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Difference between revisions of "Jacobi equation"

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A first-order ordinary differential equation
 
A first-order ordinary differential 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/j/j054/j054060/j0540601.png" /></td> </tr></table>
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$$\frac{dy}{dx}=\frac{Axy+By^2+ax+by+c}{Ax^2+Bxy+\alpha x+\beta y+\gamma}$$
  
 
or, in a more symmetric form,
 
or, in a more symmetric form,
  
<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/j/j054/j054060/j0540602.png" /></td> </tr></table>
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$$(a_1x+b_1y+c_1)(xdy-ydx)+{}$$
  
<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/j/j054/j054060/j0540603.png" /></td> </tr></table>
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$${}+(a_2x+b_2y+c_2)dx-(a_3x+b_3y+c_3)dy=0,$$
  
 
where all the coefficients are constant numbers. This equation, which is a special case of the [[Darboux equation|Darboux equation]], was first studied by C.G.J. Jacobi [[#References|[1]]]. The Jacobi equation is always integrable in closed form by using the following algorithm. First one finds by direct substitution at least one particular linear solution
 
where all the coefficients are constant numbers. This equation, which is a special case of the [[Darboux equation|Darboux equation]], was first studied by C.G.J. Jacobi [[#References|[1]]]. The Jacobi equation is always integrable in closed form by using the following algorithm. First one finds by direct substitution at least one particular linear solution
  
<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/j/j054/j054060/j0540604.png" /></td> </tr></table>
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$$y=px+q.$$
  
 
Then one makes the changes of variables
 
Then one makes the changes of variables
  
<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/j/j054/j054060/j0540605.png" /></td> </tr></table>
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$$\xi=\frac x{px-y+q},\quad\eta=\frac y{px-y+q},$$
  
 
to obtain an equation that is reducible to a homogeneous equation.
 
to obtain an equation that is reducible to a homogeneous equation.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,  "De integratione aequationis differentialis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054060/j0540606.png" />"  ''J. Reine Angew. Math.'' , '''24'''  (1842)  pp. 1–4</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,  "De integratione aequationis differentialis $(A+A'x+A''y)(x\partial y-y\partial x)-(B+B'x+B''y)\partial y+(C+C'x+C''y)\partial x=0$"  ''J. Reine Angew. Math.'' , '''24'''  (1842)  pp. 1–4</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR></table>
  
  

Latest revision as of 19:27, 31 March 2017

A first-order ordinary differential equation

$$\frac{dy}{dx}=\frac{Axy+By^2+ax+by+c}{Ax^2+Bxy+\alpha x+\beta y+\gamma}$$

or, in a more symmetric form,

$$(a_1x+b_1y+c_1)(xdy-ydx)+{}$$

$${}+(a_2x+b_2y+c_2)dx-(a_3x+b_3y+c_3)dy=0,$$

where all the coefficients are constant numbers. This equation, which is a special case of the Darboux equation, was first studied by C.G.J. Jacobi [1]. The Jacobi equation is always integrable in closed form by using the following algorithm. First one finds by direct substitution at least one particular linear solution

$$y=px+q.$$

Then one makes the changes of variables

$$\xi=\frac x{px-y+q},\quad\eta=\frac y{px-y+q},$$

to obtain an equation that is reducible to a homogeneous equation.

References

[1] C.G.J. Jacobi, "De integratione aequationis differentialis $(A+A'x+A''y)(x\partial y-y\partial x)-(B+B'x+B''y)\partial y+(C+C'x+C''y)\partial x=0$" J. Reine Angew. Math. , 24 (1842) pp. 1–4
[2] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)


Comments

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Jacobi equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_equation&oldid=15838
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article