\magnification=\magstep1
\centerline{Mathematics 304 -- Ordinary Differential Equations}
\centerline{Problem Set 1}
\centerline{September 7, 2000}
\bigskip
\noindent
{\it Information:}  This is an individual assignment, due in class
Thursday, September 14.
\bigskip
\noindent
From the text:  
\bigskip
\item{} Section 2.2 (p. 51-53)/12, 17, 19, 21, 29, 31
\bigskip
\item{} Section 2.3 (p. 59-60)/1, 3, 5, 15, 17
\bigskip
\noindent
{\it Extra Problems}
\bigskip
\noindent
I)  In question C on Lab 1, we graphed some solutions of
the ODE with piecewise-defined
slope function
$${dP\over dt} = \cases{(0.4) P (1 - P/30) & if $t \le 5$\cr
                        (0.4) P (1 - P/30) - (0.25)P & if $t > 5$\cr}
                        \leqno(1)$$
Note that the Existence and Uniqueness Theorem does not guarantee anything
about solutions on intervals containing $t = 5$, since the slope
function is {\it discontinuous} at $(t,P)$ with $P > 0$.  Nevertheless,
Maple did not seem to have any difficulty generating plots of 
approximate solutions for this equation.  We want to understand this, 
and generate formulas for the solutions Maple found.
\bigskip
\item{A)}  Explain, using the Existence and Uniqueness Theorem, how 
a {\it continuous} solution of the equation (1) could be ``pieced together'' 
from a solution to ${dP\over dt} = (0.4) P (1 - P/30)$, with an initial condition 
at $t = 0$, and a solution to 
${dP\over dt} = (0.4) P (1 - P/30) - (0.25)P$, with an initial condition
at $t = 5$.  
\item{B)}  Both equations in A are separable {\it logistic equations} as
we discussed in class on 9/7.  Find explicit solutions
of each and piece together the appropriate solution of (1) with $P(0) = 25$.
This is one of the cases you did in Maple in the lab.  How does the 
plot of your explicit analytic solution compare with the approximate 
plot Maple found?  
\item{C)}  In two or three sentences, describe the effect of the 
appearance of the parasite on the insect population at $t = 5$.
Explain the behavior as $t \to \infty$.
\bigskip
\noindent
II)  We can also get information on the second derivative
(concavity) of solutions of a first order ODE ${dy\over dx} = f(x,y)$
by differentiating
both sides with respect to $x$ and using {\it implicit differentiation}
on the right-hand side -- that is differentiate $f(x,y)$ with respect to $x$,
thinking of $y$ as the solution function, so $y$ depends on $x$.
\bigskip
\item{A)}  Using this, show that the graph of the solution of 
${dy \over dx} = xy$ with $y(0) = 1$ is concave up for all $x$.
\item{B)}  Show that every solution of the autonomous equation
${dy \over dx} = (y-2)(y-4)$ and $y(0) = y_0$ with $2 < y_0 < 4$
has an inflection point where it crosses the line $y = 3$.
\bye