\magnification=\magstep1
\hsize=7truein
\hoffset=-.25truein
\parindent=0pt

\input epsf
\def\p{{\bf p}}
\def\v{{\bf v}}
\def\u{{\bf u}}
\def\iint{\int\!\!\!\int}
\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\frac#1#2{{{#1}\over{#2}}}
\def\emph#1{{\it #1}}


\def\problemspace{0.5in}

%% TITLE
\noindent
{\bf MATH 228: Intro to Lin Alg and Diff Eqns}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl April 23, 2002}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}

\bigskip
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 10 multiple
choice questions worth 5 points each and 3 partial credits problems worth
10 points each. You start with 20 points.  On the partial credit problems
\emph{you must show your work and all important steps to receive credit}.

\bigskip

You may \emph{not} use a calculator.

\bigskip

\def\answerspace{0.25in}


%1
%% PROBLEM
Determine which of the following matrices is similar to $A =
\bmatrix{0&1&2\cr1&1&1\cr2&1&0}$.

%% CORRECT
$\bmatrix{-2&0&0\cr0&0&0\cr0&0&3}$
%% WRONG
$\bmatrix{1&0&0\cr0&1&0\cr0&0&0}$
%% WRONG
$\bmatrix{0&0&0\cr0&1&0\cr0&0&0}$
%% WRONG
$\bmatrix{-1&0&0\cr0&-2&0\cr0&0&1}$
%% WRONG
$\bmatrix{-1&0&0\cr0&1&0\cr0&0&5}$

%2
%% PROBLEM
Determine which of the following vectors is an eigenvector of the matrix
$A = \bmatrix{1&0&1\cr0&1&1\cr1&1&0}$.

%% CORRECT
$\bmatrix{1\cr1\cr1}$
%% WRONG
$\bmatrix{1\cr0\cr1}$
%% WRONG
$\bmatrix{1\cr1\cr0}$
%% WRONG
$\bmatrix{0\cr1\cr1}$
%% WRONG
$\bmatrix{0\cr1\cr0}$


%3
%% PROBLEM
The vectors $(1,1,0)$, $(1,0,-1)$, $(1,-1,1)$ are eigenvectors of a
symmetric matrix~$A$. Find an orthogonal matrix $Q$ such that
$Q^TAQ$ is a diagonal matrix.

%% CORRECT
$Q = \bmatrix{1/\sqrt{2}&\hfill1/\sqrt{6}&\hfill1/\sqrt{3}\cr
1/\sqrt{2}&-1/\sqrt{6}&-1/\sqrt{3}\cr
0&-2/\sqrt{6}&\hfill1/\sqrt{3}}$
%% WRONG
$Q = \bmatrix{1/\sqrt{2}&0&0\cr
0&1/\sqrt{3}&0\cr
0&0&-1/\sqrt{6}}$
%% WRONG
$Q = \bmatrix{1/\sqrt{2}&\hfill1/\sqrt{2}&-1/\sqrt{3}\cr
1/\sqrt{2}&0&\hfill1/\sqrt{3}\cr
0&-1/\sqrt{2}&-1/\sqrt{3}}$
%% WRONG
$Q = \bmatrix{\hfill1/\sqrt{2}&1/\sqrt{3}&\hfill1/\sqrt{6}\cr
0&1/\sqrt{3}&-2/\sqrt{6}\cr
-1/\sqrt{2}&1/\sqrt{3}&\hfill1/\sqrt{6}}$
%% WRONG
$Q = \bmatrix{1/\sqrt{2}&-1/\sqrt{3}&\hfill1/\sqrt{6}\cr
1/\sqrt{2}&\hfill1/\sqrt{3}&-1/\sqrt{6}\cr
0&\hfill1/\sqrt{3}&\hfill2/\sqrt{6}}$


%4
%% PROBLEM
Find the general solution of the differential equation $y' + \tan(t)
y = \cos(t)$.

%% CORRECT
$y = (t+c)\cos(t)$
%% WRONG
$y = t\cos(t) + c$
%% WRONG
$y = \frac{t+c}{\cos(t)}$
%% WRONG
$y = t\sin(t) + c$
%% WRONG
$y = \frac{\frac{1}{2}t + \frac{1}{4}\sin(2t) + c}{\cos(t)}$


%5
%% PROBLEM
Solve the initial value problem $y'= (1+y)^2 e^{-x}$, $y(0) = 0$.

%% CORRECT
$y = e^x - 1$
%% WRONG
$y = e^{-x} - 1$
%% WRONG
$y = e^{x-1}$
%% WRONG
$y = \frac{1}{3}e^{x/3}-1$
%% WRONG
$y = \big(e^{-x} - \frac{1}{2}\big)^{-1} - 1$


%6
%% PROBLEM
The population of moths in a certain area increases at a rate proprtional
to the current population and, in the absence of other factors, the
population doubles every week. However, birds in the area eat 10 moths
per day. If $P(t)$ denotes the number of moths after $t$ weeks, determine
which of the following equations describes $P(t)$.

%% CORRECT
$\frac{dP}{dt} = \ln(2) P - 70$
%% WRONG
$\frac{dP}{dt} = 2 P - 70$
%% WRONG
$\frac{dP}{dt} = 2 P - 10$
%% WRONG
$\frac{dP}{dt} = 2 t - 70$
%% WRONG
$\frac{dP}{dt} = \ln(2)P - 10t$


%7
%% PROBLEM
Solve the initial value problem $y'' + 2 y' - 15 y = 0$, $y(0)=1$,
$y'(0)=2$.

%% CORRECT
$\frac{7}{8} e^{3t} + \frac{1}{8} e^{-5t}$
%% WRONG
$\frac{3}{2} e^{3t} - \frac{1}{2} e^{5t}$
%% WRONG
$\frac{1}{3} e^{3t} + \frac{2}{3} e^{-5t}$
%% WRONG
$\frac{3}{8} e^{-3t} + \frac{5}{8} e^{5t}$
%% WRONG
$\frac{1}{2} e^{2t} + \frac{1}{2} e^{-15t}$


%8
%% PROBLEM
Find the general solution to the differential equation $y'' + 6 y' + 10 y = 0$.

%% CORRECT
$c_1 e^{-3t} \cos(t) + c_2 e^{-3t} \sin(t)$
%% WRONG
$c_1 e^{3t} \cos(t) + c_2 e^{3t} \sin(t)$
%% WRONG
$c_1 e^t \cos(3t) + c_2 e^t \sin(3t)$
%% WRONG
$c_1 e^{-3t} + c_2 e^t$
%% WRONG
$c_1 e^{6t}  + c_2 e^{10t}$  


%9
%% PROBLEM
Let $y_1$ and $y_2$ be solutions of $y'' + p(t)y' + q(t)y = 0$ where $p$
and $q$ are continuous on an open interval~$I$.  Four of the following
statements are equivalent, in the sense that each one implies the other
three. Find the one statement that is \emph{not} equivalent to any of the
other four.

%% CORRECT
$y_1(t) \ne y_2(t)$ for all $t$ in~$I$.
%% WRONG
$y_1$, $y_2$ are a fundamental set of solutions on~$I$.
%% WRONG
$y_1$, $y_2$ are linearly independent on~$I$.
%% WRONG
$W(y_1,y_2)(t_0) \ne 0$ for some $t_0$ in~$I$.
%% WRONG
$W(y_1,y_2)(t) \ne 0$ for all $t$ in~$I$.


%10
%% PROBLEM
Determine the longest interval in which the following initial value
problem is certain to have a unique twice differentiable solution.
$$(t^2+t-6) y'' + \cos(t) y' - \ln(t^2) y = e^t,\quad y(-1)=1,\quad
y'(-1)=2$$

%% CORRECT
$(-3, 0)$
%% WRONG
$(0, 2)$
%% WRONG
$(-3, 2)$
%% WRONG
$(-\infty, 0)$
%% WRONG
$(-2, \infty)$

%% END

\vfill\eject

\noindent\llap{11.\ }%
Find the matrix of the linear transformation $T(x,y) = (3x-y,2x+2y)$ with
respect to the basis $B = \{(5,2),(2,1)\}$.
\vfill\eject

\noindent\llap{12.\ }%
Consider the differential equation $(2 + \sin(y))dx + \cos(y)dy = 0$.
\bigskip
(a) Show that the equation \emph{not} exact.

\vfill
(b) Find an integrating factor, $\mu = \mu(x)$, for the equation.

\vfill
\vfill

(c) Use your answer in part (b) to solve the equation.

\vfill
\vfill
\eject

\noindent\llap{13.\ }%
Determine whether the functions $y_1(x) = x$ and $y_2(x) = x \cos{x}$
are linearly independent. Justify your answer.


\bye
