Suppose that news spreads through a city of fixed size of 800000people at a time rate proportional to the number of people who have not heardthe news.(a.) Formulate a differential equation and initial condition for y(t), the number ofpeople who have heard the news t days after it has happened.No one has heard the news at first, so y(0)the number of people who have heard the news is proportional to the number of= 0. The "time rate of increase inpeople who have not heard the news" translates into the differential equationdyk(|3Ddt

Total number of people in the village are =800000. No. of people who have heard the news after t…

Answered: Suppose that news spreads through a…

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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Consider a time-varying population that follows the logistic population model. The population has a limiting population of 800, and at time t = 0, its population of 200 is growing at a rate of 60 per year. Please write a differential equation for the populatoin P(t) that models this scenario.
A scientist is studying three different cultures of bacteria {labeled "A", "B" and "C"} and wants to mathematically model the bacterial change of each type. To do this, the scientist decides to collect data for 25 consecutive days and builds the following scatter diagrams where the horizontal axis corresponds to "t" and the vertical axis to the amount of bacteria present in the culture P = P (t) Help the scientist decide which mathematical model (initial value problem) to use for each of the three types of crops. To do this, associate each of the graphs above with one of the following initial value problems. Crop type A a) dP/dt = k with P(0)=P0 and k>0.b) dP/dt = k with P(0)=P0 and k<0.c) dP/dt = kt with P(0)=P0 and k>0.d) dP/dt = kt with P(0)=P0 and k<0.e) dP/dt = kP with P(0)=P0 and k>0.f) dP/dt = kP with P(0)=P0 and k<0.g) dP/dt = kP(1-P/M) with P(0)=P0 and M, k>0.h) dP/dt = kP(1-P/M) with P(0)=P0 and M, k<0. Crop type B a) dP/dt = k with P(0)=P0 and…
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity. Consider the cohort of individuals born in a given year (t = 0), and let n(t) be the number of these individuals surviving t years later. Let x(t) be the number of members of this cohort who have not had smallpox by year t and who are therefore still susceptible. Let 3 be the rate at which susceptibles contract smallpox, and let v be the rate at which people who contract smallpox die from the disease. Finally, let µ(t) be the death rate from all causes other than smallpox. Then dx/dt, the rate at which the number of susceptibles declines, is given by d = −(B+ µ(t))x. The first term on the right-hand side of this equation is the rate at which susceptibles contract smallpox, and the…
A fish hatchery has 500 fish at time ? = 0. Each year 40 fish are harvested (evenly, throughoutthe year). The population’s instantaneous growth rate is 10% per year. Thus after 1 year, if nofish were harvested, there would be slightly more than 550 fish. a) Create the modeling differential equation for the amount of the number of fish in the hatchery, y, at time t.b) Solve the differential equation.c) Is there a steady-state level of the number of fish in the hatchery?
It is estimated that t years from now, the population of a certain town will be F(t) = 40-{8/(t+2)}. Use differentials to estimate the amount by which the population will increase during the next six months
A loan account accrues interest at a rate proportional to its value. Suppose thatthis interest is compounded continuously in time at interest rate r, and supposethat payments of p dollars per year are made in a way that can be consideredto be continuous in time. write the differential equation and solve

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