dN Find the general solution to the first-order ordinary differential equation (ODE), N = rN. Supposing r = 1, find the solution of this ODE when the initial population density (t = 0) satisfies N(0) = 100. dt Suppose the per capita growth rate will decrease linearly with the population density. When the population density approaches its maximum size K, the per capita growth rate decreases to This yields the logistic equation, dN = N(1-). dt Find the general solution of this ODE by the method of the separation of variables. Note, K is a constant number not a variable here. According to this general solution, describe how the population size changes as t →∞.
dN Find the general solution to the first-order ordinary differential equation (ODE), N = rN. Supposing r = 1, find the solution of this ODE when the initial population density (t = 0) satisfies N(0) = 100. dt Suppose the per capita growth rate will decrease linearly with the population density. When the population density approaches its maximum size K, the per capita growth rate decreases to This yields the logistic equation, dN = N(1-). dt Find the general solution of this ODE by the method of the separation of variables. Note, K is a constant number not a variable here. According to this general solution, describe how the population size changes as t →∞.
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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Transcribed Image Text:Letr be the per capita growth rate of a population in the time interval dt and N be the
population density, which is the total number of individuals in this population. Note
r is a constant number not a variable here.
(a) Find the general solution to the first-order ordinary differential equation (ODE),
dN=rN. Supposing r = 1, find the solution of this ODE when the initial
population density (t = 0) satisfies N(0) = 100.
dt
(b) Suppose the per capita growth rate will decrease linearly with the population
density. When the population density approaches its maximum size K, the per
capita growth rate decreases to 0. This yields the logistic equation,
dN
dt
=
N(1 -
N
K
Find the general solution of this ODE by the method of the separation of
variables. Note, K is a constant number not a variable here. According to this
general solution, describe how the population size changes as t → ∞.
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