Information Technology Reference
In-Depth Information
so
ln.
r.t/
r
0
/
D
at
and thus
r.t/
D
r
0
e
at
:
(2.10)
From this model it follows that the number of rabbits increases exponentially in
time. Note that this is a highly nontrivial observation; we only assume that the rate
of change is proportional to the number of rabbits and then it follows that the growth
has to be exponential. And, in fact, this is often a fairly accurate description.
Uniqueness
We mentioned above that the function r is uniquely determined by the two equations
r
0
.t /
D
ar
.t /;
(2.11)
r.0/
D
r
0
:
Then we constructed the solution
r.t/
D
r
0
e
at
;
but we did not ask whether other solutions were possible. This is a general issue;
you cannot claim that a solution is unique simply by constructing one. So let us
assume that there are two solutions, r and q,wherer satisfies (
2.11
)andq satisfies
the same two conditions, i.e.,
q
0
.t /
D
aq
.t /;
(2.12)
q.0/
D
q
0
:
Define the difference
E.t/
D
r.t/
q.t/
and assume that q
0
D
r
0
: Then
E
0
D
E.0/
D
r.0/
q.0/
D
0:
Moreover,
E
0
.t /
D
r
0
.t /
q
0
.t /
D
a.r.t/
q.t//
D
aE
.t /:
