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13. Consider a population P = P(t) with constant relative birth and death rates a and B, respectively, and a constant emigra- tion rate m, where a, B, and m are positive constants. Assume that a > B. Then the rate of change of the population at time t is modeled by the differential equation dP where k- a - B kP – m dt (a) Find the solution of this equation that satisfies the initial condition P(0) - Po. (b) What condition on m will lead to an exponential expan- sion of the population? (c) What condition on m will result in a constant population? A population decline? (d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6% of the population. Because of the potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?

13. Consider a population P = P(t) with constant relative birth and death rates a and B, respectively, and a constant emigra- tion rate m, where a, B, and m are positive constants. Assume that a > B. Then the rate of change of the population at time t is modeled by the differential equation dP where k- a - B kP – m dt (a) Find the solution of this equation that satisfies the initial condition P(0) - Po. (b) What condition on m will lead to an exponential expan- sion of the population? (c) What condition on m will result in a constant population? A population decline? (d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6% of the population. Because of the potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?

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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13. Consider a population P = P(t) with constant relative birth and death rates a and B, respectively, and a constant emigra- tion rate m, where a, B, and m are positive constants. Assume that a > B. Then the rate of change of the population at time t is modeled by the differential equation dP = kP - m di where k - a - B (a) Find the solution of this equation that satisfies the initial condition P(0) = Po. (b) What condition on m will lead to an exponential expan- sion of the population? (c) What condition on m will result in a constant population? A population decline? (d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6% of the population. Because of the potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?
Transcribed Image Text:13. Consider a population P = P(t) with constant relative birth and death rates a and B, respectively, and a constant emigra- tion rate m, where a, B, and m are positive constants. Assume that a > B. Then the rate of change of the population at time t is modeled by the differential equation dP = kP - m di where k - a - B (a) Find the solution of this equation that satisfies the initial condition P(0) = Po. (b) What condition on m will lead to an exponential expan- sion of the population? (c) What condition on m will result in a constant population? A population decline? (d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6% of the population. Because of the potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?
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