713
SECTION 7.2
.
.
Separable Differential Equations
577
argon40 does not remain in molten lava. For a ratio of 0.00012,
find the age of the rock. Comment on why this method is used
to date very old rocks.
3.
Three confused hunting dogs start at the vertices of an equi
lateral triangle of side 1. Each dog runs with a constant speed
aimed directly at the dog that is positioned clockwise from it.
The chase stops when the dogs meet in the middle (having
grabbed each other by their tails). How far does each dog run?
[Hints: Represent the position of each dog in polar coordinates
(
r
,θ
) with the center of the triangle at the origin. By symme
try, each dog has the same
r
value, and if one dog has angle
θ
, then it is aimed at the dog with angle
θ
−
2
π
3
. Set up a
differential equation for the motion of one dog and show that
there is a solution if
r
(
θ
)
=
√
3
r
. Use the arc length formula
L
=
θ
2
θ
1
[
r
(
θ
)]
2
+
[
r
(
θ
)]
2
d
θ.
]
4.
To generalize exercise 3, suppose that there are
n
dogs starting
at the vertices of a regular
n
gon of side
s
. If
α
is the interior
angle from the center of the
n
gon to adjacent vertices, show
that the distance run by each dog equals
s
1
−
cos
α
. What hap
pens to the distance as
n
increases without bound? Explain this
in terms of the paths of the dogs.
7.2
SEPARABLE DIFFERENTIAL EQUATIONS
In section 7.1, we solved two different differential equations:
y
(
t
)
=
ky
(
t
)
and
y
(
t
)
=
k
[
y
(
t
)
−
T
a
]
.
These are both examples of
separable
differential equations. We will examine this type
of equation at some length in this section. First, we consider the more general
firstorder
ordinary differential equation
y
=
f
(
x
,
y
)
.
(2.1)
Here, the derivative
y
of some unknown function
y
(
x
) is given as a function
f
of both
x
and
y
. Our objective is to find some function
y
(
x
) (a
solution
) that satisfies equation (2.1). The
equation is
firstorder,
since it involves only the first derivative of the unknown function.
We will consider the case where the
x
’s and
y
’s can be separated. We call equation (2.1)
separable
if we can separate the variables, i.e., if we can rewrite it in the form
g
(
y
)
y
=
h
(
x
)
,
where all of the
x
’s are on one side of the equation and all of the
y
’s are on the other side.
EXAMPLE 2.1
A Separable Differential Equation
Determine whether the differential equation
y
=
xy
2
−
2
xy
is separable.
Solution
Notice that this equation is separable, since we can rewrite it as
y
=
x
(
y
2
−
2
y
)
and then divide by (
y
2
−
2
y
) (assuming this is not zero), to obtain
1
y
2
−
2
y
y
=
x
.
NOTE
Do not be distracted by the letter
used for the independent variable.
We frequently use the independent
variable
x
, as in equation (2.1).
Whenever the independent
variable represents time, we use
t
as the independent variable, in
order to reinforce this connection,
as we did in example 1.2. There,
the equation describing
radioactive decay was given as
y
(
t
)
=
ky
(
t
)
.
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CHAPTER 7
.
.
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 Spring '10
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 IVP, Logistic function

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