\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{amsfonts}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}


\begin{document}
\pagestyle{empty}



\begin{center}
Math 132: Calculus for Physical and Life Sciences 2\\ 
Problem Set 7 \\ 
Due Friday, April 4, 2008, at the beginning of class.
\end{center}
\vspace{10pt}

General Directions: You must show all work for credit on these problems. 

\begin{enumerate}
\item All parts of this question deal with the differential equation 
$y' = x + y$.  
\begin{enumerate}
\item[(a)]  Verify that $y = -x - 1$ is a solution of this differential equation.
\item[(b)]  Sketch the direction field of this equation by hand, showing the 
slopes at all points with $-3 \le x \le 3, -3 \le y \le -3$ with integer coordinates.
\item[(c)]  On your direction field graph, sketch approximate graphs
of the solutions of this differential equation satisfying the initial 
conditions $y(0) = -2$, $y(0) = 0$, $y(0) = 1$.  
\item[(d)]  Use Euler's method with $n = 5$, $\Delta x = .2$ 
to approximate the solution of the equation with the initial condition 
$y(0) = 1$ for $0 \le x \le 1$.  
\end{enumerate}

\item All parts of this question deal with the differential equation 
$y' = 3 - y$.  
\begin{enumerate}
\item[(a)]  Sketch the direction field of this equation by hand, showing the 
slopes at all points with $-1 \le x \le 5, 0 \le y \le 5$ with integer coordinates.
\item[(b)]  On your direction field graph, sketch approximate graphs
of the solutions of this differential equation satisfying the initial 
conditions $y(0) = 0$, $y(0) = 3$, and $y(0) = 4$.  What is the long 
term behavior of each?
\item[(c)]  Use Euler's method with $n = 5$, $\Delta x = .2$ 
to approximate the solution of the equation with the initial condition 
$y(0) = 1$ for $0 \le x \le 1$.  
\item[(d)]  This is a separable equation.  Derive the general solution
by separating variables and integrating.   
\item[(e)]  What is the exact formula for the solution of the 
differential equation with the initial condition $y(0) = 1$?
\end{enumerate}

\item Solve each of the following differential equations, finding
an explicit formula for $y$ as a function of $x$ if possible (you 
can leave the solution in implicit form if not).  In some cases
you are given initial conditions; use them to determine the arbitrary
constant in the general solution.
\begin{enumerate}
\item[(a)] $\displaystyle{y' = \frac{\tan(y)}{x^2 + 4x + 3}}$, find the general solution.
\item[(b)] $\displaystyle{y' = \frac{(x-1)y^5}{x^2(2y^3 - y)}}$, find the general solution.
\item[(c)] $y' = 2xy$ with $y(0) = 4$.
\item[(d)] $y' = x(y^2 + 1)$ with $y(0) = 1$.
\item[(e)] $y' = 2xy^2 + 3x^2y^2$ with $y(1) = -1$.  
\item[(f)] $y' = 3y(1 - y/20)$ with $y(0) = 5$.  (Note:  this is a logistic equation.)
\end{enumerate}

\item (Radiocarbon dating.) $C^{14}$ is a radioactive isotope 
of carbon with half-life of approximately $5700$ years.  It 
occurs naturally and is incorporated in living tissues by normal 
metabolic processes.   Assume that any $C^{14}$ present when the 
tissue dies decays exponentially from that time on.  
Carbon extracted from an ancient skull contains
only one-sixth as much $C^{14}$ as carbon extracted from present-day 
bone.  How old is the skull?

\item (Newton's Law of Heating/Cooling.) A pitcher of buttermilk initially 
at $25^\circ$ C is cooled
by setting it out on the front porch on a cold day when the exterior
temperature is a constant $0^\circ$ C.  Suppose that the buttermilk
has reached $15^\circ$ C after 20 minutes.  
\begin{enumerate}
\item[(a)] What will the temperature be after $25$ minutes?
\item[(b)] When will the temperature reach $5^\circ$ C?
\end{enumerate}

\item
The population of fish in a lake is attacked by a microscopic water-borne
parasite at $t = 0$, and as a result the population declines at 
a rate proportional to the {\it square root\/} of the population from that
time on.
\begin{enumerate}
\item[(a)]  Express this statement about the rate of growth of the 
population $P$ as a differential equation.
(There should be a constant of proportionality, say $-k$, 
in your equation.)  
\item[(b)]  Use separation of variables to find the general solution
of your differential equation.
\item[(c)]  At $t = 0$ there were 900 fish in the lake; 441 were
left after 6 weeks.  When did the fish population disappear entirely?
\end{enumerate}

\end{enumerate}
\end{document}
