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\ {\bf    EML 5060  \hfill Homework Set 3    \hfill     Fall 2002 }\\
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\begin{tabular}{|c|c|c|l|}\hline\hline
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\   Page &  HW      & Class   & Topic                              \\\hline
\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\
\     6  & 1.15    & 1.14    & Notations                           \\
\     6  & 1.17    &         & Notations                           \\
\     6  & 1.18    &         & Notations                           \\
\     6  & 1.22    & 1.21    & Notations                           \\
\     6  & 1.26    &         & Solution by inspection              \\
\    23  & 3.40    & 3.39    & Separation of variables             \\
\    23  & 3.44    & 3.42    & Separation of variables             \\
\    23  & 3.49    & 3.50    & Homogeneous equations               \\
\    23  & 3.52    &         & Homogeneous equations               \\
\    33  & 4.30    & 4.32    & Exact equations                     \\
\    33  & 4.34    &         & Exact equations                     \\
\    41  & 5.35    & 5.34    & Linear equations                    \\
\    41  & 5.53    & 5.38    & Bernoulli equations                 \\
\    63  & 6.33    &         & Radioactive decay                   \\
\    64  & 6.59a   &         & Air resistance                      \\
\    81  & 8.28    & 8.18    & Vibrational and growth type         \\
\    81  & 8.23    & 8.19    & Vibrational and growth type         \\
\    81  & 8.24    & 8.21    & Vibrational and growth type         \\
\    86  & 9.23    &         & Vibrational and growth type         \\
\    96  & 10.44   & 10.45   & Vibrational and growth, forced      \\
\    96  & 10.46   &         & Vibrational and growth, forced      \\
\    96  & 10.52   & 10.47   & Vibrational and growth, forced      \\
\   103  & 11.9    & 11.10   & Vibrational and growth, forced      \\
\   103  & 11.14   &         & Vibrational and growth, forced      \\
\   103  & 11.26   & 11.25   & Vibrational and growth, forced      \\
\   107  & 12.10   & 12.11   & Vibrational and growth, forced      \\
\   122  & 13.40   &         & Spring mass system                  \\
\   198  & 22.22   & 22.12   & Solve as 22.12 (required)           \\\hline
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Note: make a graph of the solution for each solved problem.
(For problems with more than one unknown parameter, draw the
solutions taking one parameter 1 and the rest 0.)

Also solve the 4 questions below:

\begin{enumerate}

\item Solve the Cauchy equation
\[ x^2 y''+ xy' - 4y = \ln x^2 \]
by taking $u=\ln |x|$ as the new independent variable.
To eliminate $x$, use the chain rule of differentiation as in
\[ y' \equiv {dy\over dx} = {dy\over du} {du\over dx}
= {dy\over du} {1\over x}, \]
and once more to find $y''$ in terms of $dy/du$ and $d^2y/du^2$.
Please {\em do not} indicate $dy/du$ also by $y'$! Solution:
\[ y = -{\textstyle{1\over 2}} \ln x + A x^2 + B x^{-2} \]

\item Solve the aerodynamically damped spring-mass system
\[ \ddot y + \left(\dot y\right)^2 + y = 0\]
by taking $y$ as the independent variable and $\dot y$ as the
dependent variable. To eliminate the remaining $dt$,
(in $\ddot y = d\dot y/dt$), use the chain rule of differentiation. Solution:
\[ \dot y^2 = -y + {\textstyle{1\over 2}} + C_0 e^{-2y} \mbox{, hence }
t = \pm \int {dy\over \sqrt{- y + {1\over 2} + C_0 e^{-2y}}} \]

\item Solve the motion of a falling body with aerodynamic drag:
\[ \ddot x + \left(\dot x\right)^2 = 1. \]
Solution:
\[\dot x = {Ce^{2t} -1 \over C e^{2t}+1} \quad
x = \ln|Ce^{2t}+1| - t + D \]

\item Solve the equation for the streamfunction in a Stokes boundary layer:
\[ y'' + 2xy' - 2y = 0. \]
Note that $y=x$ is one solution.
Solution:
\[ y = C_0 x + C_1 x \int {e^{-x^2}\over x^2} dx \]
\end{enumerate}

{\bf Also:} Make exam 3 of 1997.  Give yourself 50 minutes.  Include
your solutions with homework set I and grade yourself using the
solutions on the web after you get it back.

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