Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z): = (y∂_x - z∂_y)u(x,y,z) = f(x,y,z) ∈ 𝒪_{x,y,z}$, where $P = y∂_x - z∂_y: 𝒪_{x,y,z} → 𝒪_{x,y,z}$. For this equation, our aim is to characterize the solvability on $𝒪_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.
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We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f(x,u,D_x u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,ξ^{0}) ∈ ℂ^{n}_{x} × ℂ_{u} × ℂ^{n}_{ξ} (ξ^{0} = D_{x}u(0))$ and $f(0,0,ξ^{0}) = 0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $(ξ ∈ ℂ^{n})$. The criterion of convergence of a formal solution $u(x) = ∑_{|α| ≥ 1} u_{α}x^{α}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal solution diverges a precise rate of divergence or the formal Gevrey order is specified which can be interpreted in terms of the Newton polygon as in the case of linear equations but for nonlinear equations it depends on the individual formal solution.
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