
% extra credit ch4 2280, S2016
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 4 \\
{\small\bf S2016}
\end{center}

{\bf Submitted work}. Please submit one stapled package per chapter.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number, e.g., \fbox{Xc4.1-8}. Please attach
this printed sheet to simplify your work.

\Prob{XcL3.1}{(Numerical Solutions)}
You may submit this problem only for score increases on numerical solutions 2.4 to 2.6.

Solve symbolically by chapter 1 methods the initial value problem
$y'=2xy^2$, $y(0)=1$. Do an answer check in maple or by hand.
Answer: $y=1/(1-x^2)$. This problem has no numerical work!

\Prob{XcL3.2}{(Numerical Solutions)}
You may submit this problem only for score increases on numerical solutions 2.4 to 2.6.
This problem counts as three (3) problems.

Solve $y'=2xy^2$, $y(0)=1$ numerically
for the value of $y(0.5)$ using (1) Euler's method,
(2) Heun's method and (3) the RK4 method. Use step size $h=0.1$. Include
computer code and a print of the data.  Report the answers in a table
for $x$-values $0$, $0.1$, $0.2$, $0.3$, $0,4$, $0.5$.


\Prob{XcL4.1}{(Numerical Solutions)}
You may submit this problem only for score increases on numerical solutions 2.4 to 2.6.

Solve symbolically by chapter 1 methods the initial value problem
$y'=e^{-y}$, $y(0)=0$. Do an answer check in maple or by hand.
Answer: $y=\ln(1+x)$. This problem has no numerical work!

\Prob{XcL4.2}{(Numerical Solutions)}
You may submit this problem only for score increases on numerical solutions 2.4 to 2.6.
This problem counts as three (3) problems.

Solve $y'=e^{-y}$, $y(0)=0$ numerically
for the value of $y(1.0)$ using (1) Euler's method,
(2) Heun's method and (3) the RK4 method. Use step size $h=0.001$. Include
a computer code appendix in the report, but do not print the data.
Report the answers in a table for $x$-values $0$, $0.2$, $0,4$, $0.6$,
$0.8$, $1.0$. Include the percentage error
$E=100|\ln(2)-y(1.0)|/|\ln(2)|$ in your report, one error report for
each of the three methods.

\Prob{Xc4.1-8}{(Transform to a first order system)}
 Use the position-velocity substitution $u_1=x(t)$, $u_2=x'(t)$,
 $u_3=y(t)$, $u_4=y'(t)$ to transform the system below into
 vector-matrix form $\.u'(t)=A\.u(t)$. Do not attempt to solve the
 system.

 $$ x''-2x'+5y=0,\quad y'' + 2y' -5x=0.$$

\Prob{Xc4.1-20a}{(Dynamical systems)}
 Prove this result for system
 \begin{equation}\label{ch7.xc.dynamical-system}
 \begin{array}{lcrcrcr}
  x' &=& ax &+& by, \\
  y' &=& cx &+& dy.
 \end{array}
 \end{equation}

\begin{quote}\sf
\hrule

 \BF{Theorem}.  Let $A=\twobytwo{a&b}{c&d}$ and define
 ${\bf trace}(A)=a+d$. Then $p_1=-{\bf trace}(A)$, $p_2=\det(A)$ are the coefficients
 in the determinant expansion
  $$\det(A-rI)=r^2+p_1r+p_2$$
 and $x(t)$ and $y(t)$ in equation (\ref{ch7.xc.dynamical-system}) are both solutions of the differential
 equation $u''+p_1u'+p_2u=0$.

 \hrule
 \end{quote}

\Prob{xC4.1-20b}{(Solve dynamical systems)}

 (a) Apply the previous problem to solve
 \begin{displaymath}
  \begin{array}{lcrcrcr}
  x' &=& 2x &-& y, \\
  y' &=& x &+& 2y.
   \end{array}
 \end{displaymath}

 (b) Use first order methods to solve the system
 \begin{displaymath}
  \begin{array}{lcrcrcr}
  x' &=& 2x &-& y, \\
  y' &=&    & & 2y.
   \end{array}
 \end{displaymath}


\Prob{Xc4.2-12}{(General solution answer check)}
 (a) Verify that $\.x_1(t)=e^{3t}\colvector{1\\ -1}$ and
 $\.x_2(t)=e^{2t}\colvector{1\\ -2}$ are solutions of $\.x'=A\.x$,
 where
 $$A=\twobytwo{4&1}{-2&1}.$$

 (b) Apply the Wronskian test $\det(\AUG(\.x_1,\.x2))\ne 0$ to
 verify that the two solutions are independent.

 (c) Display the general solution of $\.x'=A\.x$.


 \vfill {\bf End of extra credit problems chapter 4.}

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