Elementary differential equations 7th edition  Boyce W.E
ISBN 0471319996
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To construct the general solution of the system (1) we proceed by analogy with the treatment of second order linear equations in Section 3.1. Thus we seek solutions of Eq. (1) of the form
x = (3)
where the exponent r and the constant vector ? are to be determined. Substituting from Eq. (3) for x in the system (1) gives
r?ert = A?ert.
Upon canceling the nonzero scalar factor ert we obtain A? = r?, or
(A — rI)? = 0, (4)
where I is the n x n identity matrix. Thus, to solve the system of differential equations (1), we must solve the system of algebraic equations (4). This latter problem is precisely the one that determines the eigenvalues and eigenvectors of the matrix A. Therefore the vector x given by Eq. (3) is a solution of Eq. (1) provided that r is an eigenvalue and ? an associated eigenvector of the coefficient matrix A.
The following two examples illustrate the solution procedure in the case of 2 x 2 coefficient matrices. We also show how to construct the corresponding phase portraits. Later in the section we return to a further discussion of the general n x n system.
374
Chapter 7. Systems of First Order Linear Equations
EXAMPLE
1
Consider the system
x
1 1 4 1
(5)
Plot a direction field and determine the qualitative behavior of solutions. Then find the general solution and draw several trajectories.
A direction field for this system is shown in Figure 7.5.1. From this figure it is easy to see that a typical solution departs from the neighborhood of the origin and ultimately has a slope of approximately 2 in either the first or third quadrant.
\ \ \ V
I I I \ i I I I i I I i I
l i I

HI /. /
\ X* \ \ V \
\ \ N \ \ \ i \ \
I i v / t \
/ I I t I I
2
1 
/ / / / / / / / / / / / / / / / / / / /
/ / / / / ////
/ / / /
/ / / /
/ / / /
/ / / /
/ / / t t / / t t / / ?
t / / t
/ / / / / / / / / / / / / / / / / /
1 /
/
/
/
/ / / / / / / / / / / /
/ 1 / /
/ /
/ / .
2 / ? ? ^
{ I \
 V ^ N
1
? t 1 \ ? \ t \ \ \
s' 
V •
N N \ \
? V \ \ \ \ \ '*N \ \
FIGURE 7.5.1 Direction field for the system (5).
To find solutions explicitly we assume that x = ^ert, and substitute for x in Eq. (5). We are led to the system of algebraic equations
1  r 1 4 1  r
=
%2,
(6)
Equations (6) have a nontrivial solution if and only if the determinant of coefficients is zero. Thus allowable values of r are found from the equation
1  r 1 4 1  r
= (1  r)2  4
r2 2r 3 0.
(7)
Equation (7) has the roots r1 = 3 and r2 = 1; these are the eigenvalues of the coefficient matrix in Eq. (5). If r = 3, then the system (6) reduces to the single equation
2f1 + ?2 = °.
Thus ?2 = 2?1 and the eigenvector corresponding to r1 = 3 can be taken as
t(1) =
(8)
(9)
2
7.5 Homogeneous Linear Systems with Constant Coefficients
375
Similarly, corresponding to r2 — 1, we find that ?2 — — 2f1, so the eigenvector is
?(2) — ( . (10)
—2J'
The corresponding solutions of the differential equation are
x'"(.t)—(2) ?, x2'(t)—(. (11)
The Wronskian of these solutions is
W[x(1), x(2)](t) —
e3t e—t
2e3t —2e~t
= — 4e2t, (12)
which is never zero. Hence the solutions x(1) and x(2) form a fundamental set, and the general solution of the system (5) is
x — c1x(1)(t) + c2x(2)(t)
— q(T)e3t+ci( ^e—t’ (13)
^2) ^ 2\—2,
where c1 and c2 are arbitrary constants.
To visualize the solution (13) it is helpful to consider its graph in the x1 x2plane for various values of the constants c1 and c2. We start with x — c1x(1)(t), or in scalar form
x1 — c1e3t, x2 — 2c1e3t.
By eliminating t between these two equations, we see that this solution lies on the straight line x2 — 2^1; see Figure 7.5.2a. This is the line through the origin in the direction of the eigenvector ?(1). If we look on the solution as the trajectory of a moving particle, then the particle is in the first quadrant when c1 > 0 and in the third quadrant when c1 < 0. In either case the particle departs from the origin as t increases. Next consider x — c2x(2)(t), or
x1 — c2e—t, x2 — — 2c2e— t.
This solution lies on the line x2 — — 2x1, whose direction is determined by the eigenvector ?(2). The solution is in the fourth quadrant when c2 > 0 and in the second quadrant when c2 < 0, as shown in Figure 7.5.2a. In both cases the particle moves toward the origin as t increases. The solution (13) is a combination of x(1) (t) and x(2) (t). For large t the term c1x(1)(t) is dominant and the term c2x(2)(t) becomes negligible. Thus all solutions for which c1 — 0 are asymptotic to the line x2 — 2x1 as t ^ro. Similarly, all solutions for which c2 — 0 are asymptotic to the line x2 — — 2x1 as t ^ — cro. The graphs of several solutions are shown in Figure 7.5.2a. The pattern of trajectories in this figure is typical of all second order systems x; — Ax for which the eigenvalues are real and of opposite signs. The origin is called a saddle point in this case. Saddle points are always unstable because almost all trajectories depart from them as t increases.