Consider the initial value problem for logistic growth with a threshold: dy dt = −r(1 −y/T)(1 −y/K)y, y(0) = y0 where r > 0 and 0 < T < K are given constants. (a) Solve the IVP in implicit form (ie. as an algebraic equation relating y and t). [Partial fractions are recommended for the ODE.] (b) Without giving an explicit solution, use the direction field to characterize the limit- ing behavior limt→∞y(t) of the different solutions y(t) for all possible values of y0.
Consider the initial value problem for logistic growth with a threshold: dy dt = −r(1 −y/T)(1 −y/K)y, y(0) = y0 where r > 0 and 0 < T < K are given constants. (a) Solve the IVP in implicit form (ie. as an algebraic equation relating y and t). [Partial fractions are recommended for the ODE.] (b) Without giving an explicit solution, use the direction field to characterize the limit- ing behavior limt→∞y(t) of the different solutions y(t) for all possible values of y0.
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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Question
Consider the initial value problem for logistic growth with a threshold:
dy
dt = −r(1 −y/T)(1 −y/K)y, y(0) = y0
where r > 0 and 0 < T < K are given constants.
(a) Solve the IVP in implicit form (ie. as an algebraic equation relating y and t). [Partial
fractions are recommended for the ODE.]
(b) Without giving an explicit solution, use the direction field to characterize the limit-
ing behavior limt→∞y(t) of the different solutions y(t) for all possible values of y0.
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