%% %% Ein Beispiel der DANTE-Edition %% Mathematiksatz mit LaTeX %% 3. Auflage %% Beispiel 09-20-1 auf Seite 201. %% Copyright (C) 2018 Herbert Voss %% %% It may be distributed and/or modified under the conditions %% of the LaTeX Project Public License, either version 1.3 %% of this license or (at your option) any later version. %% See http://www.latex-project.org/lppl.txt for details. %% %% ==== % Show page(s) 1 %% %% \documentclass[10pt]{exaartplain} \pagestyle{empty} \setlength\textwidth{352.81416pt} \AtBeginDocument{\setlength\parindent{0pt}} %StartShownPreambleCommands \usepackage{amsmath,mathastext} \MTDeclareVersion[n]{lmvtt}{T1}{lmvtt}{m}{n} %StopShownPreambleCommands \begin{document} \MTversion{lmvtt} Let $(X,Y)$ be two functions of a variable $a$. If they obey the differential system $(VI_{\nu,n})$: \begin{align*} a\frac{d}{da} X &= \nu X - (1 - X^2)\frac{2n a}{1 - a^2}\frac{aX+Y}{1+a XY} \\ a\frac{d}{da} Y &= -(\nu+1) Y + (1-Y^2)\frac{2n a}{1 - a^2}\frac{X+aY}{1+a XY} \end{align*} then the quantity $q=a\frac{aX+Y}{X+aY}$ satisfies as function of $b=a^2$ the $P_{VI}$ differential equation: \begin{equation*} \begin{split} \frac{d^2 q}{db^2} = \frac12\left\{\frac1q+\frac1{q-1} +\frac1{q-b}\right\}\left(\frac{dq}{db}\right)^2 - \left\{\frac1b+\frac1{b-1} +\frac1{q-b}\right\}\frac{dq}{db}\\+\frac{q(q-1)(q-b)}{b^2(b-1)^2}\left\{\alpha+\frac{\beta b}{q^2} + \frac{\gamma (b-1)}{(q-1)^2}+\frac{\delta b(b-1)}{(q-b)^2}\right\} \end{split} \end{equation*} with parameters $(\alpha,\beta,\gamma,\delta) = (\frac{(\nu+n)^2}2, \frac{-(\nu+n+1)^2}2, \frac{n^2}2, \frac{1 - n^2}2)$. \end{document}