Singularities of implicit ordinary differential equations

This paper concerns quasi-linear implicit differential equations of form 0=A₁(x)x˙-g₁(x), 0=g₂(x), where A₁: U→L(Rⁿ,R^n-m)∈C¹, g_l: U→R^n-m∈C¹, g₂: U→R^m∈C², U⊆Rⁿ is open, n, m∈N, and m<n. In particular, (1) is considered about impasse points x₀∈U, i.e., points x₀ beyond which solutions are not continuable. Under appropriate assumptions, it is shown that there is a diffeomorphism that transforms solutions of the implicit differential equation (1) near such points into solutions of the normal form x₁^rx˙₁=σ, x˙₂=0,...,x˙_n-m=0, x_n-m+1=0,...,x_n=0, near 0, and vice versa, where σ=±1=const. In particular, standard impasse points in the sense of RABIER and RHEINBOLDT lead to (2) with r=1. A practical example for r=2 is also given