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Difference between revisions of "D'Alembert equation"

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A differential equation of the form
 
A differential equation of the form
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030040/d0300401.png" /></td> </tr></table>
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$$y=x\phi(y')+f(y'),$$
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030040/d0300402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030040/d0300403.png" /> are the functions to be differentiated; first studied in 1748 by J. d'Alembert. Also known as the Lagrange equation.
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where $\phi$ and $f$ are the functions to be differentiated; first studied in 1748 by [[DAlembert|J. d'Alembert]]. Also known as the Lagrange equation.
  
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For $\phi(y')=y'$ the d'Alembert equation specializes to the [[Clairaut equation|Clairaut equation]]. For some results on (solving) the d'Alembert equation cf., e.g., [[#References|[a1]]].
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====Comments====
 
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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030040/d0300404.png" /> the d'Alembert equation specializes to the [[Clairaut equation|Clairaut equation]]. For some results on (solving) the d'Alembert equation cf., e.g., [[#References|[a1]]].
 
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,  "Integration of ordinary differential equations" , Oliver &amp; Boyd  (1963)  pp. 43ff</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,  "Integration of ordinary differential equations" , Oliver &amp; Boyd  (1963)  pp. 43 {{ZBL}0612.34002}}</TD></TR>
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</table>

Latest revision as of 11:24, 22 March 2023

A differential equation of the form

$$y=x\phi(y')+f(y'),$$

where $\phi$ and $f$ are the functions to be differentiated; first studied in 1748 by J. d'Alembert. Also known as the Lagrange equation.

For $\phi(y')=y'$ the d'Alembert equation specializes to the Clairaut equation. For some results on (solving) the d'Alembert equation cf., e.g., [a1].

References

[a1] E.L. Ince, "Integration of ordinary differential equations" , Oliver & Boyd (1963) pp. 43 {{ZBL}0612.34002}}
How to Cite This Entry:
D'Alembert equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_equation&oldid=13539
This article was adapted from an original article by BSE-2 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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