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Consider the differential equation OP - kpl +c where k > 0 and c2 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, 0), that is, P(t) o as t - 00. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation dP = 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 10 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) - o as t-T. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what P(90)? P(180)? (Round your answers to the nearest whole number.) P(90) = P(180) =

Consider the differential equation OP - kpl +c where k > 0 and c2 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, 0), that is, P(t) o as t - 00. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation dP = 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 10 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) - o as t-T. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what P(90)? P(180)? (Round your answers to the nearest whole number.) P(90) = P(180) =

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 2EQ
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Consider the differential equation dP kp1 + c = dt where k > 0 and c 2 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, 0), that is, P(t) → o as t → ∞. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation dP kp1.01, k > 0, dt 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 10 months. (Round the coefficient of t to six decimal places.) P(t) = (b) The differential equation in part (a) is called a doomsd equation because the population P( exhibits unbounded growth over finite time interval (0, T), that is, there is som time T such that → o as t –→T¯. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(90)? P(180)? (Round your answers to the nearest whole number.) P(90) P(180)
Transcribed Image Text:Consider the differential equation dP kp1 + c = dt where k > 0 and c 2 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, 0), that is, P(t) → o as t → ∞. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation dP kp1.01, k > 0, dt 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 10 months. (Round the coefficient of t to six decimal places.) P(t) = (b) The differential equation in part (a) is called a doomsd equation because the population P( exhibits unbounded growth over finite time interval (0, T), that is, there is som time T such that → o as t –→T¯. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(90)? P(180)? (Round your answers to the nearest whole number.) P(90) P(180)
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