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\centerline{Mathematics 304, Section 2 -- Ordinary Differential Equations}
\centerline{Midterm Exam 1 -- October 1, 2004}
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{\it Directions:}
Do all work in the blue exam booklet.  You may use a calculator and
the table of integrals at any point.  There are 100 total regular
points and 10 Extra Credit points.
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I.  Consider the first order ODE 
$$y' = 2t(y-2)^{2/3}\leqno(1)$$
\item{A)}  (15) Find a solution of this equation that satisfies
the initial condition $y(0) = 3$.
\item{B)}  (10) What does the Existence and Uniqueness Theorem say
about solutions of the initial value problem for (1) with
$y(0) = 2$?  Show that the constant function $y(t) = 2$ for all $t$ is 
one solution.  Is it the only one?
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II. (25) A tank with capacity 200 gal. originally contains 50 gal. water
with 30 lb of salt in solution.  Water containing .5 lb of 
salt per gallon is entering at 3 gal./minute and the 
tank is thoroughly mixed so the salt concentration is 
uniform at all times.   Contents of the tank are also flowing out of an 
exit pipe at 2 gal/minute.  Find the amount of salt in the tank at all 
times prior to the time the tank overflows.  What is the amount of
salt at the instant the tank starts to overflow?
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III.  Let $T,M$ be positive constants with $T < M$, and consider the 
autonomous 1st order ODE
$$P' = -P\left(1 - {P\over T}\right)\left(1 - {P\over M}\right)\leqno(2)$$
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\item{A)} (15) What are the equilibrium solutions of this equation?
Which of the equilibria are sinks and 
which are sources?
\item{B)} (10)  Suppose the equation (2) models growth of a population
of a species of organisms in a particular environment.
Based on your answer to part A, what happens to solutions of initial value 
problems for (1) if $P(0) < T$? if $T < P(0) < M$? if $P(0) > M$?  
What are the meanings of the $P$ values $P = T$
and $P = M$? 
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IV. (25) Sketch the bifurcation diagram for the family of ODE 
$$y' = y^3 - 3ay$$
containing the parameter $a$.  Describe all bifurcations that occur.
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{\it Extra Credit}
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(10) The ODE $y' = a(t)y + b(t)$ has solutions
$y = t^2 + e^{-t}$ and $y = 3t^2 + e^{-t}$.  What are the 
functions $a(t)$ and $b(t)$?  Is there only one possible
answer?
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