MATH2860U:
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
first-order 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: