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

ISBN 0-471-31999-6

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x,

x(0) =

-1

2

1

1

-i)x

by using the fundamental matrix O(t) found in Problem 3.

12. Solve the initial value problem

x'

-1 -4

11

x(0) =

by using the fundamental matrix O(t) found in Problem 6.

Show that O(t) = 0(t)0-1 (t0), where O(t) and O(t) are as defined in this section.

The fundamental matrix O(t) for the system (3) was found in Example 2. Show that O(t)O(s) = O(f + s) by multiplying O(t) and O(s).

Let O(t) denote the fundamental matrix satisfying O; = AO, O(0) = I. In the text we also denoted this matrix by exp (A t). In this problem we show that O does indeed have the principal algebraic properties associated with the exponential function.

(a) Show that O(t)O(s) = O(t + s); that is, exp(At) exp(As) = exp[A(t + s)].

Hint: Show that if s is fixed and t is variable, then both O(t)O(s) and O(t + s) satisfy the initial value problem Z = AZ, Z(0) = O(s).

(b) Showthat O(t)O(-t) = I; that is, exp (At) exp[A(-t)] = I. Then show that O(-t) =

O-1 (t)•

(c) Show that O (t - s) = O (t)O 1 (s)

x

x

x

x

x

x

1

x

2

3

x

1

7.8 Repeated Eigenvalues

401

16. Show that if A is a diagonal matrix with diagonal elements a1, a2,..., an, then exp(At) is also a diagonal matrix with diagonal elements exp(a11), exp(a2t), ..., exp(ant).

17. The method of successive approximations (see Section 2.8) can also be applied to systems of equations. For example, consider the initial value problem

where A is a constant matrix and x0 a prescribed vector.

(a) Assuming that a solution x = ^ (t) exists, show that it must satisfy the integral equation

We conclude our consideration of the linear homogeneous system with constant coefficients

with a discussion of the case in which the matrix A has a repeated eigenvalue. Recall that in Section 7.3 we noted that a repeated eigenvalue with multiplicity k maybe associated with fewer than k linearly independent eigenvectors. The following example illustrates this possibility.

Find the eigenvalues and eigenvectors of the matrix

The eigenvalues r and eigenvectors ? satisfy the equation (A — rI)? = 0, or

x' = Ax, x(0) = x0,

(i)

(b) Start with the initial approximation ^(0)(t) = x0. Substitute this expression for ^(s) in the right side of Eq. (ii) and obtain a new approximation ^(1) (t). Show that

^(1)(f) = (I + At)x0.

(iii)

(c) Repeat this process and thereby obtain a sequence of approximations ^(0'1, ^(1), . Use an inductive argument to show that

(iv)

(d) Let n — ^ and show that the solution of the initial value problem (i) is

^(f) = exp(Af)x°.

(v)

7.8 Repeated Eigenvalues

x' = Ax

(1)

EXAMPLE

1

(2)

(3)

402

Chapter 7. Systems of First Order Linear Equations

EXAMPLE

2

The eigenvalues are the roots of the equation

det(A — rI) =

1 — r —1 1 3- r

= r2 — 4r + 4 = 0.

(4)

Thus the two eigenvalues are r1 = 2 and r2 = 2; that is, the eigenvalue 2 has multiplicity 2.

To determine the eigenvectors we must return to Eq. (3) and use for r the value 2. This gives

—1 —1

1 1

M = (°

(5)

Hence we obtain the single condition f1 + ?2 = 0, which determines ?2 in terms of f1, or vice versa. Thus the eigenvector corresponding to the eigenvalue r = 2 is

1

1

(6)

or any nonzero multiple of this vector. Observe that there is only one linearly independent eigenvector associated with the double eigenvalue.

Returning to the system (1), suppose that r = p is a k-fold root of the determinantal equation

det(A — rI) = 0.

(7)

Then p is an eigenvalue of multiplicity k of the matrix A. In this event, there are two possibilities: either there are k linearly independent eigenvectors corresponding to the eigenvalue p, or else there are fewer than k such eigenvectors.

In the first case, let ?(1),..., ?(k) be k linearly independent eigenvectors associated with the eigenvalue p of multiplicity k. Then x(1) (t) = ?(1) ept,, x(k)(t) = ?(k) ept are k linearly independent solutions of Eq. (1). Thus in this case it makes no difference that the eigenvalue r = p is repeated; there is still a fundamental set of solutions of Eq. (1) of the form ?ert. This case always occurs if the coefficient matrix A is Hermitian.

However, if the coefficient matrix is not Hermitian, then there may be fewer than k independent eigenvectors corresponding to an eigenvalue p of multiplicity k, and if so, there will be fewer than k solutions of Eq. (1) of the form ?ept associated with this eigenvalue. Therefore, to construct the general solution of Eq. (1) it is necessary to find other solutions of a different form. By analogy with previous results for linear equations of order n, it is natural to seek additional solutions involving products of polynomials and exponential functions. We first consider an example.

Find a fundamental set of solutions of

x = Ax =

11

(8)

and draw a phase portrait for this system.

7.8 Repeated Eigenvalues

403

NN\\\\\\\V

NN\\\\\\\

2

*~ ??^'^ ^ N \ \ \ \ \

— — \ \ \ \

1

-----— ^ N \ \

— ~ \ \

_l________________

i I i

I I

\ V

n

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