We assume that a person's chances of getting married (in some small time interval Delta t ) is proportional
to the fraction of people who are already married by that age; this can be considered peer pressure[1]
We first assume that the rate of change of m(t) is proportional to the number of interactions
(the fraction of) people married (for a first time) by age t have with (the fraction of) people of
age t who have not yet married.
(a) Using our notation above, what fraction of people are married by age t ? What fraction are
unmarried at age t ? How many interactions are there between these groups of people?
(b) Write and solve a differential equation modeling the situation with these assumptions.
(c) What is the long-term behavior of this solution? Does this make sense for the scenario?
Hint: Based on the situation, is the proportionality constant positive or negative? Why?
(d) This differential equation is autonomous.
i. Find the equilibrium point(s).
ii. Analyze the stability of the equilibrium point(s).
(e) Your general solution to Problem 1b should have two unknown constants in it; the propor-
tionality constant and another constant that arose when solving the first-order differential
equation. Choose three different combinations of values for these constants. Plot the
solution with these values plugged in.
i. Do these graphs support your answers from Problem 1c and 1d?
ii. What do you notice about solution at t=0 ? Does this make sense?
(f) One criticism of this model is that it assumes people experience that peer pressure to the
same degree at every age. Come up with at least two other criticisms.
We assume that a person's chances of getting married (in some small time interval t) is proportional
to the fraction of people who are already married by that age; this can be considered peer pressure[1]
1. We first assume that the rate of change of m(t) is proportional to the number of interactions (the fraction of) people married (for a first time) by age t have with (the fraction of) people of age t who have not yet married.
(a) Using our notation above, what fraction of people are married by age t? What fraction are unmarried at age t? How many interactions are there between these groups of people?
(b) Write and solve a differential equation modeling the situation with these assumptions.
(c) What is the long-term behavior of this solution? Does this make sense for the scenario?
Hint: Based on the situation, is the proportionality constant positive or negative? Why?
(d) This differential equation is autonomous.
i. Find the equilibrium point(s).
ii. Analyze the stability of the equilibrium point(s)
(e) Your general solution to Problem 1b should have two unknown constants in it; the propor-
tionality constant and another constant that arose when solving the first-order differential
equation. Choose three different combinations of values for these constants. Plot the solution with these values plugged in.
i. Do these graphs support your answers from Problem 1c and 1d? ii. What do you notice about solution at t = 0? Does this make sense?
(f) One criticism of this model is that it assumes people experience that peer pressure to the same degree at every age. Come up with at least two other criticisms.