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\begin{document}
\begin{center}
{\bf Homework 5}\\
(due Wednesday, February 29)
\end{center}
\vspace*{.3in}

\noindent 

\vspace*{.2in}
\vspace*{.3in}
\noindent{\bf Do the following problems from Polking, Boggess and Arnold (on reserve)}
\begin{description}
 \item[p100] 9, 10, 20, 21 (classify equilibria as stable or unstable according to our conventions in class; also, illustrate your answers by sketching a `phase portrait' for each system, i.e. a graph showing several solutions to the ODE with various initial values $y(0)$.)
 \item[p156] 1, 11, 19, 31.
 \item[p172] 3, 5, 19
\end{description}
\vspace*{.2in}


\begin{prob}
Recall the logistic equation $y' = ry(K-y)$.  Find a formula for the solution of this equation and then use your formula to determine the asymptotic behavior (for increasing \emph{and} decreasing $t$) of solutions satisfying
\begin{itemize}
 \item $y(0) > K$
 \item $0<y(0)<K$
 \item $y(0) < 0$.
\end{itemize}
Your conclusions should agree with the ones given by qualitative analysis.  Do they?
\end{prob}

\begin{prob}
Civilization collapses and in particular, everything about trig functions is 
forgotten.  Some time after the collapse, we rediscover the existence and
uniqueness theorem for solutions to second order linear ODEs.  We then define 
functions $y=C,S:\R\to \R$ to be the unique solutions of $y''+ y=0$ 
satisfying the initial conditions $C(0) = 1$, $C'(0) = 0$ and $S(0) = 0$
and $S'(0) = 1$.  From this, prove each of the following.
\begin{enumerate}
\item $C' = -S$ and $S' = C$.
\item $C(t)^2 + S(t)^2 = 1$ for all $t\in\R$.
\item $C(t+s) = C(t)C(s) - S(t)S(s)$ and $S(t+s) = S(t)C(s) + C(t)S(s)$ for
  all $t,s\in\R$.
\item (Extra credit) There is a number $t_0>0$ such that  $C(t) > 0$ when
$t\in[0,t_0)$ but $C(t_0) = 0$.  The hard part here is to show that there
exists \emph{any} point $s$ where $C(s)=0$, let alone a first such point.
Proving the existence of $s$ tests your understanding of differential calculus
from math 165 (or wherever you learned that stuff).
\item $C(t)$ decreases from $1$ to $0$ on $[0,t_0]$ whereas $S(t)$ increases
  from $0$ to $1$. 
\item Work out similar statements for $C$ and $S$ on the intervals
  $[t_0,2t_0]$, $[2t_0,3t_0]$, and $[3t_0,4t_0]$.
\item $C$ and $S$ are both periodic with period $4t_0$.  
\end{enumerate}
\end{prob}

\begin{prob}
Show that no solution $y(t)$ of the homogeneous linear equation 
$y'' - t^2y = 0$ can be found by solving the first order equations
$y' - ty = 0$, $y'+ty =0$.  Hence the technique of factoring differential
operators (at least when it is applied naively) does not extend to help solve
linear homogeneous equations with non-constant coefficients.  Why not?
\end{prob}


\begin{prob} {\bf Yet more extra credit:} This is analysis rather than algebra, but since I mentioned it in class... Let $y:\R\to\R$ be a function that is differentiable at every point. 
 \begin{enumerate}
  \item Show by example that $y'$ need not be continuous.
  \item Give an example where $y'(0) >0$ but (nevertheless) $y$ fails to be increasing on $(-\epsilon,\epsilon)$ no matter how small you choose $\epsilon>0$.
  \item Prove that $y'$ satisfies the conclusion of the intermediate value theorem; say specifically, if $y'(0) < 0$ and $y'(1) > 0$, then there exists $t\in (0,1)$ such that $y'(t) = 0$.
 \end{enumerate}
I'll give credit for solutions to each of these separately.
\end{prob}


\end{document}
\end

