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\begin{center}\Large\bf
Applied Differential Equations 2280\\
Midterm Exam 1\\
Thursday, 16 February 2006 \\
\end{center}

{\bf Instructions}: This in-class exam is 50 minutes. No
calculators, notes, tables or books.
No answer check is expected.
Details count $75$\%. The answer counts $25$\%.


\begin{exercises}
\prob{1.}{(Quadrature Equation)} ~\newline
        Solve for the general solution $y(x)$ in the equation 
 $\dd y'=
 2\cot x
 + \frac{1250 x^{3}}{1+25x^{2}} 
 + x\ln(1+x^2)$.
 %  f:=1250*x^(3)/(1+25*x^(2));

\iftrue {\small [The required integration talent includes
 basic formulae, integration by parts, substitution and college algebra.]
}\fi

\end{exercises}
\EJECT


\begin{exercises}
\prob{2.}{(Separable Equation Test)} ~\newline
 The problem $y'=f(x,y)$ is said to be separable provided
$f(x,y)=F(x)G(y)$ for some functions $F$ and $G$. 

 (a) [75\%]  Check (\fbox{X}) the problems that can be put into separable form,
 but don't supply any details.
\begin{center}\large\begin{tabular}{|lp{0.45\linewidth}|lp{0.3\linewidth}|}
\hline
 \fbox{$\phantom{X}$} & $ y'=-y(2xy+1)+(2x+3)y^2$ & \fbox{$\phantom{X}$} &   $yy'=xy^2+5x^2y$ \\[2ex]  \hline
  \fbox{$\phantom{X}$} &  $y'=e^{x+y}+e^y$ & \fbox{$\phantom{X}$} &  $3y'+5y=10y^2$ \\[2ex] \hline
  \end{tabular}
\end{center}

 (b) [25\%] State a test which can verify that an equation is not separable.
 Use the test to verify that $y'=x+\sqrt{|xy|}$ is not separable.

\end{exercises}
\EJECT

\begin{exercises}
\prob{3.}{(Solve a Separable Equation)} ~\newline
 % u:=(2*x^2+4*x)/(1+x); int(u,x);
 Given $\dd y^2y' = \frac{2x^2+3x}{1+x^2}\left(\frac{125}{64}-y^3\right)$. ~\newline
 (a) Find all equilibrium solutions. ~\newline
 (b) Find the non-equilibrium solution in implicit form. ~\newline
 To save time, {\bf do not solve} for $y$ explicitly.

\end{exercises}
\EJECT

\begin{exercises}
\prob{4.}{(Linear Equations)} ~\newline
% de:=2*diff(v(t),t)=-32+(2/(3*t+1))*v(t); ic:=v(0)=-8; dsolve({de,ic},v(t));
 (a) [60\%] Solve $\dd 2v'(t)=-32 + \frac{2}{3t+1}v(t)$, $v(0)=-8$. Show
 all integrating factor steps.\newline
 (b) [30\%]  Solve $\dd 2\sqrt{x+2}\,\frac{dy}{dx}=y$. The answer contains symbol $c$.
\newline  
 (c) [10\%] The problem $\dd 2\sqrt{x+2}\,y'=y-5$ can be solved using
 the answer $y_h$ from part (b) plus superposition $y=y_h+y_p$. Find $y_p$. Hint:
 If you cannot write the answer in a few seconds, then return here after
 finishing all problems on the exam.

\end{exercises}
\EJECT

\begin{exercises}
\prob{5.}{(Stability)} ~\newline
 (a) [50\%] Draw a phase line diagram for the differential equation
 $$dx/dt = 1000\left(2-\sqrt[5]{x}\right)^3(2+3x)(9x^2-4)^{8}.$$
 Expected in the diagram are equilibrium
 points and signs of $x'$ (or flow direction markers $<$ and
 $>$).\newline
 (b) [40\%] Draw a phase diagram using the phase line diagram of (a). Add these labels
 as appropriate: funnel, spout, node, source, sink, stable, unstable. Show at
 least 8 threaded curves. A direction field is not expected or required.
 \newline
 (c) [10\%] Outline how to solve for non-equilibrium solutions, without doing any
  integrations or long details.
\end{exercises}
\EJECTEND

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