Question

We want to use differential equations to model the change in the proportion of individuals who support a bill that will be up for a vote in the future. Let x(t) be the proportion of individuals who support this measure, so x(t) will be a value between 0 and 1 , and so that the proportion of individuals who do not support the measure is given by 1−x(t). The people in this population are generally apathetic, so they lose interest over time, but the issue is popular when people hear about it. The assumptions we have about this model are that the proportion of individuals who support this measure will increase at a rate of 5 times the proportion who support the measure, multiplied by the proportion who do not support the measure per day, but the supporting proportion decreases at a constant rate of (21/20) per day. (a) Write a differential equation model for the function x(t) using this information. Assume that t is given in days.

(b) Are there any proportion values where the value will not change in time? If so, find them. (c) Using this information from the last part, assume that the support currently sits at 50%. What is going to happen over time? Do you need to intervene if you want the measure to pass in the upcoming vote?

Answer 1

The differential equation model for the function x(t) is given by dx/dt = 5x(1-x) - (21/20)x. There are no proportion values where the value will not change over time. Assuming the support currently sits at 50%, the value of x(t) will decrease over time.

Step-by-step explanation:

(a) The differential equation model for the function x(t) can be written as:

dx/dt = 5x(1-x) - (21/20)x

(b) No, there are no proportion values where the value will not change over time.

(c) Assuming the support currently sits at 50%, the value of x(t) will decrease over time. You do not need to intervene if you want the measure to pass in the upcoming vote.