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u
0
.t /
u
.t /
D
v
0
.t /
v
.t /
(3.102)
and use direct integration to show that
u
2
.t /
C
v
2
.t /
D
u
0
C
v
0
:
(3.103)
(g) Suppose we consider system (
3.95
) with slightly different initial conditions, i.e.,
we consider the following system
u
0
.t /
D
v
.t /
u
.0/
D
u
0
;
v
0
.t /
D
u
.t /
(3.104)
v
.0/
D
v
0
:
Define
U
D
u
u
;
V
D
v
v
(3.105)
and show that
U
0
.t /
D
V.t/;
V
0
.t /
D
(3.106)
U.t/;
and use this to conclude that
.
u
.t /
u
.t //
2
C
.
v
.t /
v
.t //
2
D
.
u
0
u
0
/
2
C
.
v
0
v
0
/
2
:
(3.107)
What does this equation tell you about the stability of system (
3.95
) with respect
to perturbations in the initial conditions?
(h) Consider the system
u
0
.t /
D
v
.t /
u
.0/
D
1;
v
0
.t /
D
u
.t /
(3.108)
v
.0/
D
0:
Show that
u
.t /
D
cos.t /;
v
.t /
D
sin.t /
(3.109)
solves this system.
(i) Use the analytical solution (
3.109
) to test the accuracy of the scheme considered
in (a). More specifically, compute
j
u
.5/
u
N
j
j
u
.5/
j
j
v
.5/
v
N
j
j
v
.5/
j
e
t
D
C
;
(3.110)
