4
MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 9
so that we have the equation
0=(1
.
01)
48
·
10000+
1
−
(1
.
01)
48
1
−
(1
.
01)
b
=
⇒
0 = 16122
.
261+61
.
223
b
=
⇒
b
=
−
16122
.
261
61
.
223
=
−
263
.
338
.
Hence we must pay at least $263.34 each month in order to pay of the loan in 4 years.
Nonlinear Equations.
Recall the logistic diferential equation
dy
dt
=
ry
°
1
−
y
K
±
For some (positive) constants
r
and
K
. The associated diference equation is in the Form
∆
y
n
∆
t
n
=
ry
n
°
1
−
y
n
K
±
where
∆
t
n
=
h.
We can place this diference equation in the Form
y
n
+1
=
y
n
+
ry
n
°
1
−
y
n
K
±
h
=(1+
hr
)
y
n
²
1
−
hr
(1 +
hr
)
K
y
n
³
.
Make the substitution
u
n
=
hr
(1 +
hr
)
K
y
n
=
⇒
u
n
+1
=
ρu
n
(1
−
u
n
)
in terms oF the constant
ρ
=1+
hr
. We call this equation the
logistic diference equation
. Note that this
equation is not a linear equation.
We ask two questions:
#1: What are the equilibrium solutions
u
L
?
#2: Can we ±nd a Formula For the exact solution
u
n
, such as For linear equations?
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.
 Spring '09
 EdrayGoins

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