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and one unstable. The issue of stability will also be discussed many times later in
this text.
(a) Consider the logistic model
s
0
D
s.1
s/;
s.0/
D
x;
where x
>
0 is the given initial state. Show, by direct differentiation, that
x
x
C
e
t
.1
x/
s.t/
D
is the analytical solution of this problem.
(b) Make a graph of the solution of this problem for x
D
0;0:2;0:4;:::;2:0 and
for 0
5: Discuss the stability with respect to changes of the initial data
based on the graphs.
(c) Let S
D
S.x/ denote the solution at t
D
1.Verifythat
6
t
6
x
x
C
e
1
.1
x/
;
S.x/
D
and make a plot of S as a function of x for 0
6
x
6
5:
(d) Show that
1
e.e
1
x
x
e
1
/
2
S
0
.x/
D
;
and plot S
0
for 0
5:
(e) Let " be a very small number. Then, by the Taylor series, we have
6
x
6
S.x
C
"/
S.x/
C
"S
0
.x/:
Use this observation to discuss the stability of the solution at time t
D
1 with
respect to perturbations in the initial data.
(f) Next we consider the initial value problem
u
0
D
u
.
u
1/;
u
.0/
D
x;
where x
0 again is the given initial state. Show that the analytical solution of
this problem is given by
>
x
x
C
e
t
.1
x/
:
u
.t /
D
