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\begin{document}   \setlength{\baselineskip}{20pt}     \centerline {\bf
Math 226T \ Test III  } \centerline { November 30, 1988}      

\noindent Name:\ {\underline {\hspace* {3in}}}      

















\vspace {1in}   $\begin{array}{rrr}      

















\hspace* {4in} &  \rm Poss. & \rm Lost \\  & & \\  1\  \& \ 2 & \rm 10 \
pts. &  {\underline {\hspace* {.5in}}}\\   & & \\  3 & \rm 15\ pts. & 
{\underline {\hspace* {.5in}}}\\  & & \\   4 & \rm 15\ pts. & {\underline
{\hspace* {.5in}}}\\  & & \\  5 & \rm 15\  pts. &  {\underline {\hspace*
{.5in}}}\\  & & \\  6 & \rm 15\  pts. &  {\underline {\hspace* {.5in}}}\\  & &
\\  7 & \rm 15\  pts. &  {\underline {\hspace* {.5in}}}\\  & & \\  8 & \rm
15\  pts. &  {\underline {\hspace* {.5in}}}\\  & & \\  & & \\ \rm Total\
Lost & &  {\underline {\hspace* {.5in}}}\\  & & \\  & & \\  & & \\  \rm
TOTAL & &  {\underline {\hspace* {.5in}}}  \end{array}$      

\vspace {2in}   

\noindent  I have not violated the Honor Code in any way with regard to 
this work.    

\noindent Signature \ {\underline {\hspace* {3in}}}      

\pagebreak   \begin{itemize}  \item[1. (5 pts.)] Do the functions $e^x$ and
$e^{x+2}$ form a fundamental  system of solutions of the equation
$y^{\prime\prime} - y = 0$?     

\hspace{2in}YES \hspace{1in}    NO   

\vspace {1.5in}    

\item[2. (5 pts.)] Are the functions $\cos x$ and  $\cos (x + \frac{
\textstyle \pi}{ \textstyle 6})$ linearly independent?     

\hspace{2in}YES \hspace{1in}    NO  

\vspace {1.5in}   

\item [3. (15 pts.)] Find the explicit solution to  $\frac {\textstyle
dy}{\textstyle dx}   =     \frac{\textstyle x^2  +  2y^2}{\textstyle 2xy}$
subject to the initial  condition $y(e) = e$.   To receive partial credit for
an incorrect answer,  you must make a serious effort to check  that your
answer is correct.   

\pagebreak     

 \item [4. (15 pts.)]  Solve $y^{\prime \prime } y^{\prime } = x-1$  subject
to  $y^{\prime }(0) = 1 \mbox { ; } y(0) = 1$. To receive partial credit for an
incorrect answer, you must  make a serious effort to check that your
answer is correct.     

\vspace {3.5in}   

\item [5. (15 pts.)]  Write a first order differential equation with initial 
condition for the  amount of salt present at time t in the following
problem. DO NOT SOLVE YOUR  EQUATION.   A tank with unlimited capacity
originally contains 600 gallons of water with 100 pounds of  salt in it. 
Water containing 1 pound of salt per gallon  is entering the tank at a rate
of 5  gallons per minute  and the mixture leaves  at the rate of 3 gallons
per minute.   

\pagebreak     

\item [6. (15 pts.)]  Solve $3xy^2 dx  +  2x^2y  dy  =  0$ implicitly, by first
finding an  integrating factor.   To receive partial credit for an incorrect
answer, you must make a  serious effort to check that the integrating
factor that you  use is a correct one.   

\vspace {4in}   

\item [7. (15 pts.)]  Knowing that $y = x$ is a solution of the linear
homogeneous differential equation  $$x^2y^{\prime\prime} - 3xy^{\prime}
+ 3y = 0$$  find the particular solution which satisfies the initial
conditions   

















\noindent $y(1) = 0 $, $y^{\prime}(1) = -2$.   

\pagebreak  

\item[8. (15 pts.)]  Find the general solution of the equations     

















\noindent  (a)\hfill$y^{\prime\prime} + y^{\prime} - 6y = 0$\hfill\     

\vspace {2.5in}    









\noindent  (b)\hfill$y^{\prime\prime} - 6y^{\prime} + 9y = 0$\hfill\      

\vspace {2.5in}    









\noindent  (c)\hfill$y^{\prime\prime} - 2y^{\prime} + 2y = 0$\hfill\     

































\end{itemize}     

 \end{document}     































