\magnification=\magstep1
\centerline{Mathematics 304, section 1 -- Ordinary Differential Equations}
\centerline{Discussion 1 -- ``More Mixed Up''}
\centerline{\it September 8, 2004}
\bigskip
\noindent
{\it Background}
\bigskip
Last week, recall that we considered an example involving cholesterol
levels in the bloodstream and modeled that real world system by 
a (separable) first order ODE.  Today, we will consider other similar
examples involving the idea of ``mixing''.  The basic patterns behind
all of these examples are:
\item{$\bullet$} we have a system in which we are tracking
the amount of some quantity present,
\item{$\bullet$} there one or more ways
in which that quantity is being added and subtracted, so
\item{$\bullet$}  writing $y$ for the amount as a function of 
$t$, we have:  {\it the rate of change of $y$ is equal to 
the (total) in-flow rate of $y$, minus the total out-flow rate of $y$}:
$${dy\over dt} = ({\rm inflow\ rate}) - ({\rm outflow\ rate})\leqno(1)$$
\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\noindent
I.   (Imagine that we are
in the Heinz pickle factory preparing to produce a batch of
Polish dills.)  A large tank holds 1000 liters of pure water when 
a salt solution begins to flow 
in at the rate of 6 liters per minute.  The salt concentration in 
the liquid entering is 1 kilogram/liter.  The mixture in the tank 
is kept well-stirred at all times (so the salt concentration is 
{\it kept uniform throughout the tank}).  At the same time, brine is
being removed from the tank at the same rate: 6 liters per minute.
\bigskip
\item{A)}  Let $y(t)$ denote the amount (i.e. mass) of the salt in the 
tank as a function of time.  Explain why (1) leads to the ODE
$${dy\over dt} = 6 - {3\over 500} y.$$
\item{B)}  When will the concentration of salt in the tank reach $1/2$ 
kilogram/liter?
\item{C)}  What happens to the concentration of salt in the tank if we
let $t \to \infty$?  Is your answer reasonable?  Why or why not?
\bigskip
\noindent
II.  Now, suppose we have the same situation as in 
question I, but the out-flow rate is 5 liters per minute, rather
than 6 liters per minute.  
\item{A)}  What changes in the differential equation?  (Be careful;
note what happens to the volume of the mixture in the tank as
$t$ increases.)
\item{B)}  Solve your new differential equation and 
determine what happens to the concentration of salt in the tank if we
let $t \to \infty$ again.  
\item{C)} If the capacity of the tank is (only) 8000 
liters, though, is our differential equation a good model?  Why?
How might we include this feature of the real-world system?
\bigskip
\noindent
III.  The equation 
$${dy\over dt} = -2y + ty^{-2}\leqno(2)$$
is not first order linear.  (Why not?)
\item{A)}  Show however, that the substitution $v = y^3$
(and some algebra) reduces the equation to the first order linear
form:
$${dv\over dt} = -6v + 3t.$$
\item{B)}  Solve this equation, then substitute $v = y^3$ back in to 
derive the general solution of (2).  This is an example of a 
{\it Bernoulli equation}; Bernoulli equations can always be reduced
to first order linear equations by a similar process.
\bigskip
\noindent
{\it Assignment}
\bigskip
Group write-ups, due in class, Monday September 13.
\bye
 
