%125 points so far
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I.  Consider the first order ODE 
$${dy\over dt} = \sin(t)(y-1)^{1\over 3}\leqno(1)$$
\item{A)}  (15)  Find a solution of the initial value problem 
for (1) with $y(0) = 3$. 
\item{B)}  (10) What does the Existence and Uniqueness Theorem say
about solutions of the initial value problem for (1) with
$y(0) = 1$?  Show that the constant function $y(t) \equiv 1$ is 
one solution.  Is it the only one?
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II.  
%Qualitative 1st order/bifurcations

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III.  The temperature $y$ of an insulated shed with no internal 
heating or cooling varies according to the following ODE
(from Newton's Law of Cooling):
$$y' = (1.5)(A(t) - y).\leqno(2)$$
In this equation, $A(t)$ is the outside temperature, which varies as
a scaled and shifted cosine function with a minimum of 
$40^\circ$ F at 12 am and a maximum of $70^\circ$ F at 12 pm.  
\item{A)} (10) Taking $t = 0$ corresponding to 12pm, write
a formula for $A(t)$.
\item{B)} (10) Using your answer from part A, solve (2) for $y(t)$. 
\item{C)} (10) Your answer from part B should have an arbitrary
constant.  Does the value of that constant in a particular
solution affect the {\it long-term behavior} of $y$?  Explain.

IV.
%First order systems/LA

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V.  Consider the following 2nd order ODE.
$$y'' + 8 y' + 16y = 3e^{-4t} + t^2\leqno(2)$$
\item{A)} (15) Find the general solution of the associated
homogeneous equation.
\item{B)} (15) Find a particular solution of (2) by the method 
of undetermined coefficients.
\item{C)} (15) How would the form of the solutions of (2)
change if the coefficient of the $y$ term was increased
to $16 + \varepsilon$ for $\varepsilon > 0$? 

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VI.  All parts of the following problem refer to the 
1st order system
$$\cases{{dx\over dt} = y - x - 4&\cr
         {dy\over dt} = x(x - 3y)&\cr}\leqno(3)$$
\item{A)} (10) Find all the equilibrium points of (3).
\item{B)} (15) Compute the linearization of (3) at each
critical point and use that information to determine the type of each 
(sink, source, saddle, or center).  

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VII.
%Hamiltonian system
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