DIFFERENTIAL EQUATIONS

Question

Shortly after the last elections in Utopia Chisambama (Chisa) announced that he had won the election. This spread like a wildfire to all the 3 million people who voted for him. Just 2 days after his announcement, 2 million had spread the news and they relayed to their MDC - A friends until all those who voted for Chisa heard the news.

The rate at which this news spread was proportional to both the number N of MDC - A voters who had heard this news, and to the number of voters who had not yet heard the news.

Note: In this question, work in millions

(1 000 000 people = 1 unit)

a. Set up the Differential Equation (DE) that models this situation to resemble the following general solution:

N = Et + D, where E and D are some constants.

b. Find the value of D using the given information.

c. Find the value of E given that after 30 days, 1.5 million of MDC - A voters heard this news.

Answer 1

N implies the number of those who have voted ✅ and heard

D denotes the number of those who voted ✅ but haven't heard the news

But judging from the information, 3 million people voted for him but only 2 million people heard after 2 days

E implies the rate of those who have heard and those who haven't heard per day

Hence, E α (N-D)

E = (N-D)/t

D = N-Et = 2(1-E) units

E = (N-D)/t = (3-1.5)/30 = 1.5/30 = 0.05unit/days

That mean after 2 days, the amount of those that didn't hear the new

D = 2(1-0.05) = 0.95 unit

=> after 2 days, 950,000 voted but didn't hear the news