Differential Equation - Rumor Spreading

Question

There are M = 700 students attending school. On Wednesday at 8am, 4 students arrive at school with a rumor that school will be closed on Monday due to the upcoming snowstorm. On the average each student talks to other students at a rate of 5 students/hour (24 hours/day), passing on the rumor to others - some having heard it and others not. The rate at which students hear the rumor for the first time is proportional to the number of students who have already heard it, times the fraction of students who have not yet heard it (given below). If y is the number of students who have heard the rumor since school started Wednesday morning, then image {it is determined that image for the students rate of passing a rumor}

b) Use the space below to estimate the number of students who have heard the rumor after two hours with image using Euler's Method by hand. (show all of your work)

c) Use technology and Euler's Method with image hours to approximate how many students have heard the rumor after four hours.

d) Analytically determine y as a function of t.

I am lost!! Please help!

Answer 1

Let $y(t)$ be the number of students who have already heard the rumor at time $t$. The number of students who have not heard the rumor would then be $M - y$, and the fraction of such students would be $\frac{M - y}{M} = 1 - \frac yM$. In the next $dt$ hours, the number tellings of the rumor is $5dt$ per student who knows it, for a total of $5ydt$ tellings in all. But of those, only $1 - \frac yM$ of those told have not already heard it. So the total increase in the number of students who know the rumor is $$dy = 5y\left(1 - \frac yM\right)dt$$ Dividing by $dt$, $$\dot y = 5y\left(1 - \frac yM\right)$$

You are also given the initial condition $y(0) = 4$.

Now you just have to look over your information on Euler's method and apply it to that equation.