In this paper we introduce the concept of the index of an implicit differential equation F (x, y, p) = 0, where F is a smooth function, p = dy dx , Fp = 0 and Fpp = 0 at an isolated singular point. We also apply the results to study the geometry of surfaces in R 5 .
In this paper we introduce the concepts of multiplicity and index of first order partial differential equations. In particular, the concept of multiplicity coincides with the multiplicity of implicit differential equations given by Bruce and Tari in [2]. We also show that these concepts are invariants by smooth equivalences. Following the works Hayakawa, Ishikawa, Izumiya and Yamaguchi on implicit differential equations with first integrals, we introduce a definition of multiplicity for this class of equations
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