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\textbf{Math 365: Honors Analysis I}
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\textbf{Final Exam}\quad \textsl{December 11, 2000}

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\emph{There are 12 problems, each worth 15 points. You will receive
partial credit for all problems attempted, up to 150 points total.}

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1. It began to snow on a certain morning, and the snow continued to fall
steadily throughout the day. At noon a snow plow started to clear a road at
a constant rate in terms of the volume of snow removed per hour. The
snow plow cleared 2 miles by 3~\textsc{p.m.} and 1 more mile by
5~\textsc{p.m.}. When did the snow start falling?

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2. Find the general solution of $y^{(4)} - y = \cos(x)$.

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3. Show that any solution of $x y'' - y' + x y = 0$ has infinitely many
positive zeros.

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4. Find two independent power series solutions of $y'' - x y = 0$.

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5. Find two independent Frobenius series solutions of $2x y'' + y' +
y = 0$.

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6. a) Determine the Fourier series expansion of $f(x) = x$ on $[-\pi,\pi]$.

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b) Determine the cosine series expansion of $f(x) = x$ on $[0,1]$.

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7. Derive the solution to the heat equation
$a^2\frac{\partial^2w}{\partial x^2} = \frac{\partial w}{\partial t}$
subject to the boundary conditions $w(0,t) = w_1$, $w(\pi,t) = w_2$,
and $w(x,0) = f(x)$.

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8. Find a function $w(r,\theta)$ that is harmonic in the unit disk, $r <
1$, and satisfies $w(1,\theta) = \theta$ for $-\pi < \theta < \pi$.

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9.  Solve the initial value problem $y'' -y' = \sin(x)$, $y(0) = 0$,
$y'(0) = 0$, using Laplace Transforms.

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10. Find the general solution of the system of equations
\[\X' = \left(\begin{array}{cc} 3 & -4 \\ 2 & -3 \end{array}\right) \X +
\left(\begin{array}{c} e^t \\ e^{-t} \end{array}\right)\]

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11. Find the critical points of the system
\begin{eqnarray*}
\frac{dx}{dt} &=& x - x^2 - xy\\
\frac{dy}{dt} &=& 2y - y^2 - 3xy\\
\end{eqnarray*}
and discuss the stability of the system at each.

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12. State the existence and uniqueness theorems for 

\begin{itemize}
\item[a)] the solution of a general system of first order equations and
\item[b)] the solution of a \emph{linear} system of first order equations.
\end{itemize}

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