MA 36600 LECTURE NOTES: MONDAY, APRIL 13
Basic Theory of Systems of First Order Linear Equations
Solutions to Systems of Linear Differential Equations.
We continue to focus on a system of first
order differential equations in the form
x
0
1
=
p
11
(
t
)
x
1
+
p
12
(
t
)
x
2
+
· · ·
+
p
1
n
(
t
)
x
n
+
g
1
(
t
)
x
0
2
=
p
21
(
t
)
x
1
+
p
22
(
t
)
x
2
+
· · ·
+
p
2
n
(
t
)
x
n
+
g
2
(
t
)
.
.
.
.
.
.
x
0
m
=
p
m
1
(
t
)
x
1
+
p
m
2
(
t
)
x
2
+
· · ·
+
p
mn
(
t
)
x
n
+
g
m
(
t
)
Such a system contains
m
equations and
n
variables. When
m
=
n
, we can express this as
d
dt
x
=
P
(
t
)
x
+
g
(
t
)
.
We focus on a few special cases to gain intuition about the solutions of this system.
Example #1.
Consider first when
m
=
n
= 1. The first order equation
x
0
=
p
(
t
)
x
+
g
(
t
)
can be solved using integrating factors:
μ
(
t
) = exp

Z
t
p
(
τ
)
dτ
=
⇒
x
(
t
) =
1
μ
(
t
)
Z
t
μ
(
τ
)
g
(
τ
)
dτ
+
C
.
Hence the general solution is in the form
x
(
t
) =
C x
(1)
(
t
) +
x
(
p
)
(
t
), where
x
(1)
(
t
) =
1
μ
(
t
)
= exp
Z
t
p
(
τ
)
dτ
=
⇒
d
dt
x
(1)
=
p
(
t
)
x
(1)
is a solution to the homogeneous equation, and
x
(
p
)
(
t
) =
1
μ
(
t
)
Z
t
μ
(
τ
)
g
(
τ
)
dτ
is a particular solution to the nonhomogeneous equation.
Example #2.
Consider now when
m
=
n
= 2. The equation
x
00
=
q
(
t
)
x
+
p
(
t
)
x
0
+
g
(
t
)
can be transformed into a system of first order equations through the substitution
x
=
x
x
0
=
⇒
d
dt
x
=
0
1
q
(
t
)
p
(
t
)
x
+
0
g
(
t
)
.
Say that we have three functions
x
1
=
x
1
(
t
),
x
2
=
x
2
(
t
), and
X
=
X
(
t
) such that
i.
x
1
and
x
2
are solutions to the homogeneous equation
x
00

p
(
t
)
x
0

q
(
t
)
x
= 0
.
ii. Their Wronskian is nonzero, in terms of the function
W
(
x
1
, x
2
)
(
t
) =
x
1
(
t
)
x
0
2
(
t
)

x
0
1
(
t
)
x
2
(
t
) = det
x
1
(
t
)
x
2
(
t
)
x
0
1
(
t
)
x
0
2
(
t
)
=
C
exp
Z
t
p
(
τ
)
dτ
.
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 Spring '09
 MA
 Linear Equations, Equations, Trigraph, Elementary algebra

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