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\begin{center}
  {\Large UNIVERSITY OF TORONTO}\\
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  {\large Faculty of Arts and Sciences}\\
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  {\Large DECEMBER EXAMINATIONS 2012}\\
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  {\Large Math 267H1 Advanced Ordinary Differential Equations --- Final Exam}

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  Dror Bar-Natan\par
  December 18, 2012\par
\end{center}

\vfill

Solve all of the following 5 questions. The questions carry equal weight
though different parts of the same question may be weighted differently.

\vfill

\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!}}

\vfill

\noindent{\color{blue} Post-exam modifications in blue.}

\newpage

{\noindent\color{blue} {\bf Solve all of the following 5 problems. } Each problem
is worth 20 points. You have three hours. {\bf Neatness counts! Language counts!}}

\vfil

\noindent{\bf Problem 1. } Find the most general solutions of the following
differential equations:
\begin{enumerate}
\item $\displaystyle \frac{dy}{dt}+\frac{y}{t^2} = \frac{1}{t^2}$. {\color{blue}\hfill(I should have asked this with $1/t^3$ on the right hand side).}
\item $\displaystyle \frac{dy}{dx}-\frac{y}{x}=\frac{y^2}{x}$.
\item $\displaystyle (\sin y)dx+[(x+y)\cos y+\sin y]dy=0$.
\item $\displaystyle y^{(4)}-2y''+y=0$.
\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. }
\begin{enumerate}
\item State precisely (without proof) the theorem about existence and
uniqueness of solutions for a single first order ordinary differential
equation.
\item Show by an example that if the Lipschitz condition is dropped from
the above statement, the uniqueness of solutions may fail.
\end{enumerate}

\vfil\noindent{\bf Problem 3. }
It is given that a differentiable function $\phi\colon[a,b]\to\bbR$
is a local minimum of the functional $J(y)=\int_a^bF(x,y,y')dx$ among
all differentiable functions $y$ that satisfy $y(a)=A$ and $y(b)=B$, in the
sense that whenever a differentiable function $h\colon[a,b]\to\bbR$
satisfies $h(a)=0=h(b)$, the function $\epsilon\to J(\phi+\epsilon h)$ has
a local minimum at $\epsilon=0$. Derive the Euler-Lagrange necessary
condition that $\phi$ must satisfy.

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

\newpage

\noindent{\bf Problem 4. }
\begin{enumerate}

\item In this part $t$ is always assumed to belong to some fixed
interval $I$ in $\bbR$. Assume $v_1(t)$ and $v_2(t)$ are differentiable
$\bbR^2$-valued functions, are linearly independent for all $t$ in $I$,
and solve the system of differential equations $v'(t)=A(t)v(t)$, where
$A(t)$ is a $2\times 2$ matrix that depends continuously on $t$. Explain
how by the method of ``fundamental matrices'', $v_1$ and $v_2$ can be
used to solve the non-homogeneous version $v'(t)=A(t)v(t)+g(t)$ of the
equation. (Here $g(t)$ is a given continuous $\bbR^2$-valued function).

\item Assume $t>0$. For the following equation,
\[ v'=\frac{1}{t}\begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}v +
  \begin{pmatrix} 0 \\ 2 \end{pmatrix},
\]
it is given that a solution of the homogeneous version is
\[ v(t) = c_1\begin{pmatrix}1\\1\end{pmatrix}t +
  c_2\begin{pmatrix}1\\3\end{pmatrix}t^{-1}.
\]
Use the technique from the previous part of this question to find a
solution of the full equation.
\end{enumerate}
\par\noindent{\small {\bf Tip. } You may find it handy to note that
$
  \left(
    \begin{pmatrix} 1 & 1 \\ 1 & 3 \end{pmatrix}
    \begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix}
  \right)^{-1}
  = \frac12\begin{pmatrix} t^{-1} & 0 \\ 0 & t \end{pmatrix}
    \begin{pmatrix} 3 & -1 \\ -1 & 1 \end{pmatrix}.
$
}

\vfil\noindent{\bf Problem 5. }
\begin{enumerate}
\item State and prove the ``Sturm Comparison Theorem''.
\item Use it to decide whether solutions of $y''+x^{-3/2}y{\color{blue}\,=0}$ oscillate as
$x\to\infty$.
\end{enumerate}

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

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