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\begin{document}
\title{Solutions, 316-XIX}
\maketitle
 7.8<13,15*,21,23*,24,29*>
CAUTION: there may be errors!!!
\section{Problem 7.8.15}
Solve the given IVP:
\[ y^{\prime \prime} + 2y^\prime -3y = \delta(t-1)-\delta(t-2) \ ; \ 
y(0) = 2\ ,\ y^\prime(0) = -2 \ .\]
{\bf Solution:}\\
\begin{eqnarray*}
{\cal L}\left\{ y^{\prime \prime} + 2y^\prime -3y \right\}&= 
           & {\cal L}\left\{ \delta(t-1)-\delta(t-2)  \right\}\Rightarrow \\
(s^2 + 2s - 3)Y(s) - 2s + 2 - 2  & = & e^{s-1} - e^{s-2} \ . 
\end{eqnarray*}
Solving for $Y(s)$ (and using the partial fractions expansion
$1/(s^2 + 2s - 3) = 1/4(1/(s-1) - 1/(s+3))$ we have

\begin{eqnarray*}
Y(s) & = & \frac{2s + e^{s-1} - e^{s-2}}{s^2 + 2s - 3} \\
     & = & \frac{1}{4}\left(2s + e^{s-1} - e^{s-2}\right)
                     \left(\frac{1}{s-1} - \frac{1}{s+3}\right) \\
     & = & \frac{1}{4} \frac{2(s-1) + 2  + e^{s-1} - e^{s-2}}{s-1} -
           \frac{1}{4} \frac{2(s+3) - 6  + e^{s-1} - e^{s-2}}{s+3} \\
     & = & \frac{1}{2} + \frac{1}{4}\frac{2  + e^{s-1} - e^{s-2}}{s-1} 
       -   \frac{1}{2} + \frac{1}{4}\frac{6  - e^{s-1} + e^{s-2}}{s+3}\\
     & = & \frac{1}{2} \frac{1}{s-1} + \frac{3}{2} \frac{1}{s+3} + \frac{1}{4}
\left( e^{s-1} - e^{s-2} \right) \left(\frac{1}{s-1} - \frac{1}{s+3} \right)
\end{eqnarray*}
and inverting the transforms we find:
\begin{eqnarray*}
y(t) & = & \frac{1}{2} e^{t} + \frac{3}{2}e^{-3t}
+ \frac{1}{4} \left( f(t-1)u(t-1) - f(t-2) u(t-2) \right)
\end{eqnarray*}
where 
\[ f(t) = {\cal L}^{-1} \left\{ \frac{1}{s-1} - \frac{1}{s+3} \right\}
        = e^{t} - e^{-3t} \ .\]
Then
\[ f(t-1) = e^{t-1} - e^{-3(t-1)}  \ ,\]
and
\[ f(t-2) = e^{t-2} - e^{-3(t-2)} \ .\]
Putting it all together:
\[ y(t) = \left\{
\begin{array}{lcl}
\frac{1}{2}e^{t}+\frac{3}{2}e^{-3t} & , & 0 \le t < 1 \\
\frac{1}{2}e^{t}+\frac{3}{2}e^{-3t}+\frac{1}{4}e^{t-1}-\frac{1}{4}e^{-3(t-1)}
& , & 1 \le < t < 2 \\
\frac{1}{2}e^{t}+\frac{3}{2}e^{-3t}+\frac{1}{4}e^{t-1}-\frac{1}{4}e^{-3(t-1)}
- \frac{1}{4} e^{t-2} + \frac{1}{4} e^{-3(t-2)} & , & 2 \le t < \infty \ .  
\end{array} \right. \]

\newpage

\section{Problem 7.8.23}
Solve the IVP and sketch the solution
\[ y^{\prime \prime} +y = -\delta(t-\pi)+\delta(t-2\pi)\ ; \ 
y(0) = 0\ ,\ y^\prime(0) = 1 \ .\]
{\bf Solution:}\\
\begin{eqnarray*}
s^2 Y - 1 + Y & = &      -e^{-\pi s}   + e^{-2 \pi s}    \Rightarrow \\
(s^2 + 1) Y   & = & 1    -e^{-\pi s}   + e^{-2 \pi s}    \Rightarrow \\
          Y   & = & \sfrac{1}{s^2 + 1}
-\sfrac{e^{-\pi s}}{s^2 + 1} + \sfrac{e^{-2 \pi s}}{s^2 + 1} \Rightarrow \\
y(t) & = & \sin{t} - u(t-\pi) \sin{(t-\pi)} +u(t-2\pi) \sin{(t-2\pi)}
\end{eqnarray*}
\[ y(t) = \left\{
\begin{array}{lcl}
\sin{t} & , & 0 \le t < \pi \ , \\
2\sin{t} & , & \pi \le t < 2\pi \ , \\
3\sin{t} & , & 2\pi \le t  <\infty \end{array} \right. \ . \]

\begin{center}
\psfig{figure=y23.ps,width=4.in}
\end{center}

\newpage

\section{Problem 7.8.29}
A mass attached to a spring is released from rest $1$ m below the equilibrium
position for the spring-mass system and it begins to vibrate. After
$\pi/2$ sec, the mass is struck by a hammer exerting an impulse on the mass.
The system is governed by the symbolic IVP:
\[ \frac{d^2x}{dt^2} + 9x = -3\delta\left(t-\frac{\pi}{2}\right) \ ;\ 
x(0) = 1\ ,\ \frac{dx}{dt}(0) = 0 \ ,\]
where $x(t)$ denotes the displacement from equilibrium at time $t$.
What happens to the mass after it is struck?\\
{\bf Solution:}\\
\begin{eqnarray*}
s^2 X - s + 9X & = & -3 e^{-s\pi/2}  \\
X  & = & \frac{s}{s^2+9} - \frac{3}{s^2+9}e^{-s\pi/2} \Rightarrow \\
x = \cos{3t} - \sin{3(t-\frac{\pi}{2})} u(t-\frac{\pi}{2}) 
  & = & \cos{3t} - \cos{3t} u(t-\frac{\pi}{2})
\end{eqnarray*}
\[ x(t) = \cos{3t}\left( 1- u(t-\frac{\pi}{2})\right) = \left\{
\begin{array}{lcl}
\cos{3t} & , & 0 \le t < \frac{\pi}{2} \ , \\
0 & , & 2\pi \le t  <\infty \end{array} \right. \ . \]

\begin{center}
\psfig{figure=x24.ps,width=4.in}
\end{center}

\end{document}

