Continuous dynamic systems
39
Furthermore, if we have the initial condition that
y
(0)
=
y
0
, then we can
solve for
c
y
0
=−
a
b
+
c
c
=
y
0
+
a
b
Hence, we have the solution to the initial value problem of
y
(
t
)
a
b
+
±
y
0
+
a
b
²
e
bt
Example 2.11
Suppose we have the initial value problem
dy
dt
=
2
y
+
4
ty
(0)
=
1
Applying the fourstep procedure we have
Step 1
−
2
y
=
4
t
Step 2
µ
(
t
)
=
e
³
−
2
=
e
−
2
t
Step 3
e
−
2
t
y
=
´
4
te
−
2
t
+
c
Step 4
e
−
2
t
y
2
te
−
2
t
−
e
−
2
t
+
c
Or
y
2
t
−
1
+
ce
2
t
Since
y
(0)
=
1, then
1
1
+
c
c
=
2
Hence,
y
(
t
)
2
t
−
1
+
2
e
2
t
2.3 Compound interest
If an amount
A
is compounded annually at a market interest rate of
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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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