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(k) Show that the requirement
@F
@F
@
D
D 0
leads to the linear equations
10˛ C 9 X
nD0
! D
9 X
d n
c n
nD0
(5.166)
9 X
! ˛ C 9 X
nD0
! D
9 X
d n
d n
c n d n :
nD0
nD0
(l) Solve the linear system ( 5.166 ) with respect to ˛ and .Verifythat
˛ 0:0368
and
0:0040:
(5.167)
(m) Use the fact that D ˛=ˇ to show that
ˇ 9:2 :
(5.168)
(n) Show that the solution of the logistic model with an initial data y 0 D 6:08,
which corresponds to the population on the Earth in the year 2000, is given by
55:94
6:08 C 3:12 e 0:0368 t
y.t/ D
:
(5.169)
(o) Plot y.t/ given by ( 5.169 )andp.t/ given by ( 5.101 ).
 
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