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\title{\textcolor{red}{\textbf{MAT 274 \\
Elementary Differential Equations \\
Final Examination}}}
\date{}

\markboth{Mathematics 274}{Mathematics 274}


\begin{document}

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            \strut P\#1    &   10          \\\hline
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            \strut P\#4    &   15          \\\hline
            \strut P\#5    &   15          \\\hline
            \strut P\#6    &   15          \\\hline
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            \strut Total   &  100          \\\hline
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   \textbf{WARNING}:
   This test is designed to be completed without notes or cheat
   sheets at the Department of Mathematics \& Statistics testing
   center. There is no time limit.
   It requires the use of a calculator with basic operations and
   functions. If additional space is needed, write your name on
   every page submitted, and staple them to your exam. Show your
   work and circle your answers.
   \newpage

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\begin{exam}
\item
Suppose that a bacterial population  is growing according to the
logistic equation
\[\frac{dP}{dt}=rP(1-\frac{P}{k}).\]
Assume that this population begins with $1000$ bacteria and
doubles in the first $10$ hours. The population is observed to
stabilize at $20,000$ bacteria.
\begin{enumerate}
\item What is the value of the parameter $k$?
\item Determine an expression for $P$ in terms of $t$ and the parameter $r$.
\item Use the given information and the result of part 2 to determine $r$.
\item Find the number of bacteria present after $20$ hours.
\end{enumerate}

\newpage

\item
Consider the initial value problem
\[ \left\{\begin{array}{l} \ds y'=\frac{2x}{2y+1},\\[4pt]
y(1)=0.  \end{array}\right. \]
\begin{enumerate}
\item
Determine the solution of this problem in implicit form (i.e., a relation
of the form $F(x,y)=0$).
\item
Find an explicit form of the solution (i.e., $y = y(x)$. If more than one
solution are obtained from part 1, you need to justify your choice).
For what range of $x$ values is this solution valid?
\item
Calculate the approximation of $y(2)$ obtained by applying two (equal)
steps of Euler's method to the above problem.
\end{enumerate}

\newpage

\item
\rule{0pt}{1pt}
\begin{enumerate}
\item
Find the inverse Laplace transform for $\ds \frac{1}{(s+2)(s^2+2s+1)}$.
\item Use your result to solve the initial value problem
\[ \left\{\begin{array}{l} y''+3y'+2y=e^{-t},\\[4pt]
y(0) = 1, \\[4pt] y'(0)= \frac{1}{2}.  \end{array}\right. \]
\end{enumerate}

\newpage

\item
Find the solution of the initial value problem
\[ {\bf y}'= \left[\begin{array}{cc} 2&3\\2&1 \end{array}\right]{\bf y},
\quad
   {\bf y}(0) = \left[\begin{array}{c}2\\-1\end{array}\right].  \]

\newpage

\item
Determine the solution of the IVP $t^2y'+2ty=\cos(t)$, $y(-\pi)=0$.
What is the interval of validity of the solution?

\newpage

\item
Consider the ODE $t^2y''-ty'+y=t$ for $t>0$.
\begin{enumerate}
\item Verify that the functions $y_1(t)=t$ and $y_2(t)=t\ln(t)$ are
solutions of the associated homogeneous problem $t^2y''-ty'+y=0$ for $t>0$.
\item Prove or disprove that the functions $y_1$ and $y_2$ are linearly
independent on the interval $t>0$.
\item Determine the general solution of the ODE.
\end{enumerate}

\newpage

\item
Solve the initial value problem
\[ \left\{
\begin{array}{l} 2y''+4y'+4y=e^{-t} \\[4pt]
y(\pi)=0 \\[4pt]
y'(\pi)=0
\end{array}
\right. \]
(any method you want).


\end{exam}

\end{document}