Page 350 - Advanced engineering mathematics

330 CHAPTER 10 Systems of Linear Differential Equations

Given a solution, we can think of the point (x(t), y(t)) as moving along a curve or trajectory
in the plane as t, often thought of as time, increases. A copy of the plane, with trajectories drawn
through various points, is called a phase portrait for X = AX. Phase portraits provide visual

insight into how the trajectories move and how solutions behave. Because of the uniqueness of
solutions of initial value problems, there can be only one trajectory through any given point in the
plane. Furthermore, two distinct trajectories cannot intersect, because at the point of intersection
there would be two trajectories through the same point, and these would both be solutions of the
same initial value problem.
Often phase portraits are drawn within a direction field. Recall from Chapter 5 that a direc-
tion field consists of short line segments of tangents to trajectories. These tangent segments
outline the way solution curves move in the plane, and provide a flow pattern for the trajectories.
Arrows drawn along these segments indicate the direction of the flow as t increases.

For the system X =AX, the origin (0,0) plays a special role. This point is actually the graph
of the constant solution
x(t) = 0, y(t) = 0 for all t
which is the solution of the unique initial value problem

0

X = AX;X(0) = .
0
No other trajectory can pass through the origin, because then two distinct trajectories would
intersect.
We will now examine trajectories of X = AX, paying particular attention to their behavior

near the origin. Because solutions are determined by the eigenvalues of A,wewill usetheseto
distinguish cases.
Case 1: Real Distinct Eigenvalues λ and μ of the Same Sign

Let associated eigenvectors be E 1 and E 2 . Because λ and μ are distinct, these eigenvectors are
linearly independent and the general solution is

x(t) λt μt
X(t) = = c 1 E 1 e + c 2 E 2 e .
y(t)
Represent the vectors E 1 and E 2 as vectors from the origin, as in Figure 10.14. Draw L 1 and L 2 ,
respectively, through the origin along these vectors. These will serve as guidelines in drawing
trajectories.

Case 1(a): The Eigenvalues are Negative, say λ<μ< 0
μt
λt
Now e → 0 and e → 0as t →∞,so X(t) → (0,0) and each trajectory approaches the ori-
gin. This can happen in three ways, depending on an initial point P 0 : (x 0 , y 0 ) we choose for a
trajectory to pass through at time t = 0. These possibilities are as follows.
1. If P 0 is on L 1 , then c 2 = 0 and
λt
X(t) = c 1 e .
For any t this is a scalar multiple of E 1 , so the trajectory through P 0 is part of L 1 , with arrows
along it pointing toward the origin because the trajectory moves toward the origin as time
increases. This is the trajectory T 1 of Figure 10.15.
2. If P 0 is on L 2 , then c 1 = 0 and now
μt
X(t) = c 2 e .
This trajectory is part of the line L 2 , with arrows of the direction field indicating that it also
approaches the origin as t increases. This is the trajectory T 2 of Figure 10.15.

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October 14, 2010 20:32 THM/NEIL Page-330 27410_10_ch10_p295-342