4 3d , f AtB ! c"11

"5dAx AtB # f AtB, where A and f AtB x¿ AtB !

In Problems 7–10, use the method of undetermined coeffi-cients to determine only the form of a particular solutionfor the system , where and aregiven.

7.

8.

9.

10.

In Problems 11–16, use the variation of parameters for-mula (11) to find a general solution of the system

are given.

11. A ! c 0 1"1 0

d , f AtB ! c 10dx¿ AtB ! Ax AtB # f AtB, where A and f AtB

A ! c 2 "11 5

d , f AtB ! c te"t

3e"t dA ! £ 0 "1 0

"1 0 10 0 1

§ , f AtB ! £ e2t

sin tt§

A ! c"1 02 2

d , f AtB ! c t2

t # 1d

A ! c 0 12 0

d , f AtB ! c sin 3ttd

f AtBAx¿ AtB ! Ax AtB # f AtB

12.

13.

14.

15.

16.

In Problems 17–20, use the variation of parameters for-mulas (11) and possibly a linear algebra software pack-age to find a general solution of the system

are given.

17.

18.

19.

20.

In Problems 21 and 22, find the solution to the given sys-tem that satisfies the given initial condition.

21.

(a) (b)

(c) (d) x A!1B " c!45dx A5B " c 1

0d

x A1B " c 01dx A0B " c 5

4d

x¿ AtB " c 0 2!1 3

d x AtB # c et

!et d ,

A " D0 1 0 00 0 1 00 0 0 18 !4 !2 !1

T , f AtB " Det

010

TA " D 0 1 0 0

!1 0 0 00 0 0 10 0 1 0

T , f AtB " D t0

e!t

t

TA " £ 1 !1 10 0 10 !1 2

§ , f AtB " £ 0et

et§

A " £ 0 1 11 0 11 1 0

§ , f AtB " £ 3et

!et

!et§

Ax AtB # f AtB, where A and f AtB x¿ AtB "

A " c 0 1!1 0

d , f AtB " c 8 sin t0d

A " c!4 22 !1

d , f AtB " c t!1

4 # 2t!1 dA " c 0 !1

1 0d , f AtB " c t2

1d

A " c 2 1!3 !2

d , f AtB " c 2et

4et dA " c 1 2

3 2d , f AtB " c 1

!1d


22.

(a) (b)

23. Using matrix algebra techniques and the method of undetermined coefficients, find a general solutionfor

Compare your solution with the solution in Example4 in Section 5.2.

24. Using matrix algebra techniques and the method ofundetermined coefficients, solve the initial valueproblem

Compare your solution with the solution in Example1 in Section 7.9.

25. To find a general solution to the system

proceed as follows:(a) Find a fundamental solution set for the corre-

sponding homogeneous system.(b) The obvious choice for a particular solution

would be a vector function of the formhowever, the homogeneous system

has a solution of this form. The next choicewould be Show that this choicedoes not work.

(c) For systems, multiplying by t is not alwayssufficient. The proper guess is

Use this guess to find a particular solution of thegiven system.

(d) Use the results of parts (a) and (c) to find ageneral solution of the given system.

26. For the system of Problem 25, we found that aproper guess for a particular solution is "

In some cases a or b may be zero.(a) Find a particular solution for the system of

Problem 25 if col(b) Find a particular solution for the system of

Problem 25 if col Aet, etB.f AtB "

A3et, 6etB.f AtB "

teta # etb.xp AtB

xp AtB " teta # etb .

xp AtB " teta.

xp AtB " eta;

x!AtB " c 0 1#2 3 d x AtB $ f AtB , where f AtB " c et

0 d ,y A0B " !5 .y¿AtB # 2y AtB ! 4x AtB " !4t ! 2 ,

x¿ AtB ! 2y AtB " 4t , x A0B " 4 ;

x¿ AtB # y¿ AtB ! x AtB " t2 .x– AtB # y¿ AtB ! x AtB # y AtB " !1 ,

x A2B " c 11dx A0B " c 4

!5d

x¿ AtB " c 0 24 !2

d x AtB # c 4t!4t ! 2

d ,

27. Find a general solution of the system

Hint: Try

28. Find a particular solution for the system

Hint: Try

In Problems 29 and 30, find a general solution to thegiven Cauchy–Euler system for (See Problem 42in Exercises 9.5.) Remember to express the system in theform before using the variationof parameters formula.

29.

30.

31. Use the variation of parameters formula (10) toderive a formula for a particular solution to thescalar equation in termsof two linearly independent solutions ofthe corresponding homogeneous equation. Showthat your answer agrees with the formulas derived inSection 4.6. [Hint: First write the scalar equation insystem form.]

32. Let be the invertible matrix

Show that

33. RL Network. The currents in the RL networkgiven by the schematic diagram in Figure 9.8 aregoverned by the following equations:

I3 AtB ! I4 AtB " I5 AtB .I1 AtB ! I2 AtB " I3 AtB ,60I5 AtB # 30I4 AtB ! 0 ,I¿3 AtB " 30I4 AtB # 90I2 AtB ! 0 ,2I¿1 AtB " 90I2 AtB ! 9 ,

U#1 AtB !13 a AtBd AtB # b AtBc AtB 4 c d AtB #b AtB

#c AtB a AtB d .U AtB J c a AtB b AtB

c AtB d AtB d .2 $ 2U AtB

y1 AtB, y2 AtBy– " p AtBy¿ " q AtBy ! g AtByp

tx¿ AtB ! c 4 #38 #6

d x AtB " c t2td

tx¿ AtB ! c 2 #13 #2

d x AtB " c t#1

1d

x¿ AtB ! A AtBx AtB " f AtBt 7 0.

xp AtB ! ta " b. 43 x¿ AtB ! c 1 #1#1 1

d x AtB " c#31d .

xp AtB ! e#ta " te#tb " c. 43x¿ AtB ! £ 0 1 1

1 0 11 1 0

§ x AtB " £ #1#1 # e#t

#2e#t§ .

Section 9.7 Nonhomogeneous Linear Systems 549

Assume the currents are initially zero. Solve for the five currents Hint: Eliminate allunknowns except and and form a normal sys-tem with and

34. Conventional Combat Model. A simplistic modelof a pair of conventional forces in combat yields thefollowing system:

where col The variables and represent the strengths of opposing forces at time t.The terms and represent the operationalloss rates, and the terms and representthe combat loss rates for the troops and respectively. The constants p and q represent therespective rates of reinforcement. Let , ,

, and By solving theappropriate initial value problem, determine whichforces will win if(a)(b)(c)

35. Mixing Problem. Two tanks A and B, each hold-ing 50 L of liquid, are interconnected by pipes. Theliquid flows from tank A into tank B at a rate of 4 L/min and from B into A at a rate of 1 L/min (seeFigure 9.9). The liquid inside each tank is kept well

x1 A0B ! 20 , x2 A0B ! 21 .x1 A0B ! 21 , x2 A0B ! 20 .x1 A0B ! 20 , x2 A0B ! 20 .

p ! q ! 5.d ! 2,c ! 3b ! 4a ! 1

x2,x1

#cx1#bx2

#dx2#ax1

x2 AtBx1 AtBAx1, x2B.x !

x¿ ! c#a #b#c #d

d x " c pqd ,

x2 ! I5. 4x1 ! I2

I5,I2

3I1, . . . , I5.

9 volts

I 1

I 1

I 4 I 5

2 henries

90 ohms

I 2

I 2

I 3 1 henry I 3

I 3

30 ohms 60 ohms

Figure 9.8 RL network for Problem 33

4 L/min 0.2 kg/L

1 L/min 0.1 kg/L x 1 (t)

50 L

x 1 (0) = 0 kg

A 4 L/min

x 2 (t)

50 L

x 2 (0) = 0.5 kg

B

4 L/min

1 L/min

1 L/min

Figure 9.9 Mixing problem for interconnected tanks

stirred. A brine solution that has a concentration of0.2 kg/L of salt flows into tank A at a rate of 4 L/min.A brine solution that has a concentration of 0.1 kg/Lof salt flows into tank B at a rate of 1 L/min. Thesolutions flow out of the system from both tanks—from tank A at 1 L/min and from tank B at 4 L/min.


If, initially, tank A contains pure water and tank Bcontains 0.5 kg of salt, determine the mass of salt ineach tank at time After several minutes haveelapsed, which tank has the higher concentration ofsalt? What is its limiting concentration?

t ! 0.

In this chapter we have developed various ways to extend techniques for scalar differentialequations to systems. In this section we take a substantial step further by showing that with theright notation, the formulas for solving normal systems with constant coefficients are identicalto the formulas for solving first-order equations with constant coefficients. For example, weknow that a general solution to the equation where a is a constant, is Analogously, we show that a general solution to the normal system

(1)

where A is a constant matrix, is Our first task is to define the matrixexponential

If A is a constant matrix, we define by taking the series expansion for andreplacing a by A; that is,

(2)

(Note that we also replace 1 by I.) By the right-hand side of (2), we mean the matrixwhose elements are power series with coefficients given by the corresponding entries in thematrices

If A is a diagonal matrix, then the computation of is straightforward. For example, if

then

and so

More generally, if A is an matrix with down its main diagonal,then is the diagonal matrix with down its main diagonal (see Problem 26).If A is not a diagonal matrix, the computation of is more involved. We deal with this impor-tant problem later in this section.

eAter1t, er2t, . . . , ernteAt

r1, r2, . . . , rnn " n diagonal

eAt # aq

n#0An

tn

n!# Daqn#0

A$1Bn t n

n!0

0 aq

n#0 2n

t n

n!

T # c e$t 00 e2t d .

A2 # AA # c1 00 4

d , A3 # c$1 00 8

d , . . . , An # c A$1Bn 00 2n d ,

A # c$1 00 2

d ,eAt

I, A, A2/2!, . . . .

n " n

eAt :! I " At " A2 t2

2!" p " An

tn

n!" p .

eateAtn " neAt.

x AtB # eAtc.n " n

x¿ AtB # Ax AtB ,x AtB # ceat.x¿ AtB # ax AtB,

9.8 THE MATRIX EXPONENTIAL FUNCTION

In Problems 1–6, (a) show that the given matrix A satis-fies for some number r and some positiveinteger k and (b) use this fact to determine the matrix [Hint: Compute the characteristic polynomial and usethe Cayley–Hamilton theorem.]

1. 2.

3.

4.

5.

6.

In Problems 7–10, determine by first finding afundamental matrix and then usingformula (3).

7. 8.

9. 10.

In Problems 11 and 12, determine by using general-ized eigenvectors to find a fundamental matrix and thenusing formula (3).

11.

12. A ! £ 1 1 12 1 "10 "1 1

§A ! £ 5 "4 0

1 0 20 2 5

§eAt

A ! £ 0 2 22 0 22 2 0

§A ! £ 0 1 00 0 11 "1 1

§A ! c 1 1

4 1dA ! c 0 1

"1 0d

X AtB for x¿ ! AxeAt

A ! £ 0 1 00 0 1

"1 "3 "3§

A ! £"2 0 04 "2 01 0 "2

§A ! £ 2 1 3

0 2 "10 0 2

§A ! £ 2 1 "1

"3 "1 19 3 "4

§A ! c 1 "1

1 3dA ! c 3 "2

0 3d

eAt.AA " rIBk ! 0

Section 9.8 The Matrix Exponential Function 557

9.8 EXERCISES

In Problems 13–16, use a linear algebra software pack-age for help in determining

13.

14.

15.

16.

In Problems 17–20, use the generalized eigenvectors ofA to find a general solution to the system where A is given.

17. 18.

19.

20.

21. Use the results of Problem 5 to find the solution tothe initial value problem

x A0B ! £ 11

"1§ .x¿ AtB ! £"2 0 0

4 "2 01 0 "2

§ x AtB ,

A ! £"1 "8 1"1 "3 2"4 "16 7

§A ! D1 0 1 2

1 1 2 10 0 2 00 0 1 1

T A ! £ 0 0 10 1 20 0 1

§A ! £ 0 1 00 0 1

"2 "5 "4§

x¿ AtB ! Ax AtB,

A ! E"1 0 0 0 00 0 1 0 00 "1 "2 0 00 0 0 0 10 0 0 "4 "4

UA ! E 0 1 0 0 0

0 0 1 0 0"1 "3 "3 0 0

0 0 0 0 10 0 0 "4 "4

UA ! E1 0 0 0 0

0 0 1 0 00 "1 "2 0 00 0 0 0 10 0 0 "1 0

UA ! E0 1 0 0 0

0 0 1 0 01 "3 3 0 00 0 0 0 10 0 0 "1 0

UeAt.

22. Use your answer to Problem 12 to find the solutionto the initial value problem

23. Use the results of Problem 3 and the variation ofparameters formula (16) to find the solution to theinitial value problem

24. Use your answer to Problem 9 and the variation ofparameters formula (16) to find the solution to theinitial value problem

25. Let

and

(a) Show that (b) Show that property (d) in Theorem 7 does not

hold for these matrices. That is, show that

26. Let A be a diagonal matrix with entriesdown its main diagonal. To compute

proceed as follows:eAt,r1, . . . , rn

n ! ne AA"BBt # eAteBt.

AB # BA.

B $ c 2 10 1

d .A $ c 1 2%1 3

dx A0B $ £ 1

%10§ .

x¿ AtB $ £ 0 1 00 0 11 %1 1

§ x AtB " £ 00t§ ,

x A0B $ £ 030§ .

x¿ AtB $ £ 2 1 %1%3 %1 1

9 3 %4§ x AtB " £ 0t

0§ ,

x A0B $ £%103§ .x¿ AtB $ £ 1 1 1

2 1 %10 %1 1

§ x AtB ,


(a) Show that is the diagonal matrix with entriesdown its main diagonal.

(b) Use the result of part (a) and definition (2) toshow that is the diagonal matrix with entries

down its main diagonal.27. In Problems 35–40 of Exercises 9.5, page 535, some

ad hoc formulas were invoked to find general solu-tions to the system when A had repeatedeigenvalues. Using the generalized eigenvectorprocedure outlined on page 554, justify the ad hocformulas proposed in(a) Problem 35, Exercises 9.5.(b) Problem 37, Exercises 9.5.(c) Problem 39, Exercises 9.5.

28. Let

(a) Find a general solution to x Ax.(b) Determine which initial conditions

yield a solution thatremains bounded for all that is, satisfies

for some constant M and all 29. For the matrix A in Problem 28, solve the initial

value problem

x A0B $ £ 01

%1§ .

x¿ AtB $ Ax AtB " sin A2tB £ 204§ ,

t & 0.

0 0x AtB 0 0 :$ 2x21 AtB " x2

2 AtB " x23 AtB ' M

t & 0;x AtB $ col Ax1 AtB, x2 AtB, x3 AtB Bx A0B $ x0

¿ $

A $ s 5 2 %40 3 04 %5 %5

t

x¿ $ Ax

er1t, . . . , ern teAt

rk1, . . . , rk

n

Ak

Chapter Summary

In this chapter we discussed the theory of linear systems in normal form and presentedmethods for solving such systems. The theory and methods are natural extensions of the devel-opment for second-order and higher-order linear equations. The important properties andtechniques are as follows.

Homogeneous Normal Systems

The matrix function is assumed to be continuous on an interval I.A AtBn ! n

x¿ AtB $ A AtBx AtB

Nagle R.K., Saff E.B., Snider A.D. Fundamentals of differential ...

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