\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\topmargin}{-1in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}

\usepackage{amsfonts}


\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\ra}{\rightarrow}


\begin{document}

\begin{center}
{\Large TEST 2

Math 2280 - 7}
\end{center}

Instructions:
\begin{itemize}
\item  Due: Beginning of Class, Tuesday July 24.
\item  Show all your work and justify all answers.
\item  You may use your book, notes and calculator but you cannot you anyone else or a computer.
\item  You have several days to do this, so what you turn in should be neat.
You will be marked down if I have trouble reading or following your work.
(On the last test, the neatest tests had each problem on a separate page).
\end{itemize}


\begin{enumerate}



\item  (10 points)  Find a particular solution to the non-homogeneous differential equation:
\[
y'' + y = 5e^x \sin x
\]

\item  (15 points)  Consider a mass - spring - shock absorber system as in figure 3.4.1 of the book.
This is the standard picture I've drawn many times.
The mass weighs 1 kg and the spring is stretched 2 meters by a force of 20 Newtons.
The shock absorber provides 2 Newtons of resistance for each meter per second of speed of the mass.
Let $x$ be the distance of the mass from its equilibrium position.
\begin{enumerate}
\item  What is the differential equation that represents this set up?
\item  Find the general solution to this system.
\item  Now suppose the mass is acted on by a constant force of 30 Newtons to the left.
What is the new differential equation and its general solution??
\item  Determine (mathematically -- do not use a a physical argument) what the limiting behavior of this system
now is for any initial conditions.  
(Discuss $\lim_{t \ra \infty} x(t)$ for any solution $x(t)$).
\item  Explain this limiting behavior from a physical viewpoint 
(why does the mass-spring system act this way?)
\end{enumerate}

\item  (15 points)  Consider the following system of differential equations
(where prime (') denotes derivative with respect to $t$):
\begin{eqnarray*}
x'' & = & 6x +12t -7z + 5t\cos t \\
y^{(3)} & = & 87ty+6z' \\
z'' & = & 8t^3z - y' \sin t
\end{eqnarray*}
\begin{enumerate}
\item  Convert this system to a first order system.
\item  Is this first order system linear?  Why or why not?  
Is it homogeneous?  Why or why not?
\end{enumerate}

\item  (20 points) Consider the system:
\[
t \mbf{X}'=\mbf{A}\mbf{X}
\]
To find a solution to this system, we try the solution $\mbf{x}=\mbf{v}t^r$ where
$\mbf{v}$ is a constant vector and $r \in \R$.
\begin{enumerate}
\item  Find conditions on the constant vector $\mbf{v}$, 
and the scalar $r$ so that choice of $\mbf{x}$ gives a solution.
\item  Apply the procedure above to find the general solution to
\[
t \mbf{X}' = 
\left[
\begin{array}{cc}
5 & -1 \\
3 & 1 
\end{array}
\right] \mbf{X}
\]
\item  Prove that what you found in the previous part is really the general solution.
\\
(What I'm looking for here is an argument appealing to Theorem 3 of section 5.1.)

\end{enumerate}



\item  (20 points) Consider the system
\begin{eqnarray*}
\frac{dx}{dt} & = & y-x(x^2+y^2) \\
\frac{dy}{dt} & = & -x-y(x^2+y^2)
\end{eqnarray*}
\begin{enumerate}
\item  Show that $(0,0)$ is an isolated critical point of this system and that 
this is an almost linear system in a neighborhood of $(0,0)$.
\item  What type of critical point is $(0,0)$ of the linear system? (and why of course)
\item  What type of critical point is $(0,0)$ of the non-linear system?
\\
Hints: Note that $r^2=x^2+y^2$.  
As a first step, take the derivative of this implicitly and see if you can show $\frac{dr}{dt}<0$.
\end{enumerate}

\item  (20 points)  Consider the system of equations
\begin{eqnarray*}
\frac{dx}{dt} & = & x-x^2-xy \\
\frac{dy}{dt} & = & 3y-xy-2y^2
\end{eqnarray*}
\begin{enumerate}
\item  Determine all critical points of this system.
\item  Determine what type of critical points these all are (proper node, saddle point, etc.)
\item  Determine the stability of each critical point.
\item  Draw a phase portrait for this system.
\end{enumerate}



\end{enumerate}

\end{document}