systems_intro_real - System of Eq Distinct and Real...

System of Eq. – Distinct and Real eigenvalues April 16, 2011
1 Introduction/Motivation
Suppose we’re given the follow system of differential equations and asked to solve it:
Note: here
x is the
dependent
variable,
and t is the
indepen-
dent
variable
x
0
1
( t ) = 1 x
1
( t ) + 1 x
2
( t ) (1)
x
0
2
( t ) = 4 x
1
( t ) + 1 x
2
( t ) (2)
If x
2
( t ) did not appear in the first equation, we’d have a simple first-order equation, and
could certainly solve it. Therefore, it is only reasonable to try and eliminate x
2
( t ) from
Eq. ( 2 ). Using Eq. ( 1 ), we find
x
2
( t ) = x
0
1
( t ) - x
1
( t ) . (3)
Plugging this expression into Eq. ( 2 ) we have
( x
0
1
( t ) - x
1
( t ))
0
= 4 x
1
( t ) + 1 ( x
0
1
( t ) - x
1
( t )) .
Expanding the left-hand-side:
x
00
1
( t ) - x
0
1
( t ) = 4 x
1
( t ) + x
0
1
( t ) - x
1
( t ) ,
or finally,
x
00
1
- x
0
1
- 3 x
1
= 0 . (4)
We can easily find the solution to above equation. The answer ends up being
x
1
( t ) = c
1
e
3 t
+ c
2
e
- t
.
Using above expression and Eq. ( 3 ), we can easily solve for x
2
( t ) . The result turns out to
!
We won’t
absorb the
2’s into the
c
1
and c
2
here.
be
x
2
( t ) = 2 c
1
e
3 t
- 2 c
2
e
- t
. (5)
To make a few observations, we will re-write our results in the following vector form:
x
1
x
2
=
c
1
e 3 t
2 c
1
e 3 t
+
c
2
e - t
- 2 c
2
e - t
= c
1
e 3 t
2 e
3 t
+ c
2
e - t
- 2 e
- t
= c
1
1
2
e
3 t
+ c
2
1
- 2
e
- t
Key observations:
1 c HF, 2011

System of Eq. – Distinct and Real eigenvalues April 16, 2011
1. We started with a system of first -order equations, and ended up with a second -order
equation.
2. Our solution is in the following form:
x
1
x
2
=
ξ
1
ξ
2
e
rt
,
where ξ
1
and ξ
2
and constants. In a more compact form, above equation may be
written as
x ( t ) = ξ e
rt
, (6)
where x (t) (bold) denotes a vector-valued function.
This should motivate us to seek two solutions having the form of above equation. In other
words, we seek a general solution of the form
x ( t ) = c
1
ξ
(1)
e
r
1
( t )
+ c
2
ξ
(2)
e
r
2
( t )
.