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 \centerline{\large Math.\ 401, Sec.\ 501
\hfill    Spring, 2006}
 
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\centerline{\bfseries\large Homework 1, due January 25} 
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\noindent
In Exercises 1 and 2,
   find approximate solutions of the form 
$x\approx x_0+\epsilon x_1$ 
 ($\epsilon$~small).
 
\begin{enumerate}
 \item \quad $x^3 +\epsilon x^2 +1=0$

 \item \quad $x^5 +\epsilon x -32 =0$ 

 \item Consider $x^2 +2\epsilon x -1=0$.
\begin{enumerate}
 \item Find approximate solutions of the forms $x\approx 
x_0+\epsilon x_1$
and $x\approx x_0+\epsilon x_1 +\epsilon^2 x_2\,$.
 \item Check the consistency of your answers to (a) with the 
Taylor expansion of the exact solution.
 \item Compare the first-order, second-order, and exact 
solutions numerically, for $\epsilon = 10$, 1, 0.1, and 0.01. 
\end{enumerate}

\item  Find the second-order solutions 
($x\approx x_0+\epsilon x_1+ \epsilon^2 x_2$)
to $x^4 + \epsilon x -1=0$.

\end{enumerate}

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