Under the same assumptions that underlie the model in (1),
determine a differential equation for the population $P(t)$ of
a country when individuals are allowed to immigrate into
the country at a constant rate $r > 0$. What is the differential
equation for the population $P(t)$ of the country when
individuals are allowed to emigrate from the country at a
constant rate $r > 0$?

2.The population model given in (1) fails to take death into
consideration; the growth rate equals the birth rate. In
another model of a changing population of a community
it is assumed that the rate at which the population changes
is a $net$ rate—that is, the difference between the rate of
births and the rate of deaths in the community. Determine
a model for the population $P(t)$ if both the birth rate and
the death rate are proportional to the population present
at time $t > 0$.

3.Using the concept of net rate introduced in Problem 2,
determine a model for a population $P(t)$ if the birth rate is
proportional to the population present at time t but the death
rate is proportional to the square of the population present at
time $t$.

4.Modify the model in Problem 3 for net rate at which the
population $P(t)$ of a certain kind of sh changes by also
assuming that the sh are harvested at a constant rate
$h > 0$.

5.A cup of coffee cools according to Newton’s law of cooling (3).
Use data from the graph of the temperature $T(t)$ in Figure 1.3.4
to estimate the constants $T_m$, $T_0$, and $k$ in a model of the form
of a rst-order initial-value problem: $\frac{dT}{dt} = k(T − T_m)$,
$T(0) = T_0$


6.The ambient temperature $T_m$ in (3) could be a function of time $t$.
Suppose that in an articially controlled environment, $T_m(t)$ is
periodic with a 24-hour period, as illustrated in Figure 1.3.5.
Devise a mathematical model for the temperature $T(t)$ of a body
within this environment.

7.Suppose a student carrying a u virus returns to an isolated
college campus of 1000 students. Determine a differential
equation for the number of people $x(t)$ who have contracted the
u if the rate at which the disease spreads is proportional to the
number of interactions between the number of students who
have the u and the number of students who have not yet been
exposed to it.


8.At a time denoted as $t = 0$ a technological innovation is
introduced into a community that has a xed population of
$n$ people. Determine a differential equation for the number of
people $x(t)$ who have adopted the innovation at time $t$ if it is
assumed that the rate at which the innovations spread through
the community is jointly proportional to the number of people
who have adopted it and the number of people who have not
adopted it.

9. Suppose that a large mixing tank initially holds 300 gallons
of water in which 50 pounds of salt have been dissolved. Pure
water is pumped into the tank at a rate of 3 gal/min, and when
the solution is well stirred, it is then pumped out at the same
rate. Determine a differential equation for the amount of salt
$A(t)$ in the tank at time $t > 0$. What is $A(0)$?

10. Suppose that a large mixing tank initially holds 300 gallons
of water in which 50 pounds of salt have been dissolved.
Another brine solution is pumped into the tank at a rate
of 3 gal/min, and when the solution is well stirred, it
is then pumped out at a slower rate of 2 gal/min. If the
concentration of the solution entering is 2 lb/gal, determine
a differential equation for the amount of salt $A(t)$ in the tank
at time $t>0$.

11. What is the differential equation in Problem 10, if the well
stirred solution is pumped out at a $faster$ rate of 3.5 gal/min?

12. Generalize the model given in equation 12 of this section by
assuming that the large tank initially contains $N_0$ number of
gallons of brine, $r_{in}$ and $r_{out}$ are the input and output rates of the
brine, respectively (measured in gallons per minute), $c_{in}$ is the
concentration of the salt in the inflow, $c(t)$ the concentration of
the salt in the tank as well as in the outow at time t (measured
in pounds of salt per gallon), and $A(t)$ is the amount of salt in
the tank at time $t>0$.

















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