# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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x(2)(t) = gtept + ^, (23)

where g satisfies Eq. (22) and ^ is determined from

(A - pI)^ = g. (24)

Even though det(A pI) = 0, it can be shown that it is always possible to solve

Eq. (24) for ^. The vector ^ is called a generalized eigenvector corresponding to the

eigenvalue p.

Fundamental Matrices. As explained in Section 7.7, fundamental matrices are formed by arranging linearly independent solutions in columns. Thus, for example, a

406

Chapter 7. Systems of First Order Linear Equations

fundamental matrix for the system (8) can be formed from the solutions x(1)(t) and x(2)(t) from Eqs. (9) and (19), respectively:

te2t

-te2t-e2t

= e

2t

1 t

-1 -1-1

(25)

The matrix O that satisfies 0(0) = I can also be readily found from the relation O(t) = 0(t)0-1 (0). ForEq. (8) we have

0(0) =

1

0

11

0-1(0) =

1

0

11

(26)

and then

O(t) = 0(t)0-1(0) = e'

2t

e

2t

1 0

-1 -1- t/\-1 -1

1 t t

t 1 + t}'

The latter matrix is also the exponential matrix exp (At).

(27)

Jordan Forms. An n x n matrix A can be diagonalized as discussed in Section 7.7 only if it has a full complement of n linearly independent eigenvectors. If there is a shortage of eigenvectors (because of repeated eigenvalues), then A can always be transformed into a nearly diagonal matrix called its Jordan5 form, which has the eigenvalues of A on the main diagonal, ones in certain positions on the diagonal above the main diagonal, and zeros elsewhere.

Consider again the matrix A given by Eq. (2). Form the transformation matrix T with the single eigenvector ? from Eq. (6) in its first column and the generalized eigenvector ^ from Eq. (17) with k = 0 in the second column. Then T and its inverse are given by

T =

1 0 11

As you can verify, it follows that

T-1AT =

T- 1

2 1 0 2

1

0

11

J.

(28)

(29)

The matrix J in Eq. (29) is the Jordan form of A. It is typical of all Jordan forms in that it has a 1 above the main diagonal in the column corresponding to the eigenvector that is lacking (and is replaced in T by the generalized eigenvector).

If we start again from Eq. (1),

X = Ax,

the transformation x = Ty, where T is given by Eq. (28), produces the system

y = Jy, (30)

where J is given by Eq. (29). In scalar form the system (30) is

/1 = 2 /1 + T2> /2 = 2 /2 (31)

Camille Jordan (1838-1921), professor at the Ecole Polytechnique and the College de France, made important contributions to analysis, topology, and especially to algebra. The Jordan form of a matrix appeared in his influential book Trait? des substitutions et des ?quations alg?briques, published in 1870.

7.8 Repeated Eigenvalues

407

These equations can be solved readily in reverse order. In this way we obtain

y = C1e2t > /1 = C1te2t + C2e2t (32)

Thus two independent solutions of the system (30) are

y"'(t) = (0) e2, yŽ = (;)eJt (33)

and the corresponding fundamental matrix is

^(t) = (*0 S) . (34)

Since \P (0) = I, we can also indentify the matrix in Eq. (34) as exp (Jt). The same result

can be reached by calculating powers of J and substituting them into the exponential series (see Problems 19 through 21). To obtain a fundamental matrix for the original system we now form the product

^(t) = T exp(Jt) = (^_ee2t _Jte_ te2^ > (35)

which is the same as the fundamental matrix given in Eq. (25).

PROBLEMS In each of Problems 1 through 6 find the general solution of the given system of equations. In

s each of Problems 1 through 4 also draw a direction field, sketch a few trajectories, and describe how the solutions behave as t -> oo.

? 1. x' =

? 3. x' =

5. x' =

'3 4

11

1 -1 I x

? 2. x' =

? 4. x' =

6. x'

' 4 2

18 -4

(0 -1

1

2 x

2 V

0 1 A

1 0 1 1 x

1 1 0

In each of Problems 7 through 10 find the solution of the given initial value problem. Draw the trajectory of the solution in the x1 x2-plane and also the graph of x1 versus t.

7-x = l 4 x.

x(0) =

x(0) =

3

? 9. x'

3

2

1

x(0) =

3

2

x

x

5

1

x

x

x

22

2

x

? 10. x'=(3 3)x x(0) = (2)

408

Chapter 7. Systems of First Order Linear Equations

In each of Problems 11 and 12 find the solution of the given initial value problem. Draw the corresponding trajectory in x1 x2x3-space and also draw the graph of x1 versus t.

In each of Problems 13 and 14 solve the given system of equations by the method of Problem 19 of Section 7.5. Assume that t > 0.

approach zero as t if and only if a + d < 0 and ad bc > 0. Compare this result

with that of Problem 38 in Section 3.5.

16. Consider again the electric circuit in Problem 26 of Section 7.6. This circuit is described by the system of differential equations

(b) Suppose that R = 1 ohm, C = 1 farad, and L = 4 henrys. Suppose also that I(0) = 1 ampere and V(0) = 2 volts. Find I(t) and V(t).

Eigenvalues of Multiplicity 3. If the matrix A has an eigenvalue of multiplicity 3, then there may be either one, two, or three corresponding linearly independent eigenvectors. The general solution of the system x' = Ax is different, depending on the number of eigenvectors associated with the triple eigenvalue. As noted in the text, there is no difficulty if there are three eigenvectors, since then there are three independent solutions of the form x = ?ert. The following two problems illustrate the solution procedure for a triple eigenvalue with one or two eigenvectors, respectively.

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