Chap
.
4
Systems of ODEs
.
Phase Plane
.
Qualitative Methods
Sec
.
4
.
1
Systems of ODEs as Models
Example 2
.
Spend time on Fig. 79 on p. 134 until you feel that you fully understand the difference between
(a) (the usual representation in calculus) and (b), because trajectories will play an important role
throughout this chapter. Try to understand the reasons for the following. The trajectory starts at the
origin. It reaches its highest point where
I
2
has a maximum (before
t
u003d
1). It has a vertical tangent where
I
1
has a maximum, shortly after
t
u003d
1. As
t
increases from there to
t
u003d
5, the trajectory goes downward
until it almost reaches the
I
1
axis at 3; this point is a limit as
t
u2192
u221e
. In terms of
t
the trajectory goes up
faster than it comes down.
Problem Set 4
.
1
.
Page 135
7
.
Electrical network
.
The problem amounts to the determination of the two arbitrary constants in a
general solution of a system of two ODEs in two unknown functions
I
1
and
I
2
, representing the currents
in an electrical network shown in Fig. 78 in Sec. 4.1. You will see that this is quite similar to the
corresponding task for a single secondorder ODE. That solution is given by (6), in components
I
1
ue0a2
t
ue0a3
u003d
2
c
1
e
u2212
2
t
u002b
c
2
e
u2212
0.8
t
u002b
3,
I
2
ue0a2
t
ue0a3
u003d
c
1
e
u2212
2
t
u002b
0.8
c
2
e
u2212
0.8
t
.
Setting
t
u003d
0 and using the given initial conditions
I
1
ue0a2
0
ue0a3
u003d
0,
I
2
ue0a2
0
ue0a3
u003d
u2212
3 gives
I
1
ue0a2
0
ue0a3
u003d
2
c
1
u002b
c
2
u002b
3
u003d
0
I
2
ue0a2
0
ue0a3
u003d
c
1
u002b
0.8
c
2
u003d
u2212
3.
(a)
(b)
From (a) you have
c
2
u003d
u2212
3
u2212
2
c
1
. From this and (b) you obtain
c
1
u002b
0.8
ue0a2
u2212
3
u2212
2
c
1
ue0a3
u003d
u2212
0.6
c
1
u2212
2.4
u003d
u2212
3,
hence
c
1
u003d
1.
Also
c
2
u003d
u2212
3
u2212
2
c
1
u003d
u2212
3
u2212
2
u003d
u2212
5. This yields the answer
I
1
ue0a2
t
ue0a3
u003d
2
e
u2212
2
t
u2212
5
e
u2212
0.8
t
u002b
3
I
2
ue0a2
t
ue0a3
u003d
e
u2212
2
t
u002b
0.8
ue0a2
u2212
5
ue0a3
e
u2212
0.8
t
u003d
e
u2212
2
t
u2212
4
e
u2212
0.8
t
.
You see that the limits are 3 and 0, respectively. Can you see this directly from Fig. 78 for physical
reasons?
11
.
Conversion of single ODEs to a system
is an important process, which always follows the pattern
shown in formulas (9) and (10) of Sec. 4.1. The present equation
y
u2032u2032
u2212
4
y
u003d
0 can be readily solved. A
general solution is
y
u003d
c
1
e
2
t
u002b
c
2
e
u2212
2
t
. The point of the problem is not to explain a (complicated) solution
method for a simple problem, but to explain the relation between systems and single ODEs and their
solutions. In the present case the formulas (9) and (10) give
y
1
u003d
y
,
y
2
u003d
y
u2032
and
y
1
u2032
u003d
y
2
y
2
u2032
u003d
4
y
1
(because the given equation can be written
y
u2032u2032
u003d
4
y
, hence
y
1
u2032u2032
u003d
4
y
1
, but
y
1
u2032u2032
u003d
y
2
u2032
). In matrix form (as in
Example 3 of the text) this is
y
u2032
u003d
Ay
u003d
0
1
4
0
y
.
The characteristic equation is
det
ue0a2
A
u2212
u03bb
I
ue0a3
u003d
u2212
u03bb
1
4
u2212
u03bb
u003d
u03bb
2
u2212
4
u003d
0.
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Chap. 4
Systems of ODEs. Phase Plane. Qualitative Methods
31
The eigenvalues are
u03bb
1
u003d
2 and
u03bb
2
u003d
u2212
2. For
u03bb
1
you obtain an eigenvector from (13) in Sec. 4.0 with
u03bb
u003d u03bb
1
, that is,
ue0a2
A
u2212
u03bb
1
I
ue0a3
x
u003d
u2212
2
1
4
u2212
2
x
1
x
2
u003d
u2212
2
x
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 Spring '12
 M.lee
 Linear Algebra, Constant of integration, Eigenvalue, eigenvector and eigenspace, general solution

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