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\begin{document}
\begin{center}
{\bf \hfill Seema Nanda\\}
{\bf Numerical Analysis \hfill Assignment 7 \hfill
 April 18, 2011}
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\begin{center}
Please follow the instructions for assignments and homework as given in the
course web page. You may discuss the problems and solutions with anyone but the work written up and submitted must be done on your own. Also programs must be written by you.
\end{center}
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% Homework starts here.
\begin{enumerate}
\item \textbf{Boundary Value Problems:}
\begin{enumerate}
\item What is the first variational equation for :
\begin{equation}
y'' = cos(ty) + sin(t^2y')
\end{equation}

Can the first variational equation be solved numerically by itself, or must it be solved simultaneously with equation for $y''$?


\item What is  the first variational equation for the following linear differential equation:
\begin{equation}
x'' = a(t) + b(t)x + c(t)x'
\end{equation}

\item 

Solve the 2 point boundary value problem
\begin{equation}
y'' +y'+ y = -(x^2 +x +1)
\end{equation}

with $y(0)= 0$ and $y(1) =0$,  using the shooting method and your favorite method for an intial value problem.
Check your numerical results by finding the exact solution. 

\item Try to solve the following problem using the same method as in previous
problem.
\begin{equation}
u'' - 100 u = 0
\end{equation}
with $u(0) = 0$ and $u(1) = 0$. The exact solution is given by
\begin{equation}
u(t) = \frac{1}{1-e^{-20}}e^{-10t} - \frac{e^{-20}}{1-e^{-20}}e^{10t}.
\end{equation}
Discuss the discrepancies between your best possible solution and the exact solution. What difficulties did you encounter, if any,  in obtaining your numerical solution?

\end{enumerate} 
\end{enumerate}
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