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Consider the differential equation = 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 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 = 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 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, 7), that is, there is some time T such that P(t) → ∞ as t→ T. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(70)? P(140)? (Round your answers to the nearest whole number.) P(70) = P(140) =

Consider the differential equation = 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 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 = 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 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, 7), that is, there is some time T such that P(t) → ∞ as t→ T. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(70)? P(140)? (Round your answers to the nearest whole number.) P(70) = P(140) =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the differential equation = kp¹ + +C₁ dP 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, ∞), 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 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 7 months. (Round the coefficient of t to six decimal places.) P(t) = - as (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, 7), that is, there is some time T such that P(t) t → T¯. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(70)? P(140)? (Round your answers to the nearest whole number.) P(70) P(140) = =
Transcribed Image Text:Consider the differential equation = kp¹ + +C₁ dP 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, ∞), 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 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 7 months. (Round the coefficient of t to six decimal places.) P(t) = - as (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, 7), that is, there is some time T such that P(t) t → T¯. Find T. (Round your answer to the nearest month.) T = months (c) From part (a), what is P(70)? P(140)? (Round your answers to the nearest whole number.) P(70) P(140) = =
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