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\thispagestyle{empty} {\large\bf Name}
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
%{{\bf Time of your class}~\underline{~~~~~~~~~~} }
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\thispagestyle{empty} \hfill  2280 Exam 1 S2009
[8:35]\newline{\large\bf Name}.
\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill}}
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\begin{center}\Large\bf
Differential Equations 2280 \\
\small Midterm Exam 1 [8:35]\\
\small Wednesday, 25 February 2009
\end{center}

{\bf Instructions}: This in-class exam is 50 minutes. No
calculators, notes, tables or books. No answer check is expected.
Details count 3/4, answers count 1/4.

\begin{exercises}
\prob{1.}{(Quadrature Equations)} ~\newline
     \BF{(a)} [25\%] Solve $y'=\dd\frac{3+x^2}{1+x^2}$. \newline
     \BF{(b)} [25\%] Solve $y'=(2\sin x + \cos x)(\sin x - 2\cos x)$.\newline
     \BF{(c)} [25\%] Solve $y'=\dd \frac{x\tan\left(\ln(1+x^2)\right)}{1+x^2}$, $y(0)=2$.\newline
     \BF{(d)} [25\%] Find the position $x(t)$ from the velocity model $\dd\frac{d}{dt}(t^2v(t))= 0$, $v(2)=10$ and the position model $\dd\frac{dx}{dt}=v(t)$, $x(2)=-20$.

\iffalse {\small [Integral tables will be supplied
 for anything other than basic formulas. This sample problem
 would require no integral table. The exam problem will be
 shorter.]}\fi

\end{exercises}
\EJECT


\begin{exercises}
\prob{2.}{(Classification of Equations)} ~\newline
 The differential equation $y'=f(x,y)$ is defined to be {\bf separable} provided
$f(x,y)=F(x)G(y)$ for some functions $F$ and $G$.

 \BF{(a)} [40\%]  Check (\fbox{X}) the problems that can be put into separable form.
 No details expected.
\begin{center}\large\begin{tabular}{|lp{0.3\linewidth}|lp{0.4\linewidth}|}
\hline
 \fbox{$\phantom{X}$} & $ y'+xy=y(2x+e^x)+x^2y$ & \fbox{$\phantom{X}$} &   $y'=(x-1)(y+1) + (1-x)y$ \\[2ex]  \hline
  \fbox{$\phantom{X}$} &  $y'=2e^{2x-y}e^{3y}+3e^{3x+2y}$ & \fbox{$\phantom{X}$} &  $\dd y'+x^2e^y=xy$ \\[2ex] \hline
  \end{tabular}
\end{center}

 \BF{(b)} [10\%] Is $y'+x(y+1)=ye^x+x$ separable? No details expected.

 \BF{(c)} [10\%] Give an example of $y'=f(x,y)$ which is separable and linear but not quadrature. No details expected.

  \BF{(d)} [40\%] Apply tests to show that $y'=x + e^y$ is not separable and not linear. Supply all details.

\end{exercises}
\EJECT

\begin{exercises}
\prob{3.}{(Solve a Separable Equation)} ~\newline
 % u:=(2*x^2+3)/(x+2); int(u,x);
 % # 1/3*ln(y-2)+2/3*ln(y+1)= -exp(2-x)+x^2-4*x+11*ln(x+2)+C
 Given
   $\dd (x+3)(y+1)y' = \left( (x+3)e^{-x+2} +
3 x^2+2 \right)(y-1)(y+2)$.


  Find a non-equilibrium solution in implicit form. ~\newline
 To save time, {\bf do not solve} for $y$ explicitly and {\bf do not
 solve} for equilibrium solutions.

\end{exercises}
\EJECT

\begin{exercises}
\prob{4.}{(Linear Equations)} ~\newline
% v:='v':de:=5*diff(v(t),t)=-160+(25/(2*t+3))*v(t); ic:=v(0)=32; dsolve({de,ic},v(t));
 \BF{(a)} [60\%] Solve the linear model $\dd 5 x'(t)= - 160 + \frac{25}{2t+3}x(t)$, $x(0)=32$. Show
 all integrating factor steps.\newline
 \BF{(b)} [20\%]  Solve the homogeneous equation $\dd \frac{dy}{dx}-(2x)y=0$.
\newline
 \BF{(c)} [20\%] Solve $\dd 5\frac{dy}{dx}+10y=7$ using the superposition principle $y=y_h+y_p$. Expected are answers for $y_h$ and $y_p$.

\end{exercises}



\EJECT

\begin{exercises}
\prob{5.}{(Stability)} ~\newline
 \BF{(a)} [50\%] Draw a phase line diagram for the differential equation
 $$\frac{dx}{dt} = \left(\ln(1+5x^2)\right)^{1/5}\left(|2x-1|-3\right)^3(2+x)^2(4-x^2)(1-x^2)^{3}e^{\cos x}.$$
 Expected in the phase line diagram are equilibrium
 points and signs of $dx/dt$.
 \newline
 \rule{\textwidth}{0.01in}

 \vspace*{1.5in}

        \BF{(b)} [50\%] Assume an autonomous equation $x'(t)=f(x(t))$. Draw a phase diagram with at least 12 threaded curves, using the phase line diagram given below. Add these labels as appropriate: funnel, spout, node [neither spout nor funnel], stable, unstable.

 \def\AA{\rule{5ex}{0.02in}}
 \def\BB{\rule[-0.1in]{0.4ex}{0.2in}}
 \def\PLUS{\mbox{\Large$+$}}
 \def\MINUS{\mbox{\Large$-$}}
 \begin{center}\begin{tabular}{*{11}{@{}c}}
 \PLUS & & \PLUS & & \PLUS & & \MINUS & & \PLUS & & \MINUS \\
\AA\AA\AA & \AA & \AA\AA\AA & \AA & \AA & \AA & \AA & \AA & \AA & \AA& \AA\AA\AA \mbox{\Large ~$x$} \\[-3ex]
          & \BB &           & \BB &     & \BB &     & \BB &     & \BB&  \\[1ex]
    & $-9$   &     & $-5$   &     & $-3$    &     & $0$    &     & $3$    &
 \end{tabular}\end{center}
 \rule{\textwidth}{0.01in}
\end{exercises}
\EJECTEND

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