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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.

##### MSC:
 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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