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

ISBN 0-471-31999-6

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(b) Show that I (t) ^ 0 and V (t) ^ 0as t regardless of the initial values I (0) and V (0).

33. Consider the preceding system of differential equations (i).

(a) Find a condition on R1, R2, C, and L that must be satisfied if the eigenvalues of the coefficient matrix are to be real and different.

(b) If the condition found in part (a) is satisfied, show that both eigenvalues are negative. Then show that I(t) ^ 0 and V(t) ^ 0 as t regardless of the initial conditions.

(c) If the condition found in part (a) is not satisfied, then the eigenvalues are either complex or repeated. Do you think that I(t) ^ 0 and V(t) ^ 0 as t ^?in these cases as well? Hint: In part (c) one approach is to change the system (i) into a single second order equation. We also discuss complex and repeated eigenvalues in Sections 7.6 and 7.8.

/_ R1 1

d (1 \ L - L

dt y= 1 1

\ C - CR

384

Chapter 7. Systems of First Order Linear Equations

7.6 Complex Eigenvalues

In this section we consider again a system of n linear homogeneous equations with constant coefficients

x = Ax, (1)

where the coefficient matrix A is real-valued. If we seek solutions of the form x = ?ert, then it follows as in Section 7.5 that r must be an eigenvalue and ? a corresponding eigenvector of the coefficient matrix A. Recall that the eigenvalues rv ..., rn of A are the roots of the equation

det(A — rI) = 0, (2)

and that the corresponding eigenvectors satisfy

(A — r I)? = 0. (3)

If A is real, then the coefficients in the polynomial equation (2) for r are real, and any complex eigenvalues must occur in conjugate pairs. For example, if r1 = X + ix, where X and x are real, is an eigenvalue of A, then so is r2 = X — ix. Further, the corresponding eigenvectors ?(1) and ?(2) are also complex conjugates. To see that this is so, suppose that r1 and ?(1) satisfy

(A — r^)?(1) = 0. (4)

On taking the complex conjugate of this equation, and noting that A and I are realvalued, we obtain

(A — r 1I)I(1) = 0, (5)

where r 1 and ?(1) are the complex conjugates of r1 and ?(1), respectively. In other

words, r2 = r 1 is also an eigenvalue, and ?(2) = ?(1) is a corresponding eigenvector.

The corresponding solutions

x(1)(t) = ? (1)er1t, x(2)(t) = ? (1)er1t (6)

of the differential equation (1) are then complex conjugates of each other. Therefore, as in Section 3.4, we can find two real-valued solutions of Eq. (1) corresponding to the eigenvalues r1 and r2 by taking the real and imaginary parts of x(1) (t) or x(2)(t) given by Eq. (6). 1 ( ) 2

Let us write ?(1) = a + ib, where a and b are real; then we have

x(1)(t) = (a + i b^1-^1 x)t

= (a + ib)eXt(cos xt + i sinxt). (7)

Upon separating x(1) (t) into its real and imaginary parts, we obtain

x(1)(t) = eXt (a cos xt — b sin xt) + ieXt (a sin xt + b cos xt). (8)

If we write x(1) (t) = u(t) + i v(t), then the vectors

u(t) = eXt(acos xt — b sin xt),

^ X^ • (9)

v(t) = e (a sin xt + b cos xt)

7.6 Complex Eigenvalues

385

are real-valued solutions of Eq. (1). It is possible to show that u and v are linearly independent solutions (see Problem 27).

For example, suppose that r1 = X + ix, r2 = X — ix, and that r3,, rn are all real and distinct. Let the corresponding eigenvectors be ?(1) = a + ib, ?(2) = a — ib, ?(3),..., ?(n). Then the general solution of Eq. (1) is

x = qu(t) + c2v(t) + c3? (3'>er3( + ••• + cn ? {n)ernt, (10)

where u(t) and v(t) are given by Eqs. (9). We emphasize that this analysis applies only if the coefficient matrix A in Eq. (1) is real, for it is only then that complex eigenvalues and eigenvectors must occur in conjugate pairs.

The following examples illustrate the case n = 2, both to simplify the algebraic calculations and to permit easy visualization of the solutions in the phase plane.

Find a fundamental set of real-valued solutions of the system

EXAMPLE

1'

(11)

and display them graphically.

A direction field for the system (11) is shown in Figure 7.6.1. This plot suggests that the trajectories in the phase plane spiral clockwise toward the origin.

XXX\ \ \ \ \ \ \ XXX \ \ \ \ \ \ \

- 'v X \ \ \ \ \ \ \

XX X\ \ \ \ \ \ '

? N\\ \ \ \ \ \ '

?W \ \ \ \ \ \

~ \ \ \ \ I 1 | .

?N\ \ \ 1 I

/ / / /?

/ / / s

/ / / / 1---------

/ /

/ / s

/ / s

/ / I

-2

1 \ J \ \ \

t t t 1 \

1 \ \ N \ \ \ \ \ \ \ \ \ \ \ \ \ N \ \ N

Uk-

\ W'

-t

\ xx-

XXX-\ N V'

-2

X XX-\NN'

/////)//>

t 2 x

-~'s'SS/S/// ^ S s' S / /

S' s'

FIGURE 7.6.1 A direction field for the system (11).

To find a fundamental set of solutions we assume that

x =

and obtain the set of linear algebraic equations

2 r

1

— — — r 2 1

?)=(0

(12)

(13)

1

2

2

386

Chapter 7. Systems of First Order Linear Equations

for the eigenvalues and eigenvectors of A. The characteristic equation is

— 2 — r 1

1

— - — r 2 1

r2 + r + 5 = 0;

(14)

therefore the eigenvalues are r1 = — 2 + i and r2 = — 1 — i. From Eq. (13) a straightforward calculation shows that the corresponding eigenvectors are

s(1) =

s(2) =

—1

Hence a fundamental set of solutions of the system (11) is

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