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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Clairaut's equation ： ウィキペディア英語版 Clairaut's equation In mathematics, a Clairaut's equation is a differential equation of the form :$y(x)=x\backslash frac+f\backslash left(\backslash frac\backslash right).$ To solve such an equation, we differentiate with respect to ''x'', yielding :$\backslash frac=\backslash frac+x\backslash frac+f\text{'}\backslash left(\backslash frac\backslash right)\backslash frac,$ so :$0=\backslash left(x+f\text{'}\backslash left(\backslash frac\backslash right)\backslash right)\backslash frac.$ Hence, either :$0=\backslash frac$ or :$0=x+f\text{'}\backslash left(\backslash frac\backslash right).$ In the former case, ''C'' = ''dy''/''dx'' for some constant ''C''. Substituting this into the Clairaut's equation, we have the family of straight line functions given by :$y(x)=Cx+f(C),\backslash ,$ the so-called ''general solution'' of Clairaut's equation. The latter case, :$0=x+f\text{'}\backslash left(\backslash frac\backslash right),$ defines only one solution ''y''(''x''), the so-called ''singular solution'', whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as (''x''(''p''), ''y''(''p'')), where ''p'' represents ''dy''/''dx''. This equation has been named after Alexis Clairaut, who introduced it in 1734. A first-order partial differential equation is also known as Clairaut's equation or Clairaut equation: :$\backslash displaystyle\; u=xu\_x+yu\_y+f(u\_x,u\_y).$ ==Examples== Image:Solutions to Clairaut's equation where f(t)=t^2.png|Solutions to Clairaut's equation where $f(p)=p^2$ Image:Solutions to Clairaut's equation where f(t)=t^3.png|$f(p)=p^3$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Clairaut's equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.