2130_Chapter7_Notes - APMA 2130 Section 7.1 Systems of...

APMA 2130
Section 7.1: Systems of First Order Linear Equations
Post-Lecture Handout
Dr. Bernard Fulgham
University of Virginia
July 23, 2013
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Exa 2 (slide 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Exa 2 (slide 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Matrix Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1

Introduction
System of n first order ordinary differential equations:
x
′
1
= F
1
( t, x
1
, x
2
, . . . , x
n
) ,
x
′
2
= F
2
( t, x
1
, x
2
, . . . , x
n
) ,
.
.
.
x
′
n
= F
n
( t, x
1
, x
2
, . . . , x
n
) ,
where each x
k
is a function of t.
If each F
k
is a linear function of x
1
, . . . , x
n
, then the system of equations is linear ; otherwise, it is nonlinear .
Systems of higher order differential equations can similarly be defined.
2 / 9
Linear Systems
Linear system of equations:
x
′
1
= p
11
( t ) x
1
+ p
12
( t ) x
2
+ ··· + p
1 n
( t ) x
n
+ g
1
( t ) ,
x
′
2
= p
21
( t ) x
1
+ p
22
( t ) x
2
+ + p
2 n
( t ) x
n
+ g
2
( t ) ,
.
.
.
x
′
n
= p
n 1
( t ) x
1
+ p
n 2
( t ) x
2
+ + p
nn
( t ) x
n
+ g
n
( t ) .
If each g
k
( t ) = 0 , then the system is homogeneous ; otherwise, it is nonhomogeneous .
Every n th order linear equation can be converted to a system of n first order linear equations.
3 / 9
2

Example 1
Third order linear equation:
y
′′′
+ 2 y
′′
+ 3 y
′
+ 4 y = g ( t ) .
Define x
1
= y, x
2
= y
′
, x
3
= y
′′
, then
x
′
1
= y
′
= ⇒ x
′
1
= x
2
,
x
′
2
= y
′′
= ⇒ x
′
2
= x
3
,
x
′
3
= y
′′′
= - 4 y - 3 y
′
- 2 y
′′
+ g ( t )
= ⇒ x
′
3
= - 4 x
1
- 3 x
2
- 2 x
3
+ g ( t ) .
System of three first order linear equations:
x
′
1
= x
2
, x
′
2
= x
3
,
x
′
3
= - 4 x
1
- 3 x
2
- 2 x
3
+ g ( t ) .
4 / 9
Theorem
Linear system of n first order equations:
x
′
1
= p
11
( t ) x
1
+ p
12
( t ) x
2
+ ··· + p
1 n
( t ) x
n
+ g
1
( t ) ,
x
′
2
= p
21
( t ) x
1
+ p
22
( t ) x
2
+ + p
2 n
( t ) x
n
+ g
2
( t ) ,
.
.
.
x
′
n
= p
n 1
( t ) x
1
+ p
n 2
( t ) x
2
+ + p
nn
( t ) x
n
+ g
n
( t ) .
Initial conditions: x
1
( t
0
) = x
0
1
, x
2
( t
0
) = x
0
2
, . . . , x
n
( t
0
) = x
0
n
.
Questions
When does a solution exist? If a solution exists, is it unique?
Theorem 7.1.2 If the coefficient functions p
11
( t ) , p
12
( t ) , . . . , p
nn
( t ) , g
1
( t ) , . . . , g
n
( t ) are continuous on an
open interval I = ( α, β ) , then the system with initial conditions has a unique solution
x
1
= φ
1
( t ) , x
2
= φ
2
( t ) , . . . , x
n
= φ
n
( t ) throughout the entire interval ( α, β ) .
5 / 9
3

Example 2
x
′
1
= 3 x
1
- 2 x
2
, x
1
(0) = 3
x
′
2
= 2 x
1
- 2 x
2
, x
2
(0) = 1 / 2
Solve this system by converting to a second order linear equation:
x
′
1
= 3 x
1
- 2 x
2
= ⇒ x
2
=
1
2
(3 x
1
- x
′
1
) = ⇒ x
′
2
=
1
2
(3 x
′
1
- x
′′
1
) .
Now substitute for x
2
and x
′
2
in the second equation:
x
′
2
= 2 x
1
- 2 x
2
= ⇒
1
2
(3 x
′
1
- x
′′
1
) = 2 x
1
- (3 x
1
- x
′
1
)
= ⇒ 3 x
′
1
- x
′′
1
= 4 x
1
- 6 x
1
+ 2 x
′
1
= ⇒ x
′′
1
- x
′
1
- 2 x
1
= 0 .
Note: the original system was homogeneous, and so the resulting equation is also homogeneous.
6 / 9
Exa 2 (slide 2)
x
′
1
= 3 x
1
- 2 x
2
x
′
2
= 2 x
1
- 2 x
2
= ⇒ x
′′
1
- x
′
1
- 2 x
1
= 0
Characteristic equation:
r
2
- r - 2 = 0 = ⇒ ( r - 2)( r + 1) = 0 = ⇒ r = 2 , - 1
= ⇒ x
1
= c
1
e
2 t
+ c
2
e
− t
= ⇒ x
′
1
= 2 c
1
e
2 t
- c
2
e
− t
, and
x
2
=
1
2
(3 x
1
- x
′
1
) =
1
2
b
3( c
1
e
2 t
+ c
2
e
− t
) - (2 c
1
e
2 t
- c
2
e
− t
)
B
=
1
2
(
3 c
1
e
2 t
+ 3 c
2
e
− t
- 2 c
1
e
2 t
+ c
2
e
− t
)
=
1
2
c
1
e
2 t
+ 2 c
2
e
− t
.