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{\large Math 267 Advanced Ordinary Differential Equations}

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{\LARGE Sample Final Exam}

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Dror Bar-Natan, University of Toronto, December 2012
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Solve all of the following 5 questions. The questions carry equal weight
though different parts of the same question may be weighted differently.

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\noindent{\bf Duration. } You have 3 hours to write this exam.

\vskip 2mm

\noindent{\bf Allowed Material. } Basic calculators, not capable of
displaying text or sounding speech.

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\centerline{{\bf Good Luck!}}

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\noindent{\bf Problem 1. } Find the most general solutions of the following
differential equations:
\begin{enumerate}
\item $\displaystyle y(y+1)dx+x(x-1)dy=0$.
\item (pick your further favourites from HW or BDP).
\item \ldots
\item \ldots
\end{enumerate}

\par\noindent{\small {\bf Tip. } All explicit integrations that are
required above (and elsewhere in this exam) are easy; do not leave
them un-evaluated.}

\vskip 3mm

\par\noindent{\small {\bf Tip. } It is always an excellent idea to
substitute your solutions back into the equations and see if they
really work.}

\vskip 3mm

\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.}

\vfil\noindent{\bf Problem 2. } A function $f(x,y)$ is continuous in some
open set that contains a given point $(x_0, y_0)$, and in the vicinity of
$x_0$ a sequence of functions $\phi_n(x)$ is given by
the recursive definition
\[ \phi_0(x)=y_0,\qquad \phi_n(x)=y_0+\int_{x_0}^xf(t,\phi_{n-1}(t))dt. \]
You may assume that somebody else had already proven that $\phi_n(x)$ is
well defined.

Define what it means for the function $f$ to be uniformly Lipschitz in
the variable $y$, and prove that the sequence $\phi_n(x)$ is uniformly
Cauchy and that it converges to a solution of the differential equation
$y'=f(x,y)$ which passes through the point $(x_0, y_0)$.

\vskip 3mm
\par\noindent{\small {\bf Tip. } Neatness, cleanliness and
organization count, here and everywhere else!}

\vfil\noindent{\bf Problem 3. } How would you solve a functional
minimization problem for a functional of the form
$J(y)=\int_a^bF(x,y,y',y'')dx$ (assuming $y(a)=A$ and $y(b)=B$)? Note that
$F$ involves also the {\em second} derivative $y''$ of $y$!

\vfil\noindent{\bf Problem 4. }
Find two Frobenius-series solutions of the Bessel equation of order
$0$, $x^2y''+xy'+x^2y=0$ near $x=0$. There is no need to write a formula
for the general terms in the solutions, but it is necessary to display
enough terms to make the pattern clear.

\vfil\noindent{\bf Problem 5. }
\begin{enumerate}
\item State and prove a theorem about oscillatory behaviour in the case
when $\int_A^\infty q(x)dx=\infty$.
\item Use it to decide whether solutions of the Airy equation $y''+xy=0$
oscillate as $x\to\infty$.
\end{enumerate}

\vfil \centerline{\bf Good Luck!} \vfil

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