Discriminant curve

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)