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Math 305 \hfill Homework Set 12  \hfill Spring 2017

\begin{center}Due Friday, April 21\end{center}

\vspace{0.5in}

{\bf I.} Solve the initial value problems.

1. $y'' + 4y = 3 \sin 2t$, $y(0)=2$, $y'(0)=0$.

2. $y'' -2y' + y = te^t + 4$, $y(0)=1$, $y'(0)=0$.

\vspace{.2in}
{\bf II.}
 Suppose $y'(t) = F(y(t))$, where the graph of $F(y)$ is given
        below.  Carefully draw several solution curves for this equation.
        What are the equilibrium solutions? What are their stability
        types?  Describe the initial concavity of the solution curves.
        Assume $y(t)$ and $t$ are non-negative.

% @@@@@@@@@@@@@@@@@@@@@@@@@@@@ FIGURE @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
\begin{figure}[htb]
        \psfrag{y'}{$F(y)$}
        \psfrag{y}{$y$}
        \psfrag{t}{$t$}
        \psfrag{A}{$A$}
        \psfrag{B}{$B$}
        \psfrag{C}{$C$}
        \psfrag{D}{$D$}
        \psfrag{E}{$E$}
        \psfrag{F}{$F$}
        \psfrag{0}{$0$} 
        \begin{center}  
        \includegraphics[height=2in]{../Tests/Figs/phaseplot.eps}
        \end{center}
\end{figure}
% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

{\bf III.} These are similar to problems 1-6 in 10.5.

1. Consider the PDE, $p(x)u_{xx} + u_t =0$. Suppose $u(x,t)=X(x)T(t)$. 
Show that for some constant $\sigma$ it follows that
\[
p(x)X'' + \sigma X = 0 \hspace{.3in} T' - \sigma T = 0.
\]


2.  Consider the PDE, $u_{xx} + u_{yy} = u_t$. Suppose $u(x,y,t)=X(x)Y(y)T(t)$. Show that for some constants $\alpha$ and $\sigma$ it follows that

\[
T'+\sigma T = 0 \hspace{.3in} X'' + \alpha X = 0 \hspace{.3in} Y'' + (\sigma+\alpha)Y = 0.
\]

\vfill
\pagebreak
{\bf IV.} Now for the fun stuff. 

1. Consider a vibrating ideal string (that is assume the wave equation is valid) with $a=1$, $L=3$, with initial displacement $f(x)$ given below. 

\begin{center}
\includegraphics[height=1.5in]{Figs/string3.eps}
\end{center}



Find $u(x,t)$ and plot it for $t=0.0$, $t=0.001$, $t=0.01$, $t=0.1$, and $t=0.5$.  

\vspace{.2in}

2. Consider a vibrating ideal string (that is assume the wave equation is valid) with $a=1$, $L=\pi$, with initial displacement $f(x) =\sin 5x$. Find $u(x,t)$ and plot it for $t=0.0$, $t=0.001$, $t=0.01$, $t=0.1$, and $t=0.5$.  

\vspace{.2in}
3. Consider a metal rod of length 2 and $\alpha=1$ with the initial temperature shown below.

\begin{center}\psfrag{a}{$x^2$}\psfrag{b}{$(x-2)^2$}
\includegraphics[height=1.5in]{Figs/rod1.eps}
\end{center}



Find $u(x,t)$ and plot it for $t=0.0$, $t=0.001$, $t=0.01$, $t=0.1$, and $t=0.5$.  

\vspace{.2in}

4. Animate any of these for extra credit. You can just show me the animation  on your laptop/tablet or email it to me.

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