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{\large Math 267 Advanced Ordinary Differential Equations}

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{\LARGE Makeup Term Test}

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University of Toronto, November 9, 2012

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\noindent{\bf Solve the 3 of the 4 problems on the other side of this page. }\\
Each problem is worth 34 points.\\You have fifty minutes to
write this test.

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\noindent{\bf Notes.}

\begin{itemize}
\item No outside material other than stationary and a basic calculator
(not capable of displaying text) is allowed.
\item If you solve all 4 problems, you must indicate clearly which 3 are
to be graded. Failing this an arbitrary problem will be ignored in grading.
\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and made of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs and
calculations given during lectures often omits or shortens key
phrases. The ideal written solution to a problem does not do that.
\end{itemize}

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\centerline{\bf Good Luck!}

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\noindent{\bf Solve 3 of the following 4 problems. } Each problem
is worth 34 points. You have fifty minutes. {\bf Neatness counts!
Language counts!}

\vfill \noindent{\bf Problem 1. } Solve the following two differential
equations:
\begin{enumerate}
\item $y''-2y'+5y=\cos x$ (find all solutions).
\item $\displaystyle y'=-\frac{3x^2}{2y}$ with $y(0)=1$.
\end{enumerate}

\par\noindent{\small {\bf Tip. } Don't start working! Read the whole exam
first. You may wish to start with the questions that are easiest for you.}

\vfill \noindent{\bf Problem 2. }
\begin{enumerate}
\item State precisely (without proof) the theorem about existence and
uniqueness of solutions for a single first order ordinary differential
equation.
\item Rewrite the said equation as an integral equation and explain how to construct a sequence of functions that converges to a solution. No need to actually prove convergence.
\end{enumerate}

\vfill \noindent{\bf Problem 3. } Extremize the functional
$\displaystyle J(Y)=\int_{-1}^1\frac{(y')^2}{x^4}dx$, subject to the
boundary conditions $y(1)=2$ and $y(-1)=-2$.

\vfill \noindent{\bf Problem 4. }
\begin{enumerate}
\item State what is the ``improved Euler method'' for solving the
differential equation $\phi'=f(x,\phi)$ with initial condition
$\phi(x_0)=y_0$ using step size $h$.
\item Compute the single-step approximation for $y(1/3)$, given that
$y$ satisfies $\displaystyle y'=-\frac{3x^2}{2y}$ and $y(0)=1$, using the
improved Euler method.
\end{enumerate}

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\bf Good Luck!
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