\begin{homework}\
\label{homework:17}

\medskip
\exercise
Find an equation in the form $y=f(x)$ for the curve that passes through
the point $(3,1)$ and satisfies the differential equation

(a) $\frac{dy}{dx}=\frac{y^3}{x^2}$;~~
(b) $\frac{dy}{dx}=x^2y^2$;~~
(c) $\frac{dy}{dx}=\frac{\cos(\pi x)}{y}$.

\smallskip
\exercise
In modeling population growth one might take into account that the
reproduction rate (birth rate minus death rate) is not constant, but rather
diminishes as the population grows --- because the food supply is not
sufficient to sustain growth, crowded conditions cause disease and death,
etc.  The {\bf logistic model} of population growth is the simplest model
in which, instead of having a constant $k$ in $dy/dt=ky$, we have a
reproduction rate which decreases as the population increases.  In the
logistic model we insert a factor of $(1-y/M)$ on the right: 
\[
\frac{dy}{dt}=k\left(1-\frac{y}{ M}\right)y,
\] 
where $M$ is a constant which has the
practical meaning of {\it the maximum possible population}.  (Note that the
population would stop growing --- i.e., $dy/dt$ would become zero --- if $y$
were to become equal to $M$.) 

 (a) Derive a general formula for $y(t)$ which
will involve the constants $k$ and $M$ and another constant whose value depends
on $y_0$. 

 (b) Suppose that a beetle population on an island obeys the
logistic equation.  The maximum beetle population that the island can
support is 100 million.  At present there are 5 million beetles on the
island, and last year at this time there were 2 million beetles.  Find a
formula for the beetle population at time $t$.  How long will it be before
the population reaches 95 million?

\smallskip
\exercise
In Example~\ref{example:17.3}, what function do you get for the amount
$y(t)$ of salt at time $t$ if the rate of the {\it outgoing} water is 5
liters/min and the rate of the {\it incoming} fresh water is 3 liters/min?
What is the domain of this function (until the container is empty)?

\smallskip
\exercise
When an object falls, the deceleration caused by air resistance is some
function $r(v)$ of its velocity, and so $v(t)$ satisfies the differential
equation $dv/dt=g-r(v)$, where $g=9.8$ m/sec${}^2$ (if we choose downward
as the positive direction).  In the last section we assumed that $r(v)$ was
a {\it linear} function of $v$, namely, we took $r(v)\approx kv$.  For
example, if $k=0.102$, we have the differential equation
$dv/dt=9.8-0.102v$.  In many cases using a linear approximation $kv$ to
$r(v)$ is accurate enough --- that way we get a differential equation that
is particularly easy to solve.  (Using $kv$ instead of a more complicated
$r(v)$ is called ``linearizing'' the differential equation.)  Now suppose
that the linear approximation isn't good enough, and you need a quadratic
correction term $0.002v^2$, i.e., $r(v)=0.102v+0.002v^2$.  

(a) Graph the
right side of the resulting differential equation, and find the terminal
velocity.  

(b) Solve the differential equation to find a formula for $v(t)$
if the object is dropped from rest.  

(c) Write a definite integral for the
distance the object falls in the first 3 seconds (do not bother to evaluate
the integral).  

(d) Now suppose that you're interested in distance fallen
not as a function of time as in part (c), but rather as a function of
velocity.  Namely, you want to answer the question: {\it How far does the
object fall (from rest) before reaching a speed of 48 m/sec?} Thus,
consider distance $s$ as a function of velocity, and use the chain rule to
express $ds/dv$ in terms of $ds/dt$ (which is $v$) and $dv/dt$, thereby
obtaining a differential equation for $s(v)$.  Now write an integral for
the distance fallen as $v$ increases from $v_1$ to $v_2$, and answer the
question in italics.

\smallskip
\exercise
Answer the same question as in Problem 4(d) under the assumption that
the air resistance $r(v)$ is not given by a formula at all, but rather by
experimental readings.  That is, using a wind tunnel, you let air rush past
the object at velocities 8, 16, 24, 32, 40, and 48 m/sec, measuring the
force of air resistance, which you then divide by the mass of the object to
obtain $r(v)$.  The values you obtain are:


\begin{centering}
\begin{tabular}{c|ccccccc}
$v$&0&8&16&24&32&40&48\\ \hline
$r(v)$&0&0.91&1.82&2.75&3.75&4.80&5.84
\end{tabular}\\
\end{centering}

\noindent
Use the trapezoid rule to evaluate
the integral that represents the distance the object falls before reaching
a speed of 48 m/sec.

\smallskip
\exercise
A vat contains $V$ liters of water with $y_0$ grams of salt at time
$t=0$.  Fresh water enters the vat at a rate $r(t)$ that changes with time,
and the well-stirred solution leaves the vat at the same rate.  

(a) Find a differential equation for the amount of salt at time $t$.

(b) If $V=100$ liters, and the rate $r(t)$ starts at 2 liters/min and
increases steadily at 0.5 liters/min each min, derive a formula for $y(t)$.

(c) If $V=100$ liters, and the rate $r(t)$ varies sinusoidally with a
minimum value of 0 at $t=0$ and then a maximum of 2 liters/min after 1
minute (here $r(t)$ goes from 0 to 2 and back to 0 once every 2 minutes),
derive a formula for $y(t)$.

\smallskip
\exercise
Answer Problem 6(b) under the assumption that the rate at which water
leaves the vat remains at 2 liter/min, and only the incoming rate
increases.  (Note that the volume is no longer constant.)  Also find (in
terms of $y_0$) the limiting value of the amount of salt as
$t\longrightarrow\infty$.

\smallskip
\exercise
A vat contains 100 liters of water with $y_0$ grams of salt
at time $t=0$.  Water with $k$ gm/liter of salt enters the vat
at a rate that changes gradually over 10 minutes, as measured at
1-minute intervals and tabulated below:

\begin{centering}
\begin{tabular}{c|ccccccccccc}
t&0&1&2&3&4&5&6&7&8&9&10\\
\hline
r(t)&2.50&2.56&2.62&2.67&2.71&2.73&2.74&2.74&2.72&2.68&2.62
\end{tabular}\\
\end{centering}

\noindent
The well-stirred solution leaves the vat at the same rate.

(a) Find a differential equation for the amount of salt at time $t$.

(b) Find the solution $y(t)$ of the differential equation.  Your
solution should contain the constants $k$ and $y_0$, and should
include an integral involving $r(t)$.

(c) If we flush out the salt by pumping in pure water (i.e., if $k=0$),
find the {\bf percent} by which the amount of salt
decreases during the 10 minutes.  (Use the trapezoid rule to evaluate
the integral.)

(d) If $k=0.40$ gm/liter and $y_0=30$ gm, find $y(10)$.

\smallskip
\exercise
At time $t=0$ a car is traveling at 30 m/sec.  From time $t=0$ until the
car stops, a constant breaking force is applied which causes a deceleration
of 8 m/sec${}^2$.  The other force acting on the car is air resistance (and
friction), causing a deceleration which for the range of velocities we're
dealing with is approximately equal to $0.4v+0.01v^2$.  Find 

(a) the differential equation for $v(t)$; 

(b) the velocity function $v(t)$ for $0\le t\le T$;
and

(c) the time $T$ when the car stops.

\smallskip
\exercise
A rocket sled starts from rest and is propelled by a constant thrust
(force) $F$ in a horizontal path.  Suppose that, as the propellant is
burned, the mass of the rocket, which initially is m, decreases at the
constant rate $\alpha$.  Also assume that the force of air resistance and
friction is proportional to the velocity with proportionality constant $K$.

(a) Using the basic physical principle that

\medskip

\centerline{(mass of rocket at time $t$) $\cdot$ (acceleration)=total of
forces acting on rocket,}

\medskip

\noindent
find a differential equation for $v(t)$.  

(b) Solve the differential
equation in part (a) to find a formula for $v(t)$ in terms of $t$ and the
constants $F$, $m$, $\alpha$, $K$, and $v_0$.

\smallskip
\exercise
In Section 3 we derived the formula $y(t)=\frac{v_0}{\omega}\sin(\omega t)$
for harmonic oscillation, assuming zero initial position.  Find the
formula for harmonic oscillation if the initial velocity is zero, but
the initial position is $y_0\ne 0$. 

\smallskip
\exercise
Find the formula for harmonic oscillation if $y(0)=y_0$ and $\dot{y}(0)=v_0$,
where we no longer suppose that either of these initial values is zero.

\end{homework} 

 
%---------------------------------------------------------------
