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(c) Choose the value of t that you found in (b) and use the program to play around
with various values of F
0
and S
0
. Are the solutions always periodic? What
happens if you choose F
0
D
1 and S
0
D
2?
˘
Exercise 3.2.
We observed above that the solution of the simple system
F
0
D
1
S; F.0/
D
F
0
;
(3.71)
S
0
D
F
1; S.0/
D
S
0
;
satisfies
r.t/
D
r.0/;
(3.72)
where
r.t/
D
.F .t /
1/
2
C
.S.t /
1/
2
:
(3.73)
The purpose of this exercise is to derive a similar property for the system
F
0
D
.2
S/F; F.0/
D
F
0
;
(3.74)
S
0
D
.F
1/S; S.0/
D
S
0
:
(a) Show that
1
F
0
D
2
S
1
S
0
;
1
F
(3.75)
where F and S solve (
3.74
).
(b) Integrate (
3.75
) from 0 to t and show that
F.t/
ln.F .t //
.F
0
ln.F
0
//
D
2 ln.S.t //
S.t/
.2 ln.S
0
/
S
0
/: (3.76)
(c) Show that
e
F
F
e
S
S
2
e
F
0
e
S
0
D
:
(3.77)
(d) Choose F
0
D
1:9, S
0
D
0:1, and define the constant
e
F
0
e
S
0
K
0
D
:
(3.78)
