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18.03 Topic 1 Part I
Problem 1. (E&P problem 1.1/36) In a city with population P the time rate of change
of the number N of people infected with a contagious disease is proportional to the product
of the number who have the disease and the number who do not.
Write a differential equation modeling this situation.
Problem 2. (E&P problem 1.4/43) A pitcher of buttermilk is initially at 25◦ C. It is cooled
by putting it on the porch where the temperature is 0◦ C. Suppose the temperature drops
to 15◦ C after 20 minutes. When will it be 5◦ C?
Problem 3. (E&P 1.5/38) Consider the cascade of two mixing tanks shown. The volume
of brine in the top tank is 100 liters and that of the bottom tank is 200 liters. Each tank
initially contains 50kg of salt. The flow rates into and out of each tank are 5 liters/minuts,
with pure water flowing into the top tank.
Let x(t) be the amount of salt in the top tank and let y(t) be the amount in the bottom
tank.
(a) Write a DE for x(t) and solve it.
dy
5x
5y
(b) Show that
=
−
and solve for y(t).
dt
100 200
Problem 4. Find the general solution to xy 0 + 2y = x.
1