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| In [[mathematics]], a '''Clairaut's equation''' is a [[differential equation]] of the form
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| :<math>y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right).</math>
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| To solve such an equation, we differentiate with respect to ''x'', yielding
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| :<math>\frac{dy}{dx}=\frac{dy}{dx}+x\frac{d^2 y}{dx^2}+f'\left(\frac{dy}{dx}\right)\frac{d^2 y}{dx^2},</math> | |
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| so
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| :<math>0=\left(x+f'\left(\frac{dy}{dx}\right)\right)\frac{d^2 y}{dx^2}.</math>
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| Hence, either
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| :<math>0=\frac{d^2 y}{dx^2}</math>
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| or
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| :<math>0=x+f'\left(\frac{dy}{dx}\right).</math>
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| 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
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| :<math>y(x)=Cx+f(C),\,</math>
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| the so-called ''general solution'' of Clairaut's equation.
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| The latter case,
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| :<math>0=x+f'\left(\frac{dy}{dx}\right),</math>
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| defines only one solution ''y''(''x''), the so-called ''[[singular solution]]'', whose graph is the [[envelope (mathematics)|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''.
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| This equation has been named after [[Alexis Clairaut]], who has introduced it in 1734.
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| A first-order [[partial differential equation]] is also known as '''Clairaut's equation''' or '''Clairaut equation''':
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| :<math>\displaystyle u=xu_x+yu_y+f(u_x,u_y).</math>
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| ==Examples== | |
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| <gallery>
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| Image:Solutions to Clairaut's equation where f(t)=t^2.png|Solutions to Clairaut's equation where <math>f(p)=p^2</math>
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| Image:Solutions to Clairaut's equation where f(t)=t^3.png|<math>f(p)=p^3</math>
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| </gallery>
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| ==External links==
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| *{{Citation
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| | surname = Clairaut
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| | given = Alexis Claude
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| | title = Solution de plusieurs Problemes où il s'agit de trouver des Courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une Équation donnée.
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| | url = http://gallica.bnf.fr/ark:/12148/bpt6k3531x/f344.table
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| | journal = [http://gallica.bnf.fr/ark:/12148/cb32786820s/date Histoire de l'Académie royale des sciences]
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| | year = 1736 (Année 1734)
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| | pages = 196–215}}. At [[Gallica]]: the paper of Clairaut introducing the equation named after him.
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| *{{springer | title=Clairaut equation | id=C/c022350 | last=Rozov | first=N. Kh.}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Partial differential equations]]
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