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''of an ordinary first-order differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332201.png" />''
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''of an ordinary first-order differential equation $F(x,y,y')=0$''
  
The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332202.png" /> of the plane whose coordinates satisfy the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332203.png" /> obtained by the elimination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332204.png" /> from the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332206.png" /> or by elimination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332207.png" /> from the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d0332209.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322010.png" /> (on the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322011.png" /> in fact exists). If the discriminant curve for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322012.png" /> is a non-empty set and does not degenerate into individual points, the curve (or each one of its branches) may:
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The set of points $(x,y)$ of the plane whose coordinates satisfy the equation $\phi(x,y)=0$ obtained by the elimination of $y'$ from the relations $F=0$ and $F_{y'}'=0$ or by elimination of $x'$ from the relations $G=0$ and $G_{y'}'=0$, where $G(y,x,x')\equiv F(x,y,1/x')$ (on the assumption that $F_{y'}'$ in fact exists). If the discriminant curve for the equation $F=0$ is a non-empty set and does not degenerate into individual points, the curve (or each one of its branches) may:
  
1) be a solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322013.png" /> at each point of which uniqueness is violated; the discriminant curve will then form the envelope of a family of integral curves (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322015.png" /> for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322016.png" /> (Fig. a); or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322017.png" /> for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322018.png" /> (Fig. b));
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1) be a solution of the equation $F=0$ at each point of which uniqueness is violated; the discriminant curve will then form the envelope of a family of integral curves (e.g. $y=1$ and $y=-1$ for the equation $y'^2+y^2-1=0$ (Fig. a); or $y=0$ for the equation $y'^3-y^2=0$ (Fig. b));
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220a.gif" />
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Figure: d033220b
 
Figure: d033220b
  
2) be a solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322019.png" /> at each point of which there is uniqueness (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322020.png" /> for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322021.png" /> (Fig. c));
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2) be a solution of the equation $F=0$ at each point of which there is uniqueness (e.g. $y=0$ for the equation $y'^2-y^2=0$ (Fig. c));
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220c.gif" />
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Figure: d033220c
 
Figure: d033220c
  
3) not be a solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322022.png" />. In this case the discriminant curve is either the set of cuspidal points of the integral curves (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322023.png" /> for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322024.png" /> (Fig. d)) or the set of osculation points of different integral curves (e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322025.png" /> for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322026.png" /> (Fig. e)).
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3) not be a solution of the equation $F=0$. In this case the discriminant curve is either the set of cuspidal points of the integral curves (e.g. $x=0$ for the equation $y'^2-x=0$ (Fig. d)) or the set of osculation points of different integral curves (e.g. $x=0$ for the equation $y'^2-x^2=0$ (Fig. e)).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220d.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220d.gif" />
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Figure: d033220e
 
Figure: d033220e
  
The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322027.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322028.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033220/d03322029.png" />, is also studied in the complex domain [[#References|[2]]].
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The equation $F=0$, when $F$ is a polynomial in $y'$, is also studied in the complex domain [[#References|[2]]].
  
 
====References====
 
====References====

Revision as of 13:54, 10 August 2014

of an ordinary first-order differential equation $F(x,y,y')=0$

The set of points $(x,y)$ of the plane whose coordinates satisfy the equation $\phi(x,y)=0$ obtained by the elimination of $y'$ from the relations $F=0$ and $F_{y'}'=0$ or by elimination of $x'$ from the relations $G=0$ and $G_{y'}'=0$, where $G(y,x,x')\equiv F(x,y,1/x')$ (on the assumption that $F_{y'}'$ in fact exists). If the discriminant curve for the equation $F=0$ is a non-empty set and does not degenerate into individual points, the curve (or each one of its branches) may:

1) be a solution of the equation $F=0$ at each point of which uniqueness is violated; the discriminant curve will then form the envelope of a family of integral curves (e.g. $y=1$ and $y=-1$ for the equation $y'^2+y^2-1=0$ (Fig. a); or $y=0$ for the equation $y'^3-y^2=0$ (Fig. b));

Figure: d033220a

Figure: d033220b

2) be a solution of the equation $F=0$ at each point of which there is uniqueness (e.g. $y=0$ for the equation $y'^2-y^2=0$ (Fig. c));

Figure: d033220c

3) not be a solution of the equation $F=0$. In this case the discriminant curve is either the set of cuspidal points of the integral curves (e.g. $x=0$ for the equation $y'^2-x=0$ (Fig. d)) or the set of osculation points of different integral curves (e.g. $x=0$ for the equation $y'^2-x^2=0$ (Fig. e)).

Figure: d033220d

Figure: d033220e

The equation $F=0$, when $F$ is a polynomial in $y'$, is also studied in the complex domain [2].

References

[1] G. Sansone, "Equazioni differenziali nel campo reale" , 2 , Zanichelli (1949)
[2] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)


Comments

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Discriminant curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discriminant_curve&oldid=15781
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article