
% extra credit ch3 2280, S2016
\documentclass[leqno]{article}
\input{linalg.sty}
\def\rf#1{(\ref{#1})}
\def\Prob#1#2{\par\bigskip\par\noindent
{\large\sf Problem #1}. {\bf #2}\par}
\parskip=3pt
\def\MAPLE{{\tt maple}}
% \begin{topics}{template} ... \end{topics}
% used like enumerate, itemize, description environments
% The item[] is never optional!
%
\newcommand{\topicslabel}[1]{\mbox{#1}\hfil}
\newenvironment{topics}[1]{
    \begin{list}{}{%
      \let\makelabel\topicslabel
      \settowidth{\labelwidth}{\topicslabel{#1}}
      \setlength{\leftmargin}{\labelwidth}
      \advance\leftmargin by 1.5em}
}{\end{list}}
\pagestyle{plain}
\setlength{\textheight}{9.8in}
\begin{document}
\thispagestyle{empty}
 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 3 \\
{\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{XC3.2-18}. Please attach
this printed sheet to simplify your work.

\Prob{XcL2.1}{(maple lab 2)}
You may submit this problem only for score increases on maple lab 2.

Consider the linear differential equation $u'+ku=ka(t)$, $u(0)=u_0$, where
$a(t)= 1 + \sin (\pi(t-3)/12)$. Solve the equation for $u(t)$ and check your
answer in maple. Use maple assist for integration.

\Prob{XcL2.2}{(maple lab 2)}
You may submit this problem only for score increases on maple lab 2.

Consider the linear differential equation $u'+ku=ka(t)$, $u(0)=u_0$, where
$a(t)= 1 + \sin (\pi(t-3)/12)$. Find the steady-state periodic solution
of this equation and check your answer in maple.


\Prob{XC3.1-all}{(Second order DE)} This problem counts as full credit for
5.1, if 5.1 was not submitted, and $100$ otherwise. Solve the following seven
parts.

\begin{tabular}{lll}
 (a) $y''+4y'=0$  &(b) $4y''+12y'+9y=0$  &(c) $y''+2y'+5y=0$ \\
 (d) $21y''+10y'+y=0$  &(e) $16y''+8y'+y=0$  &(f) $y''+4y'+(4+\pi)y=0$
\end{tabular}

\begin{tabular}{l}
(g) Find the differential equation $ay''+by'+cy=0$, if $e^{-x}$ and
$e^{x}$ are solutions.
\end{tabular}


\Prob{XC3.2-18}{(Initial value problems)} Given
$x^3y'''+6x^2y''+4xy'-4y=0$ has three solutions $x$, $1/x^2$,
$\dd\frac{\ln|x|}{x^2}$, prove by the Wronskian test that they are
independent and then solve for the unique solution satisfying
$y(1)=1$, $y'(1)=5$, $y''(1)=-11$.

\Prob{XC3.2-22}{(Initial value problem)} Solve the problem
$y''-4y=2x$, $y(0)=2$, $y'(0)=-1/2$, given a particular solution
$y_p(x)=-x/2$.

\Prob{XC3.3-8}{(Complex roots)} Solve $y''-6y'+25y=0$.

\Prob{XC3.3-10}{(Higher order complex roots)} Solve $y^{iv}+\pi^2
y'''=0$.

\Prob{XC3.3-16}{(Fourth order DE)} Solve the fourth order
homogeneous equation whose characteristic equation is
$(r-1)(r^3-1)=0$.

\Prob{XC3.3-32}{(Theory of equations)} Solve
$y^{iv}-y'''+y''-3y'-6y=0$. Use the theory of equations [factor
theorem, root theorem, rational root theorem, division algorithm] to
completely factor the characteristic equation. You may check answers
by computer, but please display the complete details of
factorization.

\Prob{XC3.4-20}{(Overdamped, critically damped, underdamped)}

(a) Consider $2x''(t)+12x'(t)+50x(t)=0$. Classify as overdamped,
critically damped or underdamped.

(b) Solve $2x''(t)+12x'(t)+50x(t)=0$, $x(0)=0$, $x'(0)=-8$. Express
the answer in the form $x(t)=C_1e^{\alpha_1 t}\cos(\beta_1 t -
\theta_1)$.

(c) Solve the zero damping problem $2u''(t)+50u(t)=0$, $u(0)=0$,
$u'(0)=-8$. Express the answer in phase-amplitude form
$u(t)=C_2\cos(\beta_2 t - \theta_2)$.

(d) Using computer assist, display on one graphic plots of $x(t)$
and $u(t)$. The plot should showcase the damping effects. A
hand-made replica of a computer or calculator graphic`` is
sufficient.

\Prob{XC3.4-34}{(Inverse problem)} A body weighing 100 pounds
undergoes damped oscillation in a spring-mass system. Assume the
differential equation is $mx''+cx'+kx=0$, with $t$ in seconds and
$x(t)$ in feet. Observations give $x(0.4)=6.1/12$, $x'(0.4)=0$ and
$x(1.2)=1.4/12$, $x'(1.2)=0$ as successive maxima of $x(t)$. Then
$m=3.125$ slugs. Find $c$ and $k$.

\ExerciseHeader{Atoms}. An \BF{atom} is a term of the form
$x^ke^{ax}$, $x^ke^{ax}\cos bx$ or $x^ke^{ax}\sin bx$. The symbol
$k$ is a non-negative integer. Symbols $a$ and $b$ are real numbers
with $b>0$. In particular, $1$, $x$, $x^2$, $e^{x}$, $\cos x$, $\sin
x$ are atoms. Any distinct list of atoms is linearly independent.

\ExerciseHeader{Roots and Atoms}. Define $\atomRoot(A)$ as follows.
Symbols $\alpha$, $\beta$, $r$ are real numbers, $\beta>0$ and $k$
is a non-negative integer.

\begin{center}
\begin{tabular}{|c|c|}
\hline
atom $A$ & $\atomRoot(A)$ \\
\hline
  $x^ke^{rx}$ & $r$ \\
\hline
 $x^ke^{\alpha x}\cos \beta x $ & $\alpha+i\beta$ \\
\hline
 $x^ke^{\alpha x}\sin \beta x$ & $\alpha+i\beta$ \\
\hline
\end{tabular}
\end{center}

The fixup rule for undetermined coefficients can be stated as
follows:
\begin{quote}
\EM{Compute $\atomRoot(A)$ for all atoms $A$ in the trial solution.
Assume $r$ is a root of the characteristic equation of multiplicity
$k$. Search the trial solution for atoms $B$ with $\atomRoot(B)=r$,
and multiply each such $B$ by $x^k$. Repeat for all roots of the
characteristic equation.}
\end{quote}

\Prob{Xc3.5-1A}{(AtomRoot Part 1)}

\begin{exercises}
\prob{33.}{} Evaluate $\atomRoot(A)$ for $A=1$, $x$, $x^2$,
$e^{-x}$, $\cos 2x$, $\sin 3x$, $x\cos \pi x$, $e^{-x}\sin 3x$,
$x^3$, $e^{2x}$, $\cos x/2$, $\sin 4x$, $x^2\cos x$, $e^{3x}\sin
2x$.
\end{exercises}

\begin{exercises}
\prob{33.}{} Let $A=xe^{-2x}$ and $B=x^2e^{-2x}$. Verify that
$\atomRoot(A)=\atomRoot(B)$.
\end{exercises}

\Prob{Xc3.5-1B}{(AtomRoot Part 2)}

\begin{exercises}
\prob{33.}{} Let $A=xe^{-2x}$ and $B=x^2e^{2x}$. Verify that
$\atomRoot(A)\ne\atomRoot(B)$.
\end{exercises}

\begin{exercises}
\prob{33.}{} Atoms $A$ and $B$ are said to be \BF{related} if and
only if the derivative lists $A$, $A'$, \ldots and $B$, $B'$, \ldots
share a common atom. Prove: atoms $A$ and $B$ are related if and
only if $\atomRoot(A)=\atomRoot(B)$.
\end{exercises}




\Prob{XC3.5-6}{(Undetermined coefficients, fixup rule)} Find a
particular solution $y_p(x)$ for the equation
$y^{iv}-4y''+4y=xe^{2x}+x^2e^{-2x}$. Check your answer in \MAPLE{}.

\Prob{XC3.5-12}{()} Find a particular solution $y_p(x)$ for the
equation $y^{iv}-5y''+4y=xe^{x}+x^2e^{2x}+\cos x$. Check your answer
in \MAPLE{}.

\Prob{XC3.5-22}{(Fixup rule, trial solution)} Report a trial
solution $y$ for the calculation of $y_p$ by the method of
undetermined coefficients, after the fixup rule has been applied. To
save time, do not do any further undetermined coefficients steps.
$$ y^{v} + 2y''' + 2y'' = 5x^3+ e^{-x} +4\cos x.$$
Hint: Test $r^2(r^3+2r+2)=0$ when $r=\atomRoot(B)$ and $B$ is an
atom in the initial trial solution. This means a test only for
$r=0,-1,i$.

\Prob{XC3.5-54}{(Variation of parameters)} Solve by variation of
parameters for $y_p(x)$ in the equation $y''-16y=xe^{4x}$. Check
your answer in \MAPLE{}.

\Prob{XC3.5-58}{(Variation of parameters)} Solve by the method of
variation of parameters for $y_p(x)$ in the equation $(x^2-1)y''
-2xy' + 2y = x^2-1$. Use the fact that $\{ x,1+x^2\}$ is a basis of
the solution space of the homogeneous equation. Apply (33) in the
textbook, after division of the leading coefficient $(x^2-1)$. Check
your answer in \MAPLE{}.

\Prob{XC3.6-4}{(Harmonic superposition)} Write the general solution
$x(t)$ as the superposition of two harmonic oscillations of
frequencies $2$ and $3$, for the initial value problem
$x''(t)+4x(t)=16\sin 3t$, $x(0)=0$, $x'(0)=0$.

\Prob{XC3.6-8}{(Steady-state periodic solution)} The equation
$x''(t)+3x'(t)+3x(t)=8\cos 10t + 6\sin 10t$ has a unique
steady-state periodic solution of period $2\pi/10$. Find it.

\Prob{XC3.6-18}{(Practical resonance)} Use the equation
$\omega=\dd\sqrt{\frac{k}{m}-\frac{c^2}{2m^2}}$ to decide upon
practical resonance for the equation $mx''+cx'+kx=F_0\cos \omega t$
when $F_0=10$, $m=1$, $c=4$, $k=5$. Sketch the graph of
$C(\omega)=\dd\frac{F_0}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}}$ and
mark on the graph the location of the resonant frequency (if any).
See Figure 5.6.9 in Edwards-Penney.


%12 Nov, Page 227 in EPbvp 3.7: \quad \Y{4}, \Y{12}
\Prob{XC3.7-4}{(LR-circuit)}
An LR-circuit with emf $E(t)=100e^{-12t}$, inductor $L=2$, resistor $R=40$
is initialized with $i(0)=0$. Find the current $i(t)$ for $t\ge 0$ and argue physically and mathematically why the
observed current at $t=\infty$ should be zero.

\Prob{XC3.7-12}{(Steady-state of an RLC-circuit)}
Compute the steady-state current in an RLC-circuit with parameters
$L=5$, $R=50$, $C=1/200$ and emf $E(t)=30\cos 100t + 40\sin 100t$.
Report the amplitude, phase-lag and period of this solution.

 \vfill {\bf End of extra credit problems chapter 3.}

\end{document}
