chapter 14 hw

\documentclass[12pt]{article}
\title{Handout 14 Homework}
\author{Jesse Webber}
\date{\today}

\begin{document}

\textbf{Task 1}

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$\frac{dy}{dx}=\frac{1}{y+yx^2}$

$\frac{dy}{dx}=\frac{1}{y}\frac{1}{1+x^2}$

$\int ydy = \int \frac{dx}{1+x^2}$

$\frac{1}{2}y^2=arctan(x)+C$

$\frac{1}{2}1^2=arctan(0)+C$

$\frac{1}{2}=0+C$

$C=\frac{1}{2}$

$\frac{1}{2}y^2=arctan(x)+\frac{1}{2}$

$\frac{1}{2}y^2=arctan(20)+\frac{1}{2}$

$y^2=2arctan(20)+1$

$y=\sqrt{2arctan(20)+1} \approx 2.010392$

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\textbf{Task 2}

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\begin{tabular}{|r|r|}
\hline
n & Estimation of y(20) \\
\hline
1 & 1.04986 \\
\hline
5 & 1.32605 \\
\hline
10 & 1.58095 \\
\hline
100 & 1.98315 \\
\hline
200 & 1.99676 \\
\hline
300 & 2.00141 \\
\hline
400 & 2.00360 \\
\hline
\end{tabular}
\textbf{Table 1:} Estimating y(20) as n increases

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It takes around 300,000 boxes to get within .00001 of the actual value of y(20)

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\textbf{Task 3}

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See attachment

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\textbf{Task 4}

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The improved Euler's Method and 4\textsuperscript{th} Order Runge Kutta Method each required an n of about a million to be accurate within .00001 of the actual value of y(20)

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\textbf{Task 5}

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\begin{tabular}{|l|r|r|r|}
\hline
Method & n=1000 & n=2000 & n=3000 \\
\hline
Euler's Method & 0.125 & 0.251 \\
\hline
Improved Euler's Method & 0.379 \\
\hline
4\textsuperscript{th} Order Runge Kutte & 0.290 \\
\hline
\end{tabular}


\end{document}