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topic. That is not so at all. In fact, very often successful computer solutions of dif-
ferential equations rely heavily on analytical insight. In many cases we know a lot
about a solution without actually having computed it - see our arguments above
about increasing and decreasing the number of rabbits. We were able to figure out
quite a bit without solving anything. All such knowledge can be utilized either in
computing the solution or checking that the solution that we have computed is in
fact a reasonable approximation of the exact solution. Computer codes for nontriv-
ial problems almost always contain bugs, and methods for checking the validity of
a computed solution are therefore very important.
2.2.1
The Simplest Possible Model
In the simplest model above, the change in the rabbit population was given by an
explicit function that was independent of the population. More precisely, the model
reads
r
0
.t /
D
f.t/;
(2.22)
with r.0/
D
r
0
and where f
D
f.t/ is assumed to be a given function. Although
this model can be integrated directly, we will start by viewing it as a differential
equation. Suppose we want to find an approximation of r.t/ for t ranging from
9
t
D
0 to t
D
1: We discretize this problem by picking an integer N
>
1 and define
the time step
t
D
1=N
and time levels
t
n
D
nt
for n
D
0;1;:::;N: Note, in particular, that t
0
D
0 and t
N
D
1.Furthermore,we
let r
n
denote an approximation
10
of r.t
n
/: In order to derive an approximation of the
solution of (2.22), we will need the Taylor series, see Project 1.7.1 from Chap. 1.For
a sufficiently smooth function r
D
r.t/; we have
r.t
C
t/
D
r.t/
C
t r
0
.t /
C
O.t
2
/:
Consequently,
9
Suppose time t is measured in years. Then it makes sense to consider how the number of rabbits
changes from t
1:
10
It is important that you get this right: r.t/ is the correct solution, r.t
n
/ is the correct solution at
time t
D
0 to t
D
D
t
n
,andr
n
is the approximate solution at time t
n
: More specifically, we have
r
n
r.t
n
/:
Note that r
0
denotes both the correct and approximate solution at t
D
0: This is ok since the
solution is given at t
D
0.