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\begin{document}
\begin{center}
{\bf Homework 4}\\
(due Wednesday, February 15)
\end{center}
\vspace*{.3in}

\noindent 

\vspace*{.2in}
\noindent{\bf From Terrel's notes}
\begin{description}
 \item[Page 4] 1--do only the 3rd and 5th, but do them by hand, and then do them by Dfield.
\end{description}

\begin{prob}
Find explicit formulas for the solutions of the following ODES and initial value problems (which might not be expressed in the most convenient form for solving...).  Also use DField to draw a direction field for each ODE, and where there are given initial values, plot the corresponding solution curves on top of the direction field.
\begin{enumerate}
 \item $y' = e^{t+y}$;
 \item $y' = -t/y$, $y(0) = -1$ and again with $y(0) = 2$;
 \item $ty' - 4y = t^3$;
 \item $y' = y + 2te^{2t}$, $y(0) = 3$.
\end{enumerate}
\end{prob}

\begin{prob}
One morning in the near future, you have a life-changing epiphany: math is for
losers; you, on the other hand, were meant to be a millionaire.  Sure in your
new calling, you decide that the best way to realize your true inner self is 
to begin depositing money at a constant rate into an account earning 
5\% annual interest, compounded continuously.  Your last mathematical act will 
be to figure the rate (in dollars per year) at which you'll need to pour money 
into the account in order to self-actualize in twenty years time.  That is, 
derive the
differential equation satisfied by the amount of money in your account, impose
appropriate initial conditions and solve to find (what you will come to know
as) your `personal fulfillment constant.'
\end{prob}

\begin{prob}
Consider the initial value problem
$$
y' = y^{2/3},\quad y(0) = 0.
$$
Show that for any $C>0$, the function
$$
y(t) = \left\{\begin{array}{rcl} 0 & \text{if} & t\leq C \\
                                 \left(\frac{t-C}3\right)^3 & \text{if} & t>C
       \end{array}\right.
$$ 
is a solution.  Be especially careful with the point $t=C$.  Why does this not 
contradict the existence/uniqueness theorem for solutions of first order ODEs?
\end{prob}

\begin{prob}
Let $y(x)$ denote the height, at horizontal position $x$, of a cable freely 
suspended between two points.  Assume for simplicity's sake that the downward
force of gravity is equal to the mass of an object (i.e. $g=1$) and that the
density of the cable is $1$.  Note that the
forces acting on any small segment of the cable are
\begin{itemize}
\item the downward force of gravity due to the mass of the segment.
\item a force exerted by the portion of the cable to the right of
  the segment.
\item a force exerted by the portion of the cable to the left of the segment.
\end{itemize}
Each of these forces has a horizontal and a vertical component, and since the
cable is at rest, the forces must add to zero at every point of the cable.
Use these observations to write down a differential equation satisfied by
$y(x)$.  Then solve this equation to derive a formula for the shape of the 
hanging cable.
\end{prob}

\begin{prob}
Redo the previous problem for the situation of an ideal suspension bridge.
That is, assume the cable is massless but that it supports a  
bridge whose weight is spread out evenly (assume the density is 1 again) in 
the horizontal direction.
Believe it or not, this makes the problem much simpler.
\end{prob}

\end{document}
\end

