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\title{Brief Article}
\author{The Author}
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\normalafterproblemskip=2.5cm\pageannouncements={usual}
                                         \input exam.perm 
                                         \lastversion=0
                                         \date{{\large October 29, 1997}}
                                         \title{{\large\bf Math 108, Exam 2}}
\comment{\large 
\begin{minipage}[c]{6in}
This exam is worth a total of 100 points.  There are 11 problems -- 8 multiple choice 
and 3 partial credit.  Each multiple choice problem is worth 7 points.  Point values are
assigned next to the partial credit problems. Please show all work on the partial
credit section of the test inside the test booklet. Use the front sheet to answer 
the multiple choice section by putting a \fbox{$\times $} in the appropriate box.  
You will have until 9:20am to complete the exam. Good luck! 
\end{minipage}}

\answersheetfootline{\large\bf\fbox{\rule[-2mm]{0mm}{8mm}Sign
your name\hspace{6cm}}
\hspace{6cm}}\twocolumn{6}\noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm\pagenumberstyle=1
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\problem  
Find the constant solutions of the differential equation $y'=y^3+5y^2-24y$.


\correct $y=0,\ \ y=3, \ \  y=-8$
\wrong  $y=0, \ \ y=-6, \ \  y=4$
\wrong  $y=0, \ \ y=2, \ \ y=-12$
\wrong   $y=3, \ \ y=-8$
\wrong   $y=-6, \ \ y=4$

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\problem  

Determine the relationship of the two lines:
$$\eqalign{x-3y&=6,\cr 2x+y&=-1.\cr}$$


\wrong Cannot be determined.
\wrong The lines are parallel.
\wrong They are the same line.
\wrong There are infinitely many solutions.
\correct  There is a unique solution.


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\problem  
The function $y=t^2+3t+7$ is a solution to which of the following differential 
equations?


\wrong $y'-y=-t^2-4$
\correct $(y')^2-4y=-19$
\wrong $(y')^2-y=2$
\wrong  $y'-y=t^2+6$
\wrong  $y'=y^2+14$


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\problem  
If $\vec x = (1, 3, -2, 0)$ and $\vec y = (-1, 2, 4, -6)$ are two vectors in
$\Re^4$ , determine $(2\vec x) \cdot (-3\vec y)$.

\wrong $(6,-36,48,0)$
\correct  18
\wrong 54
\wrong $(5,0,-16,9)$
\wrong -42

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\problem  

An experimenter reports that a certain strain of bacteria grows at a rate 
proportional to the square of the size of the population.  Set up a differential
equation which describes the growth of the population and has a solution 
$y=f(t)$ where $f(t)$ is the size of the population at any time t.

\wrong  $(y')^2=k$
\wrong  $y'=k^2y$
\wrong  $y'=ky$
\correct $y'=ky^2$
\wrong  $y'=kt^2$

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\problem  

Determine the solution of the differential equation $y'=6t-5t^2$.

\wrong $6-10t$
\wrong $6t^2-5t^3+C$
\wrong $3t^2-5t^3+C$
\correct $3t^2-\frac{5}{3}t^3+C$
\wrong None of the above
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\problem  

What is the length of the vector $\vec x = (-2, 4, 2, 5)$?

\correct 7
\wrong 49
\wrong $\sqrt{41}$
\wrong 9
\wrong 41

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\problem  

Given the differential equation $y'=(y-2)(y+3)$, determine which of the following
is true for the constant solutions $y=2$ and $y=-3$.


\wrong $y=-3$ is unstable and $y=2$ is stable.
\wrong $y=2$ is stable and $y=-3$ is stable.
\correct $y=2$ is unstable and $y=-3$ is stable.
\wrong $y=-3$ is unstable and $y=2$ is unstable.
\wrong None of the above.


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\problem

Solve the following separable differential equation with the given initial
value.  Recall that $e^{xy}=(e^y)^x$ and that $e^{(...)+C}=Ae^{(...)}$.

(12 points)

$$y'=\frac{y^2+6}{yt}, \hspace{1cm} y(1)=3$$ 
\vspace*{17cm}

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\problem 
(20 points)
 
A certain individual decides to open an Individual Retirement Account (IRA).
This person makes continuous deposits of \$2000 each year.  The interest rate
is 5\%.

a)  Find the differential equation whose solution is given by the function 
$M(t)$ where $M(t)$ is the amount of money in the account at any time t.

\vspace{6cm}

b)  Sketch some solutions of this differential equation.

\vspace{6cm}

c)  Given that the person initially deposits \$3000, solve the differential 
equation to determine $M(t)$ - the amount of money in the account at any time t.

\vspace*{3cm}

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\problem  


Evaluate the following system of linear equations using Gaussian elimination.  Your
answer should be of the form $x_i=?$. (12 points)

$$\eqalign{x_1-3x_2+ \ \ x_3-2x_4&=1\cr 3x_1-6x_2+12x_3-6x_4&=3\cr 4x_1-9x_2+13x_3-8x_4&=4\cr
-2x_1+7x_2+ \ \ x_3+4x_4&=-2\cr}$$

\vspace*{15cm}

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