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\begin{document}

\begin{center}
{\Large Final

Math 2280 - 7}
\end{center}

\begin{itemize}
\item  No notes or books.
\item  Show all your work and justify all answers.
\item  You will be graded on the clarity of your answers.
\item  I will be done grading by Monday morning.
To get your grade you can do any, all, or none of the following:
\begin{itemize}
\item  Send me an email which I will respond to as soon as I can
\item  Give me a code name along with your name and I'll post your grade on my web page
\item  Come to my office on Monday morning between 9 and 9:15 AM. 
\end{itemize}
\end{itemize}

\begin{enumerate}

\item (5 points)  Consider the differential equation below.
How many (if any) solutions are there that satisfy the condition $y(3)=5$?
Remember to justify your answer.
\[
(2x+y) \frac{dy}{dx} -6x+y^2-x^2+2 = 0
\]



\item  (5 points)  Solve the initial value problem:
\[
y'-2xy=e^{x^2} \qquad y(0)=\pi
\]



\item  (10 points)  Recall that with only births and deaths in a population, the population
can be modeled by the differential equation
\[
\frac{dP}{dt} = (\beta - \delta)P
\]
where $\beta(t)$ is the birth rate and $\delta(t)$ is the death rate.
Suppose that the birth rate is constant
and the death rate is proportional to the population.
\begin{enumerate}
\item  What is the differential equation that models this situation
\item  What are the equilibrium solutions?
\item  Discuss the stability of these equilibrium solutions.
\item  Draw a graph of the solution curves to this differential equation.
\end{enumerate}




\item (5 points) Consider the initial value problem
\[
\frac{dy}{dx} = 2x-y \qquad y(0)=1
\]
Using Euler's method with one step, find an approximation for $y(1/2)$.


\newpage

\item  (10 points) Consider the differential equation
\begin{eqnarray} \label{DE}
y^{(3)}-2y''-y' = 18
\end{eqnarray}
\begin{enumerate}
\item  Find the general solution to the associated homogeneous differential equation.
Explain why what you found is the general solution.
\item  Find a particular solution to the original differential equation (\ref{DE}).
\item  Find the general solution to the differential equation (\ref{DE}).
Explain why this is the general solution.
\end{enumerate}


\item  (5 points) Consider the third order differential equation
\[
x'''-6x'-\sin(t)x + e^t
\]
\begin{enumerate}
\item  Is this differential equation linear?  Why or why not?
\item  Transform this differential equation into a system of first order differential equations.
\end{enumerate}

\item  (10 points) Consider the system of differential equations
\begin{eqnarray*}
\frac{dx}{dt} & = & 4x+y \\
\frac{dy}{dt} & = & 6x - y
\end{eqnarray*}
Find the general solution to this system.

\item  (10 points)  Consider the (almost linear) system of differential equations:
\begin{eqnarray*}
\frac{dx}{dt} & = & x^2-4x-3y-3 \\
\frac{dy}{dt} & = & -2xy-x-3
\end{eqnarray*}
\begin{enumerate}
\item  Show that $(1,-2)$ is a critical point of the system.
\item  Determine the type and stability of the critical point $(1,-2)$.
\item  Draw the phase portrait for this almost linear system near $(1,-2)$ 
(in other words, don't worry about any other critical points, 
just draw what the phase diagram looks like near this critical point).
\end{enumerate}


\newpage

\item  (10 points) Find the following (if possible):
(You may do these using any method, including using the tables)
\begin{enumerate}
\item 
\[
\lap  \left\{ 5 \left( e^x \right)^3 \right\}
\]

\item  
\[
\lap  \left\{ 5e^{x^3} \right\}
\]

\item
\[
\lap^{-1} \left\{ \frac{s}{s-1} \right\}
\]

\item
\[
\lap^{-1} \left\{ \frac{3s+5}{s^2-6s+25} \right\}
\]
\end{enumerate}


\item  (10 points)  Solve the following initial value problem using the Laplace transform
\[
x''+x = \cos t \qquad x'(0)=0, x(0)=1
\]



\item  (10 points) Consider the function defined on $(0,3)$ as $f(t)=-2$.
\begin{enumerate}
\item  Extend this function to be an even function of period 6.
Draw the graph of this function.  (You don't need to find a formula)
\item  Extend this function to be an odd function of period 6.
Draw the graph of this function.  (You don't need to find a formula)
\item  For the odd extension of $f$, find the Fourier series of this extended $f$
\item  Discuss the convergence of the Fourier series found.
\end{enumerate}


\newpage

\item  (10 points) Consider the mass spring system
\[
mx''+cx'+kx = F(t)
\]
where $m=2$, $k=7$, $c=0$ and $F(t)$ is the periodic function defined on one period to be
\[
F(t) = \left\{
\begin{array}{rr}
-t & \quad -5 < t < 0 \\
t &  \quad 0 < t < 5
\end{array}
\right.
\]
It can be shown (you don't have to do this) that $F(t)$ has the following Fourier series
\[
F(t) = \frac{5}{2} - \frac{20}{\pi^2} \sum_{n \mbox{ odd}} 
\frac{1}{n^2} \cos\left(\frac{n\pi t}{5} \right)
\]
\begin{enumerate}
\item  Find a formal Fourier series solution to this differential equation.
\item  Is there any type of resonance phenomena occurring in this differential equation?
\end{enumerate}




\item  (5 points; of course all answers, except none, are correct)
\begin{enumerate}
\item  Was the fact that this class was only 6 weeks long necessary for your schedule?
\item  Did you like or dislike the 6 week format of this class? Why?
\item  Do you think that over the summer this class should be offered as a 6 week class or 
as a 12 week class (there can't be both)?  Why?
\item  Any other comments?
\end{enumerate}



\end{enumerate}

\end{document}