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\begin{document}

\begin{center}
{\Large \em Math 560 (1998)}\\
\vspace{.15in}
{\Large Final Exam (Takehome)}\\
\vspace{.15in}
{\Large Due: Tuesday Dec 15, 1998.}\\
\end{center}

\vspace{.4in}

\large
\begin{itemize}
\item[1.] (DELTA DISTRIBUTION)
\begin{itemize}
\item[a)] For what constants $a,b$ is $\delta(2x-3)=a\delta(x-b)$?
\item[b)] Let $f(x)$ be twice differentiable and show
\eq
f(x)\delta''(x)=f''(0)\delta(x)-2f'(0)\delta'(x)+f(0)\delta''(x)
\endeq

\end{itemize}
\item[2.] Show the distributional derivative of $Pf(ln|x|)$ is
$Pf\left(\frac{1}{x}\right)$ where
\eq
\left< Pf(ln|x|),\phi\right>
\equiv \lim_{\varepsilon \rightarrow 0}
\int_{|x|\ge \varepsilon} ln|x| \phi(x) dx
\endeq
\item[3.] Consider the boundary value problem:
\eq
Lu=u''-u =f(x) \quad , \quad u(0)=0 \quad , \quad u(1)+u'(1)=0
\endeq
Find (2 different) Greens functions $g(x,t)$ for this boundary value problem
which satisfy 
\eq
\frac{d^2g}{dx^2}-g = \delta(x-t)
\endeq
and the ( 2 different) boundary conditions

\vspace{0.1in}
(i) $g(0,t)=g(1,t)+g_x(1,t)=0$

\vspace{0.1in}
(ii) $g(0,t)=g(1,t)-g_x(1,t)=0$

\vspace{0.1in}
For which of these Greens functions is
\eq
u(t) = Gf \equiv \int_0^1 g(x,t) f(x) dx
\endeq
a solution of the boundary value problem?
\end{itemize}


\end{document}