Consider the differential equation dP/dt = kP 1+c , where k > 0 and c 0.
Question:
Consider the differential equation
dP/dt = kP1+c,
where k > 0 and c ≥ 0. In Section 3.1 we saw that in thecase 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 on page 85.
(a) Suppose for c = 0.01 that the nonlinear differential equation
dP/dt = kP1.01, 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 5months.
(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.
(c) From part (a), what is P(50)? P(100)?
Step by Step Answer:
A First Course in Differential Equations with Modeling Applications
ISBN: 978-1305965720
11th edition
Authors: Dennis G. Zill