Local study of differential equations of the form xy′ = f(x,y) near x = 0

Abstract

In this chapter, we are studying the singularity of the solutions near the origin of a singular differential equation of the form

$$xy' = f(x,y)$$

((4.0.1))

where f (x, y) is holomorphic near the origin of ℂ². In particular, we are looking for a normal form of (4.0.1). We have a transformation z = z(x,y) which is reducing (4.0.1) to a normal form and this transformation is given as a solution of a partial differential equation of the form

$$\tau z = F(x,y,z)$$

((4.0.2))

where F(x, y,z) is holomorphic near the origin of ℂ³ and τ is a holomorphic vector field having a singular point at the origin ofℂ².