The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation dP = rP(1 − ) dt where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler, related differential equation d² = rx(1 − x) for x > 0. (a) Express (1¹) in partial fraction form; hence find ſz(1¹) dæ, expressing your answer in the form In(). (Make sure you consider the two cases x < 1 and x > 1 separately.) (b) If x= xo> 0 when t = 0, show that the solution of the equation is 1 1+ (−1)e-ri (c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run. x p-rt
The differential equation used to model population growth, under the assumption that the proportional growth rate is a decreasing linear function of population, is the logistic differential equation dP = rP(1 − ) dt where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler, related differential equation d² = rx(1 − x) for x > 0. (a) Express (1¹) in partial fraction form; hence find ſz(1¹) dæ, expressing your answer in the form In(). (Make sure you consider the two cases x < 1 and x > 1 separately.) (b) If x= xo> 0 when t = 0, show that the solution of the equation is 1 1+ (−1)e-ri (c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run. x p-rt
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:2 The differential equation used to model population growth, under the assumption that the proportional
growth rate is a decreasing linear function of population, is the logistic differential equation
dP
d = rP(1 - )
where r and M are positive constants. By the change of variables, P(t) = Mx(t) we can obtain the simpler,
related differential equation
d = rx(1-x)
for x > 0.
(a) Express (1-2) in partial fraction form; hence find (1-2) dx, expressing your answer in the form
In(). (Make sure you consider the two cases x < 1 and x >1 separately.)
(b) If x= xo> 0 when t = 0, show that the solution of the equation is
X =
1
1 + (1 - 1)e-rt
(c) Considering the two cases xo < 1 and xo > 1, determine what happens in the long run.
(d) What happens if xo = 0 or xo = 1?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
