Let y ( t ) be the percentage of the population that supports a particular gover

Question

Let y(t) be the percentage of the population that supports a particular government policy. The rate of change of the level of support for the policy is proportional to both the percentage of the population that agrees with the policy and the percentage that disagrees with the policy. Let the constant of proportionality for this relationship be written as k.

(a) Enter the differential equation that is satisfied by y. (Your answer should be given in terms of y. Even though y is a function of time, enter y, rather than y(t).)
dy/dt= ky(100-y)

Now, suppose that, in addition to people changing their support as described above, people also change from agreeing to disagreeing because they have second thoughts about the policy. Suppose that this rate is proportional to the percentage of people that agree with the policy, with constant of proportionality equal to c.

Enter the new differential equation that is satisfied by y.

dy/dt=?

Explanation / Answer

a) given

y(t) = percentage of population supports the policy

let y be the percentage of people agrees with the policy and so 100-y will be the percentage of people disagreeing with the policy

and given that the rate of change of the level of support for the policy is proportional to both the percentage of the population that agrees with the policy and the percentage that disagrees with the policy.

so dy/dt y(100-y) and k is the constant of proportionality

=> dy/dt = ky(100-y)

b) given that people are changing to support the disagreement of policy

we know that 100-y are the people disagreeing to to the policy and ky(100-y) are the people changing the policy

so by the time 't' the people change to disagree are tky(100-y)

so the total number of people changing is given by 100-y +kty(100-y) = (100-y)(kty+1)

now dy/dt (100-y)(kty+1)

=> dy/dt = c(100-y) (kty+1)

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