% maple lab 3 Numerical Methods S2013
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\begin{document}
{\bf
Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline

\begin{center}
\bf Math 2250
\bf Numerical Methods Project\\
\bf S2014
\end{center}


{\bf Due date}: Week 4.

{\bf References}: Edwards-Penney, Sections 2.4,2.5, 2.6.
This document: \href{http://www.math.utah.edu/~gustafso/s2014/2250/numericalDEproject-S2014.pdf}
{numericalDEproject-S2014.pdf}

Other related and required documents
are available at the \href{http://www.math.utah.edu/~gustafso/s2014/2250/}{course web site},
in particular
\href{http://www.math.utah.edu/~gustafso/s2014/2250/numericalDE2008.pdf}{Numerical Methods Slides}, 
\href{http://www.math.utah.edu/~gustafso/s2014/2250/numericalDE2010.pdf}{Numerical Methods Manuscript}, 
\href{http://www.math.utah.edu/~gustafso/s2014/2250/2250numerical-hints.txt}{Sample Maple Code for Euler, Heun, RK4}.



\smallskip

\Prob{Symbolic Solution}{(E \& P Exercises 2.4-6, 2.5-6, 2.6-6 Symbolic Solution)}
The symbolic solution of $y'=-2xy$, $y(0)=2$ is $y=2e^{-x^2}$.
Using methods from the textbook, Chapter 1, display the details of the
derivation for this symbolic solution, plus a full answer check.

\def\checkbox{\fbox{$\phantom{X}$}~}
\bigskip

The answer $y=2e^{-x^2}$ can be used to make the following table, which is used in the 3 problems below.

\begin{tt}
\begin{tabular}{|l|l|}
\hline
 $x$ & $2e^{-x^2}$   \\ \hline
 0.00 & 2.000000000 \\ \hline
 0.10 & 1.980099667 \\ \hline
 0.20 & 1.921578878 \\ \hline
 0.25 & 1.878826126 \\ \hline
 0.30 & 1.827862371 \\ \hline
 0.40 & 1.704287578 \\ \hline
 0.50 & 1.557601566\\ \hline
 
  $x$ & $2e^{-x^2}$   \\ \hline
 0.00 & 2.000000000 \\ \hline
 0.10 & 1.980099667 \\ \hline
 0.20 & 1.921578878 \\ \hline
 0.25 & 1.878826126 \\ \hline
 0.30 & 1.827862371 \\ \hline
 0.40 & 1.704287578 \\ \hline
 0.50 & 1.557601566\\ \hline

\end{tabular}
\end{tt}


\Prob{Euler's Method}{(E \& P Exercise 2.4-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply Euler's method to produce two dot tables, as follows. The
first has three rows, $h=0.25$. The second has six rows, $h=0.1$.
Reproduce the table below and fill in missing digits. 

The work is to be entirely hand-written, with calculator assist. Answer checks may use technology.

\begin{tt}
\begin{tabular}{|l|l|l|}
\hline
 $h$ & actual $y(.5)$  & approx $y(.5)$ \\ \hline
 0.25 & 1.558              &   1.750?????             \\ \hline
 0.1  & 1.558              &    1.627?076?            \\ \hline
\end{tabular}
\end{tt}

\Prob{Heun's Method (Modified Euler)}{(E \& P Exercise 2.5-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply Heun's method (Improved Euler) to produce one dot table of six
rows, $h=0.1$. Reproduce the table below and fill in missing digits.
Hand-written work is expected with computer assist. Hand-written work with calculator assist should end after the first calculation, which estimates $y(0.1)$ as $1.980099667$. Technology should be used for the remaining answers. 

\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
 $x$ &
 0.0 &
 0.1 &
 0.2 &
 0.3 &
 0.4 &
 0.5
\\ \hline
 actual $y(x)$  &
 2.000000000      &
 1.980099667      &
 1.921578878      &
 1.827862371      &
 1.704287578      &
 1.557601566
\\ \hline
 approx $y(x)$  &
 2.0000000        &
 1.980????        &
 1.921????        &
 1.827????        &
 1.704????        &
 1.557????
\\ \hline
\end{tabular}


\Prob{RK4 Method}{(E \& P Exercise 2.6-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply the RK4 method to produce one dot table of three rows,
$h=0.25$. Reproduce the table below, filling in the missing digits.
Hand-written work should complete the estimate of $y(0.25)$ as $1.8788xxxx$, using calculator assist. The remaining steps should use technology, with a plan to reproduce the hand-written result.

\begin{tabular}{|l|l|l|l|}
\hline
 $x$ &
 0.00 &
 0.25 &
 0.50
\\ \hline
 actual $y(x)$  &
 2.000000000      &
 1.878826126      &
 1.557601566
\\ \hline
 approx $y(x)$  &
 2.000000000        &
 1.8788????0 &
 1.557??329?
\\ \hline
\end{tabular}

\end{document}
