View source for Discriminant curve

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''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));

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220a.gif" />

Figure: d033220a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220b.gif" />

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));

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220c.gif" />

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)).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220d.gif" />

Figure: d033220d

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d033220e.gif" />

Figure: d033220e

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Sansone,   "Equazioni differenziali nel campo reale" , '''2''' , Zanichelli  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Golubev,   "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>

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