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Difference between revisions of "Clairaut equation"

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An ordinary first-order differential equation not solved with respect to its derivative:
 
An ordinary first-order differential equation not solved with respect to its derivative:
  
<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/c/c022/c022350/c0223501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$y=xy'+f(y'),\tag{1}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c0223502.png" /> is a non-linear function. Equation (1) is named after A. Clairaut [[#References|[1]]] who was the first to point out the difference between the general and the singular solutions of an equation of this form. The Clairaut equation is a particular case of the [[Lagrange equation|Lagrange equation]].
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where $f(t)$ is a non-linear function. Equation \ref{1} is named after A. Clairaut [[#References|[1]]] who was the first to point out the difference between the general and the singular solutions of an equation of this form. The Clairaut equation is a particular case of the [[Lagrange equation|Lagrange equation]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c0223503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c0223504.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c0223505.png" />, then the set of integral curves (cf. [[Integral curve|Integral curve]]) of (1) consists of: a parametrically given curve
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If $f(t)\in C^1(a,b)$ and $f'(t)\neq0$ when $t\in(a,b)$, then the set of integral curves (cf. [[Integral curve|Integral curve]]) of \ref{1} consists of: a parametrically given curve
  
<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/c/c022/c022350/c0223506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$x=-f'(t),\quad y=-tf'(t)+f(t),\quad a<t<b;\tag{2}$$
  
 
a one-parameter family of straight lines
 
a one-parameter family of straight lines
  
<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/c/c022/c022350/c0223507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$y=Cx+f(C),\quad C\in(a,b),\tag{3}$$
  
tangent to the curve (2); curves consisting of an arbitrary segment of the curve (2) and the two straight lines of the family (3) tangent to (2) at each end of this segment. The family (3) forms the [[General solution|general solution]], while the curve (2), which is the [[Envelope|envelope]] of the family (3), is the [[Singular solution|singular solution]] (see [[#References|[2]]]). A family of tangents to a smooth non-linear curve satisfies a Clairaut equation. Therefore, geometric problems in which it is required to determine a curve in terms of a prescribed property of its tangents (common to all points of the curve) leads to a Clairaut equation.
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tangent to the curve \ref{2}; curves consisting of an arbitrary segment of the curve \ref{2} and the two straight lines of the family \ref{3} tangent to \ref{2} at each end of this segment. The family \ref{3} forms the [[General solution|general solution]], while the curve \ref{2}, which is the [[Envelope|envelope]] of the family \ref{3}, is the [[Singular solution|singular solution]] (see [[#References|[2]]]). A family of tangents to a smooth non-linear curve satisfies a Clairaut equation. Therefore, geometric problems in which it is required to determine a curve in terms of a prescribed property of its tangents (common to all points of the curve) leads to a Clairaut equation.
  
 
The following first-order partial differential equation is also called a Clairaut equation:
 
The following first-order partial differential equation is also called a Clairaut equation:
  
<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/c/c022/c022350/c0223508.png" /></td> </tr></table>
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$$z=x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}+f\left(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right);$$
  
 
it has the integral
 
it has the integral
  
<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/c/c022/c022350/c0223509.png" /></td> </tr></table>
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$$x=\alpha x+\beta y+f(\alpha,\beta),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c02235010.png" /> is an arbitrary point of the domain of definition of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c02235011.png" /> (see [[#References|[3]]]).
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where $(\alpha,\beta)$ is an arbitrary point of the domain of definition of the function $f(p,q)$ (see [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Clairaut,  ''Histoire Acad. R. Sci. Paris (1734)''  (1736)  pp. 196–215</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Stepanov,  "A course of differential equations" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022350/c02235012.png" /> Ordnung für eine gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Clairaut,  ''Histoire Acad. R. Sci. Paris (1734)''  (1736)  pp. 196–215</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Stepanov,  "A course of differential equations" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen $\mathbf{1^\text{er}}$ Ordnung für eine gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR></table>
  
  

Revision as of 13:44, 10 August 2014

An ordinary first-order differential equation not solved with respect to its derivative:

$$y=xy'+f(y'),\tag{1}$$

where $f(t)$ is a non-linear function. Equation \ref{1} is named after A. Clairaut [1] who was the first to point out the difference between the general and the singular solutions of an equation of this form. The Clairaut equation is a particular case of the Lagrange equation.

If $f(t)\in C^1(a,b)$ and $f'(t)\neq0$ when $t\in(a,b)$, then the set of integral curves (cf. Integral curve) of \ref{1} consists of: a parametrically given curve

$$x=-f'(t),\quad y=-tf'(t)+f(t),\quad a<t<b;\tag{2}$$

a one-parameter family of straight lines

$$y=Cx+f(C),\quad C\in(a,b),\tag{3}$$

tangent to the curve \ref{2}; curves consisting of an arbitrary segment of the curve \ref{2} and the two straight lines of the family \ref{3} tangent to \ref{2} at each end of this segment. The family \ref{3} forms the general solution, while the curve \ref{2}, which is the envelope of the family \ref{3}, is the singular solution (see [2]). A family of tangents to a smooth non-linear curve satisfies a Clairaut equation. Therefore, geometric problems in which it is required to determine a curve in terms of a prescribed property of its tangents (common to all points of the curve) leads to a Clairaut equation.

The following first-order partial differential equation is also called a Clairaut equation:

$$z=x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}+f\left(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right);$$

it has the integral

$$x=\alpha x+\beta y+f(\alpha,\beta),$$

where $(\alpha,\beta)$ is an arbitrary point of the domain of definition of the function $f(p,q)$ (see [3]).

References

[1] A. Clairaut, Histoire Acad. R. Sci. Paris (1734) (1736) pp. 196–215
[2] V.V. Stepanov, "A course of differential equations" , Moscow (1959) (In Russian)
[3] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen $\mathbf{1^\text{er}}$ Ordnung für eine gesuchte Funktion , Akad. Verlagsgesell. (1944)


Comments

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Clairaut equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clairaut_equation&oldid=18469
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article