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

ISBN 0-471-31999-6

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<(1)(t) =

(—1/2+1 )t

x(2)(t) =

1

—i

(—1/2—1 )t

(15)

(16)

To obtain a set of real-valued solutions, we must find the real and imaginary parts of either x(1) or x(2). In fact,

x(1) (t) = ( 1 ) e v2(cos t + i sin t) =

1 1 —t/2(

e—t/2coS e—t/2 sint

—t/2 sin + ^ e—t/2cos t

Hence

u(t ) = e

e—t/2

cos t sint

v(t ) = e

e—t/2

sin t cos t

(17)

(18)

is a set of real-valued solutions. To verify that u(t) and v(t) are linearly independent, we compute their Wronskian:

W(u, v)(t) =

e—t/2 cos t e—t/2 sint

e~(/2 sin t

e—1/2 coc t

= e 1 (cos2 t + sin2 t) = e 1.

Since the Wronskian is never zero, it follows that u( t) and v( t) constitute a fundamental set of (real-valued) solutions of the system (11).

The graphs of the solutions u(t) and v(t) are shown in Figure 7.6.2a. Since

u(0) =

v(0) =

the graphs of u(t) and v(t) pass through the points (1, 0) and (0, 1), respectively. Other solutions of the system (11) are linear combinations of u(t) and v(t), and graphs of a few of these solutions are also shown in Figure 7.6.2a. In all cases the solution approaches the origin along a spiral path as t ^ to; this is due to the fact that the solutions (18) are products of decaying exponential and sine or cosine factors. Some typical graphs of x1 versus t are shown in Figure 7.6.2b; each one represents a decaying oscillation in time.

Figure 7.6.2a is typical of all second order systems x; = Ax whose eigenvalues are complex with negative real part. The origin is called a spiral point and is asymptotically stable because all trajectories approach it as t increases. For a system whose eigenvalues have a positive real part the trajectories are similar to those in Figure 7.6.2a, but the direction of motion is away from the origin and the trajectories become unbounded. In this case, the origin is unstable. If the real part of the eigenvalues is zero, then the trajectories neither approach the origin nor become unbounded but instead traverse

1

1

i

1

i

7.6 Complex Eigenvalues

FIGURE 7.6.2 (a) Trajectories of the system (11); the origin is a spiral point. (b) Plots of x1 versus t for the system (11).

repeatedly a closed curve about the origin. In this case the origin is called a center and is said to be stable, but not asymptotically stable. In all three cases the direction of motion may be either clockwise, as in this example, or counterclockwise, depending on the elements of the coefficient matrix A.

EXAMPLE

2

The electric circuit shown in Figure 7.6.3 is described by the system of differential equations

d ( A (-1 -A ( A

dt \vj~\ 2 -a W (19)

where I is the current through the inductor and V is the voltage drop across the capacitor. These equations were derived in Problem 19 of Section 7.1. Suppose that at time t = 0 the current is 2 amperes and the voltage drop is 2 volts. Find I(t) and V(t) at any time.

R = 1 ohm

FIGURE 7.6.3 The circuit in Example 2.

388

Chapter 7. Systems of First Order Linear Equations

Assuming that

(v

we obtain the algebraic equations

A = &rt, (20)

-Vr ---)(1)=(S)- (21)

The eigenvalues are determined from the condition

= r2 + 2r + 3 = 0; (22)

-1 - r -1

2 -1 - r

thus rl = : 1 + V2 i and r2 = -1 — V2 i. The corresponding eigenvectors are then found from Eq. (21), namely,

« =U/2 i) ? « (2, = (i ,)? (23)

The complex-valued solution corresponding to r1 and «(1) is

« {V)er1t e(_1+^21)1

= ^ ^ e~l(cosV21 + i smV2 t)

= _-ff cos^2 f\,/p-f/ sin V21 \ (24)

\V2sinV2 y \—V2cosV2 y

The real and imaginary parts of this solution form a pair of linearly independent real-valued solutions of Eq. (19):

,0. -t ( cosV21 \ -t f sinV21 \

U(t> = e UsmVl ,) ? V(t> = e (-V2coSV2 - (25)

Hence the general solution of Eqs. (19) is

I^ = v- (J?!.^ ) + c2e-' ( A,1 - (26)

V/ 1 \V2sinV21) 2 V^V2cosV21

Upon imposing the initial conditions

V}yJ ~ \2

we find that

c° (0)+c2 f-v2) = (2

A (0) = ( 2 ) , (27)

(28)

Thus q = 2 and c2 = —V2. The solution of the stated problem is then given by Eq. (26) with these values of q and c2. The graph of the solution is shown in Figure

7.6.4. The trajectory spirals counterclockwise and rapidly approaches the origin, due to the factor e-t.

7.6 Complex Eigenvalues

389

EXAMPLE

3

FIGURE 7.6.4 Solution of the initial value problem in Example 2.

The system

x =< -2 2) x (29)

contains a parameter a. Describe how the solutions depend qualitatively on a; in particular, find the critical values of a at which the qualitative behavior of the trajectories in the phase plane changes markedly.

The behavior of the trajectories is controlled by the eigenvalues of the coefficient matrix. The characteristic equation is

r2 — ar + 4 = 0, (30)

so the eigenvalues are

a ± Va2 — 16

r = ^ • (31)

From Eq. (31) it follows that the eigenvalues are complex conjugates for —4 < a < 4 and are real otherwise. Thus two critical values are a = —4 and a = 4, where the eigenvalues change from real to complex, or vice versa. For a < —4 both eigenvalues are negative, so all trajectories approach the origin, which is an asymptotically stable node. For a > 4 both eigenvalues are positive, so the origin is again a node, this time unstable; all trajectories (except x = 0) become unbounded. In the intermediate range, —4 < a < 4, the eigenvalues are complex and the trajectories are spirals. However, for —4 < a < 0 the real part of the eigenvalues is negative, the spirals are directed inward, and the origin is asymptotically stable, while for 0 < a < 4 the opposite is the case and the origin is unstable. Thus a = 0 is also a critical value where the direction of the spirals changes from inward to outward. For this value of a the origin is a center and the trajectories are closed curves about the origin, corresponding to solutions that are periodic in time. The other critical values, a = ±4, yield eigenvalues that are real and equal. In this case the origin is again a node, but the phase portrait differs somewhat from those in Section 7.5. We take up this case in Section 7.8.

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