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\begin{document}

\name
\vskip.25in
\noindent
{\bf Math. 423} \hfil {\bf Final exam} \hfil {\bf Due by Wednesday, May 10}

\vskip.25in \noindent
1. Given the initial value problem $\ds y' = \frac{\cos(t) - 2yt}{1+t^2}$
with $y(0) = 0.$

\medskip \noindent
Use MAPLE or any other program to answer the following questions.  Write
the answers below and attach some
documentation, such as a printout.

\bigskip
(a) Use the modified Euler method, first with step size $h=0.05$ and then
with $h=0.025$, to approximate $y(1)$.

\vskip3in
(b) Do the same using the standard Runge-Kutta method of order four.

\vfil \eject \noindent
(Continuing problem 1)
The actual solution of the initial value problem is $$y(t) =
\frac{\sin(t)}{1+t^2}\,.$$

\bigskip
(c) Determine the errors involved in using the modified Euler method with
$h=0.05$ and
$h=0.025$.  By what factor does the error improve when the step size is
reduced?  Is that about what one would expect? Explain.

\vskip2.75in
(d) Answer the same questions for the standard Runge-Kutta method of order
four.

\vfil \eject \noindent
2.  Given the initial value problem $y' = -y + t, \ \ y(0)=1$.

\medskip
(a) Use the modified Euler method with step size $h=0.2$ to compute $w_1$.

\vskip2.5in
(b) Use the Adams-Bashforth 2-step method with the $w_1$ of part (a) and
step size $h=0.2$ to approximate $y(0.4)$.

\vfil \eject \noindent
Continuing problem 2.

\bigskip
(c) Repeat parts (a) and (b) to approximate $y(0.4)$, but this time using
step size $h=0.1$; i.e., use
the modified Euler method to compute $w_1$ and then use the Adams-Bashforth
2-step method to approximate $y(0.4)$.


\vfill \eject \noindent
Continuing problem 2.

\bigskip
(d) The actual solution of the initial value problem is $y(t) = t - 1
+2e^{-t}$.  Use this information to estimate the error in the
approximations of $y(0.4)$ in
parts (b) and (c).  By what factor did the error improve using $h=0.1$
instead of $h=0.2$?  Was that close to what you expected?  Explain.

\vfil \eject \noindent
3.  Given the initial value problem $y' = e^y, \ \ y(0)=0$.

\medskip
(a)  Use the Taylor method of order 4 with $h=0.1$ to approximate $y(0.2)$.

\vskip4.5in
(b) The actual solution of the initial value problem is $y(t) = -\ln(1-t)$.
If you did part (a) correctly your answer should differ from the actual
value by
less than 0.0000062.  Check that it does.


\vfil \eject \noindent
4.  Given the initial value problem $y' = y^2 - t, \ \ y(0) = 1$

\medskip
(a) Use the 2-step Adams-Bashforth-Moulton predictor-corrector  method to
approximate $y(0.2)$
Use a step size of $h=0.1$, and use the Taylor method of order 2 to find $w_1$.

\vskip4.5in
(b) Use the standard 4-step Runge-Kutta method to approximate $y(0.2)$.




\end{document}