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(a) Derive, using Taylor series, the following explicit numerical scheme for system
(
3.95
),
D
u
n
t
v
n
;
u
nC1
(3.96)
v
nC1
D
v
n
C
t
u
n
:
Write a program that implements this scheme. The program should accept
u
0
and
v
0
and t as input parameters.
(b) Choose
u
0
D
1 and
v
0
D
0.Uset
D
1=100 and compute the solution from
t
D
0 to t
D
5. Plot the numerical solution as a function of t and make a graph
of it in the
u
-
v
space.
(c) Define
D
u
n
C
v
n
r
n
:
(3.97)
Compute r
n
for the time interval given above and plot the values. What proper-
ties do you think the solution
u
and
v
have?
(d) Let N be such that Nt
D
5. Compute
r
N
r
0
r
N
t
(3.98)
for t
D
1=100, 1=200, 1=300, 1=400,and1=500. Show that there is a constant
c such that
r
N
r
0
r
N
ct:
(3.99)
(e) You have obtained indications that
u
and
v
define a circle in the state space with
radius
u
0
C
v
0
1=2
and with center in .0; 0/. In order to investigate this from
an analytical point of view, we introduce the function
r.t/
D
u
2
.t /
C
v
2
.t /:
(3.100)
Use the differential equations in (
3.95
) to show that
r
0
.t /
D
0
for all t , and conclude that
u
2
.t /
C
v
2
.t /
D
u
0
C
v
0
(3.101)
for all t
0.
(f) As we saw in the text above, (
3.101
) can also be derived directly from system
(
3.95
). Show that
