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2250-2}}
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\begin{center}\Large\bf
Applied Differential Equations 2250\\
Sample Midterm Exam 1\\
Tuesday, 9 February 2006 \\
Wednesday, 10 February 2006 arranged
\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 $\dd y'=
xe^{-x}+\csc^2 x + 2\sec^2 x +
 \frac{16x^{4}}{1+4x^{2}}$.
 % f:=16*x^(4)/(1+4*x^(2));

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

\end{exercises}
\EJECT


\begin{exercises}
\prob{2.}{(Separable Equation Test)} ~\newline
 The problem
 % F:=x*sin(x)+exp(2*x)*x^(5/3);
 % G:=y*exp(y)-sin(y);
 $$y'=x\left(e^{2x}x^{2/3}ye^y + \sin(x)ye^y\right)-x\sin(x)\sin(y)-e^{2x}x^{5/3}\sin(y)$$
 may or may not be separable. If it is, then write formulae
 for $F$, $G$ which decompose
 the problem as $y'=F(x)G(y)$.
 Otherwise, explain in detail why it fails to be separable. Do not solve for
 $y$!

\end{exercises}
\EJECT

\begin{exercises}
\prob{3.}{(Solve a Separable Equation)} ~\newline
 % u:=(6*x^3+12*x)/(1+x); int(u,x);
 Given $\dd yy' = \frac{6x^3+12x}{1+x}(1-4y^2)$, ~\newline
 (a) Find all equilibrium solutions, ~\newline
 (b) Find the non-equilibrium solution in implicit form. ~\newline
 {\bf Do not solve} for $y$ explicitly.

\end{exercises}
\EJECT

\begin{exercises}
\prob{4.}{(Linear Equations)} ~\newline
 % de:=2*diff(v(t),t)=-64+((4)/(t+1))*v(t); ic:=v(0)=3; dsolve({de,ic},v(t));
 (a) Solve $\dd 2v'(t)=-64 + \frac{4}{t+1}v(t)$, $v(0)=3$. Show all integrating
 factor steps.\newline
 (b) Using the answer $v(t)$ from (a), solve $y'(t)=v(t)$, $y(0)=2$. Show all quadrature steps.


\end{exercises}
\EJECT

\begin{exercises}
\prob{5.}{(Stability)} ~\newline
 (a) Draw a phase line diagram for the differential equation
 $$dx/dt = (2-5x)^3(1-2x)(1-4x^2).$$
 Expected in the diagram are equilibrium
 points and signs of $x'$ (or flow direction markers $<$ and
 $>$).\newline
 (b) Draw a phase diagram using the phase line diagram of (a). Add these labels
 as appropriate: funnel, spout, node, stable, unstable. Show at
 least 10 threaded curves. A direction field is not required.

\end{exercises}
\EJECTEND

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