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Consider the differential equation = kpl + 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, co), that is, P(t) - co as t- o. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation = 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(o) = 10 and the fact that the animal population has doubled in 9 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) → co as t-T. Find T. (Round your answer to the nearest month.) T= months (c) From part (a), what is P(80)? P(160)? (Round your answers to the nearest whole number.) P(80) = P(160) =

Consider the differential equation = kpl + 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, co), that is, P(t) - co as t- o. See Example 1 in that section. (a) Suppose for c = 0.01 that the nonlinear differential equation = 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(o) = 10 and the fact that the animal population has doubled in 9 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) → co as t-T. Find T. (Round your answer to the nearest month.) T= months (c) From part (a), what is P(80)? P(160)? (Round your answers to the nearest whole number.) P(80) = P(160) =

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 5EQ
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Consider the differential equation dP = kp1 + c dt 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, 0), that is, P(t) → o as t → o. 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 9 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 is P(80)? P(160)? (Round your answers to the nearest whole number.) P(80) = P(160) =
Transcribed Image Text:Consider the differential equation dP = kp1 + c dt 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, 0), that is, P(t) → o as t → o. 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 9 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 is P(80)? P(160)? (Round your answers to the nearest whole number.) P(80) = P(160) =
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