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Difference between revisions of "D'Alembert equation"

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$$y=x\phi(y')+f(y'),$$
 
$$y=x\phi(y')+f(y'),$$
  
where $\phi$ and $f$ are the functions to be differentiated; first studied in 1748 by J. d'Alembert. Also known as the Lagrange equation.
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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.
  
 
 
====Comments====
 
 
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]]].
 
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]]].
  
 
====References====
 
====References====
<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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<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>
 +
</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=53075
This article was adapted from an original article by BSE-2 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article