Talk:PlanetPhysics/Differential Equation of the Family of Parabolas

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: differential equation of the family of parabolas %%% Primary Category Code: 02.30.Hq %%% Filename: DifferentialEquationOfTheFamilyOfParabolas.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

To find the differential equation of the family of parabolas

$$y = ax + bx^2$$

we differentiate twice to obtain

$$ y^{\prime} = a + 2bx$$ $$ y^{\prime \prime} = 2b$$

The last equation is solved for $b$, and the result is substituted into the previous equation. This equation is solved for $a$, and the expressions for $a$ and $b$ are substituted into $y = ax + bx^2$. The result is the \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} $$y = xy^{\prime} - \frac{1}{2}x^2y^{\prime \prime}$$

The elimination of the constants $a$ and $b$ can also be obtained by considering the equations

$$xa + x^2b + (-y)1 = 0$$ $$a + 2xb + (-y^{\prime})1 = 0$$ $$2b +(-y^{\prime \prime})1 = 0$$

as a \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of homogeneous linear equations in $a$,$b$,$1$. The solution $(a,b,1)$ is nontrivial, and hence the \htmladdnormallink{determinant}{http://planetphysics.us/encyclopedia/Determinant.html} of the coefficients vanishes.

$$ \left| \begin{array}{ccc} x & x^2 & -y \\ 1 & 2x & -y^{\prime} \\ 0 & 2 & -y^{\prime \prime} \end{array} \right| = 0 $$

Expansion about the third column yields the result above.

\subsection{References}

[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

\end{document}