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On a differential-algebraic problem. (English) Zbl 1058.34005
A method for solving the following system \[ x'(t) = f(t,x(t),y(t)),\;t\in J= [0,b],\;x(0) = k_0,\;y(t) = g(t,x(t),y(t)), \;t\in J, \] where \(f\in C(J,\mathbb R\times \mathbb R,\mathbb R,\;g\in C(J,\mathbb R\times \mathbb R,\mathbb R)\) and \(k_0\in \mathbb R\) are given, of differential-algebraic equations is suggested. The method is based on the idea of quasilinearisation known as a useful instrument for constructing approximate solutions of ordinary differential equations. In the present case, this approach leads to a sequence of approximate solutions which converges – under some natural assumptions – to the unique solution of the original problem. Moreover, the convergence is shown to be monotone and quadratic and hence, the method may be very efficient when solving practical problems.

34A34 Nonlinear ordinary differential equations and systems, general theory
34A45 Theoretical approximation of solutions to ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
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