\documentclass[12pt]{article}
  \setlength{\oddsidemargin}{0pt}
\setlength{\topmargin}{-0.5truein}
\setlength{\textwidth}{6.5truein}
\setlength{\textheight}{9truein}
  \pagestyle{empty}
                                          
\begin{document}
 \centerline{\large Math.\ 401, Sec.\ 500\hfill    Spring, 2005}
 \bigskip
\centerline{\bfseries\large Homework 7, due March 9}
\bigskip
                                          
\begin{enumerate}

  \item 
Classify these equations as linear homogeneous, linear 
nonhomogeneous, or nonlinear. 
(Here $u_t \equiv \partial u/\partial t$, etc.)

% old cases from Logan, p.~159:
%\zitem{(a)} $u_t u_{tt} + 3tu =0$
%\zitem{(b)} $e^t u_{tx} -x^2 u = \cos t$
%\zitem{(c)}  $u_{tt} - u_{xx} =0$
%\zitem{(d)}  $u_{tx} +u^2 = \sin x$

\begin{enumerate}
\item\quad  $u_{tt} - u_{xx} = \cos(x-t)$
\item\quad $u_t + u^2 u_x = 0$
\item\quad $u_t + 3t^2u =0$
\item\quad $u_{tt} - u_{xx} = -m^2 u$
\end{enumerate}

\item Why do we not make a distinction between ``homogeneous''
and ``nonhomogeneous'' for {\sl nonlinear\/} equations?
{\sl Hint:\/} Try to classify the equation $y'' + (\cos y)^2 =1$.

%\item  {\it [Logan, p.\ 195,  Ex.\ 3.1(d)]\/}
%Find the eigenvalues and eigenfunctions for the problem
%$$y'' + \lambda y = 0 \quad\mbox{on $0<x<L$}, $$
%$$y'(0) = 0, \qquad y(L) =0.$$

\item {\it [Schaum's, p.\ 46, Ex.\ 2.52]\/}\quad
Find the steady-state temperature in a bar whose ends are located
at $x=0$ and $x=10$, if these ends are kept at 150$^\circ$C
and 100$^\circ$C respectively.

\item Suppose that the boundary conditions (``BC:'') 
on p.~64 of the notes 
(and corresponding lecture on the heat equation with fixed end 
temperatures)
are replaced by 
$$   \frac{\partial u}{\partial x}(t,0) = F_1\,, \qquad 
\frac{\partial u}{\partial x}(t,1) = F_2\,.$$
(That is, the {\sl heat flux\/} through each end of the bars
is held constant.)
What happens when you attempt to find a steady-state solution
as on p.~66?
Distinguish between the two cases $F_1 = F_2$ and $F_1 \ne F_2\,$.
Can you give a physical explanation for your results?

% \item Suppose that the boundary conditions (``BC:'') 
%on p.~67 of the notes are replaced by 
%$$   \frac{\partial w}{\partial x(t,0) = 0, \qquad \
%frac{\partial w}{\partial x}(t,1) = 0.$$
%What changes would be necessary in pp\. 67--72
%(and corresponding lecture on separation of variables in the 
%heat equation)?
%{\sl Warning:\/}  Pay close attention to the possibility 
%that $\lambda$ may not always be positive.

\end{enumerate}
\end{document}