MATH2860U:
Chapter 8 cont.
..
1
SYSTEMS OF LINEAR FIRST ORDER
DIFFERENTIAL EQUATIONS
Preliminary Theory (Section 8.1 of Zill and Cullen, pg. 304)
Recall:
Earlier in the course, we did some work setting up systems of differential
equations, but we haven’t yet talked about how to solve them.
Being able to solve systems of equations is very important in applications.
Application (coupled springmass system):
Source of Image:
http://aix1.uottawa.ca/~jkhoury/oscillations.htm
)
(
)
(
1
1
1
1
2
2
2
1
2
1
t
F
x
k
x
x
k
dt
x
d
m
)
(
)
(
2
1
2
2
2
3
2
2
2
2
t
F
x
x
k
x
k
dt
x
d
m
Application (parallel LRC circuit):
Source of Image:
Modified from http://badger.physics.wisc.edu/lab/manual2/E9.html
L
V
dt
dI
RC
V
C
I
dt
dV
Application (double pendulum):
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View Full DocumentMATH2860U:
Chapter 8 cont.
..
2
Before we go on to solve some linear systems, let’s first introduce some terminology and
understand what it means to be a solution.
Definition:
A
firstorder system of differential equations
is a set of
n
simultaneous
differential equations in
n
unknown functions
)
(
,
),
(
),
(
2
1
t
x
t
x
t
x
n
.
It has the form
)
...,
,
,
,
(
)
...,
,
,
,
(
)
...,
,
,
,
(
2
1
2
1
2
2
2
1
1
1
n
n
n
n
n
x
x
x
t
g
x
x
x
x
t
g
x
x
x
x
t
g
x
Definition:
If each of the functions
n
g
g
g
,
,
,
2
1
is a linear function of the dependent
variables
n
x
x
x
,
,
,
2
1
, then the system of equations is said to be
linear
.
Thus, the
normal form of
n
first order linear equations is:
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
1
1
2
2
1
21
2
1
1
1
11
1
t
f
x
t
a
x
t
a
x
t
f
x
t
a
x
t
a
x
t
f
x
t
a
x
t
a
x
n
n
nn
n
n
n
n
n
n
Definition:
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 Spring '10
 Kidnan
 Differential Equations, ORDER DIFFERENTIAL EQUATIONS, Homogeneous Linear Systems

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