Consider the differential equation dP/dt = kP^(1+c) where k > 0 and c ≥ 0. In Section 3.1 we saw that in the case c = 0

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Consider the differential equation
dP/dt = kP^(1+c)
where k >
0 and c ≥ 0. In Section 3.1
we saw that in the case c = 0 the
linear differential
equation dP/dt = kP
is a mathematical model of a
population P(t) that exhibits unbounded
growth over the infinite time interval
[0, ∞), that
is, P(t) → ∞
as t → ∞.See Example 1 in that
section.
(a) Suppose for c =
0.01 that the nonlinear differential equation
dP/dt = kP^1.01, k>0
is a mathematical model for a population of small animals, where
time t is measured in months. Solve the
differential equation subject to the initial condition P(0) = 10
and the fact that the animal population has doubled
in 7 months. (Round the coefficient
of t to six decimal places.)
P(t) = _____
(b) The differential equation in part (a)
is called a doomsday equation because
the population P(t) exhibits unbounded
growth over a finite time interval (0, T), that is, there
is some time T such
that P(t) → ∞ as t → T −.
Find T. (Round your answer to the nearest month.)
T= _____
(c) From part (a), what
is P(70)? P(140)?
(Round your answers to the nearest whole number.)
P(70) =
P(140) =