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\centerline{Exam \#3}

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\noindent 
Due: Wednesday 5/29


\noindent
\underbar{On your own paper do the following problems.}
\bigskip

\noindent
You must show your work to receive credit.

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\begin{enumerate}





\item  For the dampened Spring-Mass system
$$ {d^2x\over dt^2}~+~b{dx\over dt}~+~16x~=~0 $$


\noindent
(a)  If $b=10$, $x(0)=0$, and $x'(0)=-3$, solve the differential equation.

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\noindent
(b)  For what values of $b$ are the motions of the above differential 
equation overdamped?



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\item  Find the amplitude, period and phase angle for the motion described by

$$ {10\over 32}x''~+~60x~=~0~,~x(0)=\frac1 6 ~,~x'(0)=-4~. $$

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\item  Given $${dq\over dt}=I~~ \hbox{  and  }~~{dI\over 
dt}=-\frac{37} 9 q $$ 
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\noindent
(a) Find $dI\over dq$.

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\noindent
(b)  Solve the separable differential equation you obtained in part (a) 
and find the equation of the solution curve in the phase plane given by $q(0)=6$ 
and $I(0)=0$.

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\noindent
(c)  Now solve the differential equation
$$ {d^2q\over dt^2}~+~\frac{37} 9 q~=~0  $$
subject to $q(0)=6$ and $q'(0)=0$.


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\item  Show that $x=t^{-1}$ is a solution of
$$ 2t^2x''~+~3tx'~-~x~=~0  $$

\noindent
and find the general solution for $t>0$ by the method of reduction of order.


\bigskip


\item  For the linear system

$$  {d{\bf X}\over dt} = \left[ \begin{array}{cc}
				2	&	13	\\
				-1	&	-8	
				\end{array}	\right] {\bf X}(t)  $$


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\noindent
a.  Find the eigenvalues and eigenvectors for the linear system.

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\noindent
b.  Find the general solution to the linear system.


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% \vspace{2.5in}

\begin{figure}[h]
	 \vspace{2.5in}
	\caption{coupled mass spring system}
	\label{fig:  Coupled Mass-Spring System}
\end{figure}

\bigskip

\item  Develop the system of differential equations that model the above
coupled mass spring system in Figure 1.  State what your variables represent.  Use
$k$'s for spring constants and $m$'s for masses.


\bigskip


\newpage


\item  {\bf Repeated eigenvalues, }  
Suppose the following system has non distinct eigenvalues,
$\lambda_1 = \lambda_2$.

$$ {d{\bf X} \over dt} = \mbox{ {\large {\bf A}} }~ {\bf X}(t)  $$


\noindent
Show why on the {\it luckier} attempt

$$  {\bf X}(t) = {\bf v}e^{\lambda_1 t} + t{\bf u}e^{\lambda_1 t}~~  ,$$


\noindent
the vector you need to find, satisfies the equation

$$  {\bf A}_{\lambda} {\bf v} = {\bf u} ~~.   $$


\noindent
Use this information to solve the following.

$$  {d{\bf X}\over dt} = \left[ \begin{array}{cc}
				2	&	4	\\
				-4	&	10	
				\end{array}	\right] {\bf X}(t)  $$






\end{enumerate}

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