
% extra credit ch7 2280, S2015
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 7 \\
{\small\bf S2015}
\end{center}

{\bf Submitted work}. Please submit one stapled package with this sheet
on top. Kindly check-mark the problems submitted and
label the problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number, e.g., \fbox{Xc7.3-20}.


\Prob{Xc7.3-20}{(Inverse transform)}
 Solve for $f(t)$ in the relation
 $\L(f(t))=\dd\frac{1}{s^4-8s^2+16}$. Use partial fractions in the
 details.

\Prob{Xc7.3-24}{(Inverse transform)} Solve for $f(t)$ in the
relation $\L(f(t))=\dd\frac{s}{s^4+4a^4}$, showing the details that
give the answer $f(t)=\dd\frac{1}{2a^2}\sinh at \sin at$

\Prob{Xc7.4-12}{(Inverse transform, convolution)} Solve for $f(t))$
in the relation $\L(f(t))=\dd\frac{1}{s(s^2+4s+5)}$. Instead of the
convolution theorem, use partial fractions for the details. If you
can see how, then check the answer with the convolution theorem.

\Prob{Xc7.4-26}{(Inverse transform techniques)} Use the
$s$-differentiation theorem in the details of solving for $f(t)$ in
the relation $\dd \L(f(t))= \arctan \frac{3}{s+2}$. You will need to
apply the theorem $\lim_{s\to\infty} \L(f(t)) = 0$.


\Prob{Xc7.4-40}{(Series methods for transforms)} Expand in a
series, using Taylor series formulas, the function
$f(t)=\dd\frac{\cos 2\sqrt{t}}{\sqrt{\pi t}}$. Then find $\L(f(t))$
as a series by Laplace transform of each series term, separately.
Finally, re-constitute the series in variable $s$ into elementary
functions, namely $e^{-1/s}$ divided by $\sqrt{s}$.

\Prob{Xc7.5-6}{(Second shifting theorem, Heaviside step)} Find the
function $f(t)$ in the relation
$\L(f(t))=\dd\frac{se^{-s}}{s^2+\pi^2}$.

\Prob{Xc7.5-14}{(Transforms of piecewise functions)}
 Let $\dd f(t)=\left\{ \begin{array}{ll} \cos \pi t & 0 \le t \le 2,
 \\ 0 & t > 2. \end{array}\right.$ Find $\L(f(t))$. Details should
 expand $f(t)$ as a linear combination of Heaviside step functions.

\Prob{Xc7.5-26}{(Sawtooth wave)}
 Let $f(t+a)=f(t)$ and $f(t)=t$ on $0\le t \le a$. Then $f$ is
 $a$-periodic and has a Laplace transform obtained from the periodic
 function formula. Show the details in the derivation to obtain the
 answer
 $\L(f(t))=\dd\frac{1}{as^2} - \frac{e^{-as}}{s(1-e^{-as})}$.

\Prob{Xc7.5-28}{(Modified sawtooth wave)}
 Let $f(t+2a)=f(t)$ and $f(t)=t$ on $0\le t \le a$, $f(t)=0$ on $a<t \le 2a$. Then $f$ is
 $2a$-periodic and has a Laplace transform obtained from the periodic
 function formula. Derive a formula for $\L(f(t))$.

%21 Nov, Page 510 in EPbvp 7.6, \Y{8}, \Y{22}
\Prob{Xc7.6-8}{(Impulsive DE)}
Solve by Laplace methods $x''+2x'+x=\delta(t)-2\delta(t-1)$, $x(0)=1$, $x'(0)=1$.


\Prob{Xc7.6-18}{(Switching circuit)}
A passive LC-circuit has battery 6 volts and model equation
$i''+100i=6\delta(t)-6\delta(t-1)$, $x(0)=1$, $x'(0)=1$. The switch is
closed at time $t=0$ and opened again at $t=1$. Solve the equation by
Laplace methods and report the number of full cycles observed before the
steady-state $i=0$ is reached.

 \vfill {\bf End of extra credit problems chapter 7.}

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