Formal normalization of binary differential equations

E. A. Cherepanova

Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 2, pp. 212-227

Résumé

Implicit differential equations (binary differential equations) of the form $ap^2+2bp+c=0$ are considered, where $a=a(x,y),~b=b(x,y),~c=c(x,y),~p=\frac{dy}{dx}$, such that $a(0,0)=b(0,0)=c(0,0)=0$. It is shown that a typical equation of this type by formal substitutions of coordinates $(x,y)\longmapsto(X,Y)$ can be reduced to the formal normal form $(\alpha X+\beta Y+\gamma(X))P^2+X+Y=0,~P=\frac{dY}{dX}$, where $\alpha,\beta\in \mathbb{C}\setminus\{0\}$, $\gamma$ is a formal series in the variable $X$, $\gamma(0)=0,~\gamma'(0)=0.$

Détail

Keywords: binary differential equation, implicit differential equation, formal normal form.