% LaTeX file -- AM/PM 331, Fall 2011, Assignment 6

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%%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 6\\
Due Monday November 21}
\end{center}
\vskip 2em 

\begin{Ex}

%\item For $b > 0$, define $T$ on $C[0,b]$ by
%$  Tf (x) = 1 + \int_0^x f(t) x e^{-xt} \, dt.$
%\begin{parts}
%\item Prove that $T$ is a contraction map.
%\item Hence show that there is a unique solution $f \in C[0,\infty)$ to the integral equation
%$ f(x) = 1 + \int_0^x f(t) x e^{-xt} \, dt$.
%\end{parts}


\item  Let $Sx = \frac12\big(x+\sqrt{x^2+1}\, \big)$ and $Tx = \tan^{-1} x$ on $\bR$.
\begin{parts}
\item Show that $|Sx-Sy|<|x-y|$ and $|Tx-Ty| < |x-y|$  for $x,y \in \bR$.
\item Show that $S$ and $T$ are not  contraction maps.
\item Show that $T$ has a unique fixed point, and that $S$ has no fixed point.
\end{parts}



\item \begin{parts}
\item Show that $f(x)=x^3+x+1$ has exactly one real root.
\item Use Newton's method to approximate it to eight decimal places.
(You can use a computer to do the calculations.)
Show your error estimates. 
\end{parts}


%\item  \begin{parts}
%\item Set up Newton's method for computing cube roots.
%\item Show by hand that $1.25 < \sqrt[3]{2} < 1.26$.
%\item Compute $\sqrt[3]{2}$ to eight decimal places.
%\end{parts}



\item For differential equation 
\[ y^{(3)} + y''- x(y')^2 = e^x ,\quad
 y(0)=1,\ y'(0)=-1, \AND y''(0)=0 \]
convert the DE into a first-order vector-valued DE and then
into a fixed-point problem.\\
You do not have to solve it.

\item Consider the DE $y'=1+xy$ and $y(0)=0$ on $[-1,1]$.
\begin{parts}
\item Show that the associated integral operator is a contraction
mapping.

\item Find a convergent power series expansion for the
unique solution.

\item Use the Global Picard Theorem to show that there is a unique
solution on $[-b,b]$ for any $b<\infty$.  Hence deduce that
there is a unique solution on $\bR$.
\end{parts}


\item  Consider $f'(x) + f(x)^2 = 4xf(x) - 4x^2 + 2$ for $x \in \bR$ and $f(0)=2$.
\begin{parts}
\item Convert this to a first order vector valued DE.
\item Show that this DE satisfies a  local Lipschitz condition on a smaller region; \\
and hence deduce that there is a local solution.

\item Solve this DE explicitly, and
find the maximal continuation of the solution.\\
\hint Find the DE satisfied by $g(x) = f(x) - 2x$ and solve it.
\end{parts}


\item Consider the DE $f(x) f'(x) = 1$ and $f(0) = a$ for $x \in \bR$.
\begin{parts}
 \item Solve this equation explicitly.
 \item Show that there is a unique solution on an interval about 0
  if $a \ne 0$ but that it extends to only a proper subset of
  $\bR$, even though the solution does not blow up.
  Why does this not contradict the Continuation Theorem?
 \item Show that there are two solutions
 when $a=0$ valid on $[0,\infty)$. 
 Why does this not contradict the Local Picard Theorem?
\end{parts}



\end{Ex}
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