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Department of Mathematical
Sciences
University of Delaware
Prof. T. Angell
September 23, 2009
Mathematics 351
Exercise Sheet 3
Exercise 11
: A growth model that is used in such diverse Felds as Economics and Medicine (in this
latter case, to model the size of animal tumors) is the equation
dx
dt
=
rx
ln
°
K
x
±
,
where
r
and
K
are positive constants.
(a) Using an argument similar to that for the Logistic Equation, show that any population satisfying
x
(0) =
x
o
>
0 will satisfy lim
t
→∞
x
(
t
)=
K
.
(b) Letting
y
=ln(
x
), Fnd the exact solution satisfying
x
(0) =
x
o
>K
.
Exercise 12
: (Another part of the
Principle of Superposition
.) Suppose that
x
1
(
t
) is a solution of the
di±erential equation ˙
x
=
a
(
t
)
x
+
b
1
(
t
) and that
x
2
(
t
) is a solution of ˙
x
=
a
(
t
)
x
(
t
)+
b
2
(
t
). Show that the
sum of the two functions
x
1
(
t
)+
x
2
(
t
) is a solution of the di±erential equation ˙
x
=
a
(
t
)
x
+[
b
1
(
t
)+
b
2
(
t
)].
Exercise 13:
The secretion of hormones into the blood is often a periodic activity. If a hormone
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