%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Exercises">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 16:31:46 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="begin" HREF="DM10-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="end" HREF="DM10-5.tex">}
%TCIDATA{<LINK REL="next" HREF="DM11-1.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM10.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM10-4.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{Exercises\label{Exercises}}

\begin{enumerate}
\item[1.]  \label{q1}Find the general solution of the equation $y^{\prime
\prime }-6y^{\prime }+5y=0$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a1}

\item[2.]  \label{q2}Find the general solution of the equation $%
x^{2}y^{\prime \prime }-3xy^{\prime }-6y=0$.\hfill \hyperref{\TCIIcon{%
BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a2}

\item[3.]  \label{q3}Find the general solution of the equation $%
2x^{2}y^{\prime }=xy+3y^{2}$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a3}

\item[4.]  \label{q4}Solve the initial-value problem $y^{\prime }+y=2$, $%
y(0)=0$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{%
}{}{a4}

\item[5.]  \label{q5}Solve the initial-value problem $\frac{dy}{dt}-y-3x=0$, 
$-5y+\frac{dx}{dt}-3x=0$, $y(0)=1$, $x(0)=-1$.

\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a5}

\item[6.]  \label{q6}The model $\frac{dP}{dt}=kP(M-P)$ describes a
population $P$ limited by a maximum sustainable population $M$, where $k$ is
a growth-rate constant. Predict the population ten years from now, assuming
a current population of $500$, if the maximum sustainable population is $2000
$ and the population ten years ago was $300$.\hfill \hyperref{\TCIIcon{%
BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a6}

\item[7.]  \label{q7}The height of a hanging cable satisfies the
differential equation $y^{\prime \prime }=\alpha \sqrt{1+(y^{\prime })^{2}}$%
. Find a solution for this differential equation with $\alpha =1$, assuming $%
y(0)=1$ and $y^{\prime }(0)=0$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a7}

\item[8.]  \label{q8}Newton's law of cooling states that the rate of change
in the temperature of an object is given by $\frac{dT}{dt}=k(T-R)$, where $k$
is a constant that depends on how well insulated the object is, $T$ is the
temperature of the object, and $R$ is room temperature. A cup of coffee is
initially $160^{\circ }$; 10 minutes later, it is $120^{\circ }$. Assuming
the room temperature is a constant $70^{\circ }$, give a formula for the
temperature at any time $t$. What will the temperature of the coffee be
after 20 minutes?\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in%
}{0in}}{}{}{a8}

\item[9.]  \label{q9}Coyotes ($C$), rabbits ($R$), and vegetation ($V$) are
about the only living things on a small, isolated island. The coyotes rely
on rabbits for food, and the rabbits eat the vegetation. The biomass is
governed by the differential equations \hfill \hyperref{\TCIIcon{%
BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a9} 
\begin{eqnarray*}
\dfrac{dC}{dt} &=&-0.2C+0.0004CR \\
\dfrac{dR}{dt} &=&-0.01CR+.001RV \\
\dfrac{dV}{dt} &=&-0.001RV+.001V(1000-V)
\end{eqnarray*}
where $t$ is measured in years. Assume an initial population of $100$
coyotes, $1000$ rabbits, and $1000$ units of vegetation. Predict the values
of $C$, $R$, and $V$ over the next $5$ years. (Choose \textsf{Numeric}.)

\item[10.]  \label{q10}Let $v$ and $u$ denote the horizontal and vertical
components of velocity, respectively, of a golf ball in flight, and let $%
\left( x,y\right) $ denote its position. Define the following constants.
\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a10} 
\begin{eqnarray*}
w &=&\frac{1.62}{16}\text{, weight of the ball in pounds} \\
c &=&\frac{1}{200}w\text{, drag term} \\
g &=&32\text{, acceleration due to gravity in ft/s}^{2} \\
\theta &=&\frac{\pi }{8}\text{, angle of the club head} \\
k &=&c\text{, lift due to backspin} \\
z &=&150\text{, club head velocity in ft/s}
\end{eqnarray*}
\end{enumerate}

Solve the following system of differential equations numerically, then plot $%
(x,y)$ parametrically to view the flight of a golf ball.

\begin{eqnarray*}
\frac{dv}{dt} &=&-\frac{gcv}w \\
\frac{du}{dt} &=&-\frac gw\left( w+cu-kv\right) \\
\frac{dx}{dt} &=&v \\
\frac{dy}{dt} &=&u \\
v(0) &=&z\cos ^2\theta \\
u(0) &=&z\cos \theta \sin \theta \\
x(0) &=&0 \\
y(0) &=&0
\end{eqnarray*}

\begin{enumerate}
\item[11.]  \label{q11}Solve the partial differential equation $%
2D_{x}u+3D_{xy}u=xy^{2}$. Verify the solution by defining $F_{1}(x)$ and $%
F_{2}(x)$ to be generic functions and substituting the suggested solution
back into the partial differential equation.\hfill \hyperref{\TCIIcon{%
BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a11}
\end{enumerate}

\section{Solutions\label{Solutions}}

\begin{enumerate}
\item[1.]  \label{a1}\textsf{Solve ODE + Exact} \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q1} 
\[
y^{\prime \prime }-6y^{\prime }+5y=0\text{, Exact solution is: }y\left(
t\right) =C_{1}e^{5t}+C_{2}e^{t} 
\]

\item[2.]  \label{a2}\textsf{Solve ODE + Exact }(Specify $x$) \hfill 
\hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q2} 
\[
x^{2}y^{\prime \prime }-3xy^{\prime }-6y=0\text{, Exact solution is: }%
y\left( x\right) =C_{1}x^{2+\sqrt{10}}+C_{2}x^{2-\sqrt{10}} 
\]

\item[3.]  \label{a3}\textsf{Solve ODE + Exact }(Specify $x$) \hfill 
\hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q3} 
\[
2x^{2}y^{\prime }=xy+3y^{2}\text{, Exact solution is: }\frac{1}{y\left(
x\right) }=\frac{3+C_{1}\sqrt{x}}{x} 
\]

\item[4.]  \label{a4}\textsf{Solve ODE + Laplace} \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q4} 
\[
\begin{array}{r}
y^{\prime }+y=2 \\ 
y(0)=0
\end{array}
\text{, Laplace solution is: }y\left( t\right) =2-2e^{-t} 
\]

\item[5.]  \label{a5}\textsf{Solve ODE + Exact} \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q5} 
\[
\begin{array}{rl}
\frac{dy}{dt}-y-3x & =0 \\ 
-5y+\frac{dx}{dt}-3x & =0 \\ 
y(0) & =1 \\ 
x(0) & =-1
\end{array}
\quad \text{Exact solution is: } 
\begin{array}{rl}
\QDATOP {} {}y\left( t\right) & =e^{-2t} \\ 
\QDATOP {} {}\;\,x\left( t\right) & =-e^{-2t}
\end{array}
\]

\item[6.]  \label{a6}\textsf{Solve ODE + Exact} 
\[
\frac{dP}{dt}=kP(2000-P)\text{, Exact solution is: }\frac{1}{P\left(
t\right) }=\frac{1}{2000}+e^{-2000kt}C_{1} 
\]

\textsf{Solve + Exact} \hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{%
0.2309in}{0.2802in}{0in}}{}{}{q6} 
\[
\begin{array}{rl}
\dfrac{1}{500} & =\dfrac{1}{2000}+e^{-2000k\left( 10\right) }C_{1} \\ 
\dfrac{1}{300} & =\dfrac{1}{2000}+e^{-2000k\left( 0\right) }C_{1}
\end{array}
\]
\[
\text{Solution is: }\left\{ C_{1}=\frac{17}{6000},k=-\frac{1}{20000}\ln 
\frac{9}{17}\right\} 
\]

\textsf{Define + New Definition} 
\begin{eqnarray*}
C_{1} &=&\frac{17}{6000} \\
k &=&-\frac{1}{20000}\ln \frac{9}{17} \\
P\left( t\right) &=&\left( \frac{1}{2000}+e^{-2000kt}C_{1}\right) ^{-1}
\end{eqnarray*}

\textsf{Evaluate} 
\[
P\left( t\right) =\frac{6000}{3+17\exp \left( \frac{1}{10}\left( 2\ln 3-\ln
17\right) t\right) } 
\]

\textsf{Evaluate Numerically} 
\[
P\left( 20\right) =773 
\]

\item[7.]  \label{a7}\textsf{Solve ODE + Exact, Simplify} \hfill \hyperref{%
\TCIIcon{BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q7} 
\begin{eqnarray*}
y^{\prime \prime } &=&\sqrt{1+\left( y^{\prime }\right) ^{2}} \\
y(0) &=&1 \\
y^{\prime }(0) &=&0
\end{eqnarray*}
Exact solution is: 
\begin{eqnarray*}
y\left( t\right) &=&\cosh t\cosh 0-\sinh t\sinh 0 \\
&=&\cosh t
\end{eqnarray*}

\item[8.]  \label{a8}\textsf{Solve ODE + Exact} 
\[
\frac{dT}{dt}=k\left( T-70\right) \text{, Exact solution is: }T\left(
t\right) =70+e^{kt}C_{1} 
\]

\textsf{Solve + Exact} \hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{%
0.2309in}{0.2802in}{0in}}{}{}{q8} 
\[
\begin{array}{l}
160=70+e^{k\left( 0\right) }C_{1} \\ 
120=70+e^{k\left( 10\right) }C_{1}
\end{array}
\text{, Solution is: }\left\{ C_{1}=90,k=\frac{1}{10}\ln \frac{5}{9}\right\} 
\]

\textsf{Define + New Definition} 
\begin{eqnarray*}
C_{1} &=&90 \\
k &=&\frac{1}{10}\ln \frac{5}{9}
\end{eqnarray*}
\textsf{Simplify} 
\[
T\left( t\right) =70+e^{kt}C_{1} 
\]

\textsf{Evaluate} 
\[
T\left( t\right) =70+90\exp \left( \frac{1}{10}\left( \ln 5-2\ln 3\right)
t\right) 
\]

\textsf{Evaluate Numerically} 
\[
T(20)=97.8^{\circ } 
\]

\item[9.]  \label{a9}\textsf{Solve ODE + Numeric} $
\begin{array}{l}
\dfrac{dC}{dt}=-0.2C+0.0004CR \\ 
\dfrac{dR}{dt}=-0.01CR+0.001RV \\ 
\dfrac{dV}{dt}=-0.001RV+0.001V\left( 1000-V\right) \\ 
C(0)=100 \\ 
R(0)=1000 \\ 
V(0)=1000
\end{array}
$

\textsf{Plot 2D + ODE} $C,R,V$\hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint%
}{0.2309in}{0.2802in}{0in}}{}{}{q9}

\FRAME{dtbpF}{3.0277in}{2.028in}{0pt}{}{}{plotode1.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
3.0277in;height 2.028in;depth 0pt;original-width 216.8125pt;original-height
144.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'PlotODE1.wmf';file-properties "XNPEU";}}

Observe what these functions predict for a longer term.\FRAME{dtbpF}{3.0277in%
}{2.028in}{0pt}{}{}{plotode2.wmf}{\special{language "Scientific Word";type
"GRAPHIC";display "PICT";valid_file "F";width 3.0277in;height 2.028in;depth
0pt;original-width 216.8125pt;original-height 144.5625pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'PlotODE2.wmf';file-properties "XNPEU";}}

\item[10.]  \label{a10}\textsf{Define + New Definition} 
\[
w=\frac{1.62}{16} 
\]
\textsf{Define + New Definition} \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q10} 
\[
c=\frac{1}{200}w 
\]
\textsf{Define + New Definition} 
\[
g=32 
\]
\textsf{Define + New Definition} 
\[
\theta =\frac{\pi }{16} 
\]
\textsf{Define + New Definition} 
\[
k=c 
\]
\textsf{Define + New Definition} 
\[
z=170 
\]
\textsf{Solve ODE + Numeric} 
\begin{eqnarray*}
\frac{dv}{dt} &=&-\frac{gcv}{w} \\
\frac{du}{dt} &=&-\frac{g}{w}\left( w-kv\right) \\
\frac{dx}{dt} &=&v \\
\frac{dy}{dt} &=&u \\
v(0) &=&z\cos ^{2}\theta \\
u(0) &=&z\cos \theta \sin \theta \\
x(0) &=&0 \\
y(0) &=&0
\end{eqnarray*}

\textsf{Plot 2D + Parametric} $(x,y)$ \FRAME{dtbpF}{3.0277in}{2.028in}{0pt}{%
}{}{paraplot.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 3.0277in;height 2.028in;depth 0pt;original-width
216.8125pt;original-height 144.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Paraplot.wmf';file-properties "XNPEU";}}

\item[11.]  \label{a11}\textsf{Solve PDE} 
\[
2D_{x}u+3D_{xy}u=xy^{2} 
\]
\textsf{Define + New Definition} \hfill \hyperref{\TCIIcon{%
BITMAPSETProbSolvHint}{0.2309in}{0.2802in}{0in}}{}{}{q11} 
\[
F_{1}(x) 
\]
\textsf{Define + New Definition } 
\[
F_{2}(x) 
\]
\textsf{Define + New Definition} 
\[
u(x,y)=\frac{1}{4}x^{2}y^{2}-\frac{3}{4}x^{2}y+\frac{9}{8}%
x^{2}+F_{1}(y)+e^{-2y/3}F_{2}(x) 
\]
\textsf{Evaluate } 
\[
2D_{x}u(x,y)+3D_{xy}u(x,y) 
\]
\end{enumerate}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Ordinary Differential Equations}{}{}{%
DM10-1.tex#Differential equation}

\item  \hyperref{Systems of Ordinary Differential Equations}{}{}{%
DM10-2.tex#Exact solutions}

\item  \hyperref{Numerical Methods for ODEs}{}{}{DM10-3.tex#Solve ODE,
numeric}

\item  \hyperref{Partial Differential Equations}{}{}{DM10-4.tex#PDE}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Differential
equations}{}{}{DMIndex.tex#Differential equations}, \hyperref{Plotting 2D}{}{%
}{DMIndex.tex#Plotting 2D}, \hyperref{Plotting 3D}{}{}{DMIndex.tex#Plotting
3D}, \hyperref{Solve ODE}{}{}{DMIndex.tex#Solve ODE}, \hyperref{Partial
differential equations}{}{}{DMIndex.tex#Partial differential equations}, 
\hyperref{Table of values}{}{}{DMIndex.tex#Table of values}}

\end{document}
