Talk:PlanetPhysics/Differential Equation of the Family of Parabolas
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%%% 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}
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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].
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