On a differential-algebraic problem. (Q1582528)

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.