## The Differential Equations of Dynamics ... |

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approximate solutions furnished ARTHUR CONSTANT LUNN Axn0 Bendixson Cauchy-Lipschitz Approximations closed interval t0 closed set complex considered continuous functions continuous partial derivatives converge uniformly corresponding defined and lie derivatives with respect determinant differential equations dXJDxj equations 12 equations dx exact solution existence finite number func functions X functions Xi holomorphic with respect homogeneous linear equations implicit functions increments inequalities 19 infinite sequence initial conditions initial constants initial values xi0 interior points intervals less Jacobians JST/s Lemma limit-point Lindklof lower bound distinct Math method of variation moduli neighborhood normal integrals Nouv Painleve periodic solution Poincare point iVj positive numbers Pr(t preceding theorems proof real variables second members single-valued functions solution P(t solutions as functions solutions of equations t0 f t tions transformations of variables uniformly convergent series uniquely defined upper bound independent values t0 variation of parameters xi(x xri0 Xt(x