One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. Write the differential equation that is satisfied by y.Write the general solution of the differential equation. Hint: This should be familiar!A small town has 1000 inhabitants. At 8am, 80 people have heard the rumor. By noon, half the town has heard it. At what time will 90% of the population have heard the rumor?Do you think this is a reasonable model of rumor spread? Why or why not?Suppose that we modify the model by multiplying the right hand side of the differential equation by e-at for some positive constant a. What would this mean in terms of rumor spreading over time? Try to make an educated guess before completing the next part of the problem.Solve this new differential equation- it is separable. You may use Sage to integrate. Fixing any other constants besides a, graph the solution using Sage for several different values of a. Do your graphs match your intuition from (e)? What does the parameter a control?Solve the specific problem in (c) using the new model. Do you have enough information to even do so? If not, feel free to add additional conditions.

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Answered: One model for the spread of a rumor is…

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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One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor.

Write the differential equation that is satisfied by y.
Write the general solution of the differential equation. Hint: This should be familiar!
A small town has 1000 inhabitants. At 8am, 80 people have heard the rumor. By noon, half the town has heard it. At what time will 90% of the population have heard the rumor?
Do you think this is a reasonable model of rumor spread? Why or why not?
Suppose that we modify the model by multiplying the right hand side of the differential equation by e^-at for some positive constant a. What would this mean in terms of rumor spreading over time? Try to make an educated guess before completing the next part of the problem.
Solve this new differential equation- it is separable. You may use Sage to integrate.
Fixing any other constants besides a, graph the solution using Sage for several different values of a. Do your graphs match your intuition from (e)? What does the parameter a control?
Solve the specific problem in (c) using the new model. Do you have enough information to even do so? If not, feel free to add additional conditions.

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