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Math 305 \hfill Homework Set 1  \hfill Spring 2017

\begin{center}Due Monday, April 10\end{center}

\vspace{0.5in}

{\bf I.} Find the general solution to each of the following.

1. $(e^x \sin y - 2 y \sin x) + (e^x \cos y + 2 \cos x) y' = 0$.

2. $t^2y' + 2t y - y^3 = 0$.

3. $y' - 2y = 4-x$.

4. $2y' - y= e^{x/3}$.

\vspace{1in}

{\bf II.} Find all solutions to the following boundary value problems (this is based on Section 10.1).

1. $y'' + y = 0$  with $y(0)=0$, $y(\pi)=0$.

2. $y'' + y = 0$  with $y(0)=0$, $y(\pi/3)=0$.

3. $y'' + 3y = 0$ with $y(0)=0$, $y(\pi)=0$.

4. For which values of $\gamma$ will $y'' + \gamma y = 0$,  $y(0)=0$, $y(\pi)=0$ have nontrivial solutions? Explain.

5.  $y'' + y = 0$  with $y'(0)=0$, $y'(\pi)=0$.

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More on next page.

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{\bf III.} Find the Fourier series of each of the functions shown below. (Make sure you are using the correct value for $L$.) You can do the integration 
with a computer or calcuator, but indicate when you have done so. Write out at least the first 10 terms. 


1. 
\begin{center}
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\psfrag{10}{10}
\psfrag{-2}{$-2$}
\psfrag{-4}{$-4$}
\psfrag{-6}{$-6$}
\psfrag{-8}{$-8$}
\psfrag{-10}{$-10$}
\includegraphics[width=5in]{Figs/FS1.eps}
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2.
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\psfrag{-2}{$-2$}
\psfrag{-4}{$-4$}
\psfrag{-6}{$-6$}
\psfrag{-8}{$-8$}
\psfrag{-10}{$-10$}
\includegraphics[width=5in]{Figs/FS2.eps}
\end{center}

3.
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\psfrag{1}{1}
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\psfrag{-1}{$-1$}
\psfrag{-2}{$-2$}
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\includegraphics[width=5in]{Figs/FS3.eps}
\end{center}

\vspace{1in}

{\bf III.} For \#1 above use a compute to plot the partial sums for 
$N=1, 2, 5,$ and 10.




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