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F. Takens (1976)
Constrained equations; a study of implicit differential equations and their discontinuous solutions
P. Rabier, W. Rheinboldt (1991)
A general existence and uniqueness theory for implicit differential-algebraic equationsDifferential and Integral Equations
G. Sansone (1948)
Equazioni differenziali nel campo reale
Differential Equations, Vol. 41, No. 1, 2005, pp. 90–95. Translated from Differentsial'nye Uravneniya, Vol. 41, No. 1, 2005, pp. 87–92. Original Russian Text Copyright c 2005 by Filippov. ORDINARY DIFFERENTIAL EQUATIONS Uniqueness of the Solution of a System of Di erential Equations Unsolved for the Derivatives A. F. Filippov Moscow State University, Moscow, Russia Received December 25, 2003 For the system of equations F (t;x ;:::;x ;dx =dt;:::;dx =dt)=0;i =1;:::;n; (1) i 1 n 1 n we show that the uniqueness of a solution with initial conditions at a nonsingular point depends on n and on whether solutions with discontinuous derivatives are admitted. We prove a theorem on sucient conditions for the uniqueness. 1. STATEMENT OF THE PROBLEM A solution of system (1) is a vector function x(t)= (x (t);:::;x (t)) that is de ned on some 1 n interval, has a derivative x (t) at each point, and satis es this system. Consider the problem F (t;x;p)= 0; x (t )= x ; x (t )= p ; (2) 0 0 0 0 where F =(F ;:::;F ), x =(x ;:::;x ), p =(p ;:::;p ), p = dx =dt (i =1;:::;n), and 1 n 1 n 1
Differential Equations – Springer Journals
Published: Apr 14, 2005
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