Continuous dynamic systems
65
(3)
Complex conjugate roots
y
c
=
c
1
e
α
t
cos(
β
t
)
+
c
2
e
α
t
sin(
β
t
)
In fnding a solution to a linear nonhomogeneous equation, Four steps need to
be Followed:
Step 1
±ind the complementary solution
y
c
.
Step 2
±ind the general solution
y
h
by solving the higherorder equation
L
h
(
y
h
)
=
0
where
y
h
is determined From
L
(
y
) and
g
(
t
).
Step 3
Obtain
y
q
=
y
h
−
y
c
.
Step 4
Determine the unknown constant, the
undetermined coefFcients
,inthe
solution
y
q
by requiring
L
(
y
q
)
=
g
(
t
)
and substituting these into
y
q
,
giving the particular solution
y
p
.
Example 2.20
Suppose
y
±±
(
t
)
+
y
±
(
t
)
=
t
Step 1
This has the complementary solution
y
c
, which is the solution to the aux
iliary equation
x
2
+
x
=
0
x
(
x
+
1)
=
0
with solutions
r
=
0 and
s
=−
1 and
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
 Fall '11
 Dr.Gwartney
 Economics

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