The Newton method for solving degenerate systems of ordinary differential equations (Q1288071)

The author studies the Cauchy problem \[ f\bigl(\dot{x}(t),x(t),t\bigr)=0, \quad t \in T=[0,b], \qquad x(0)=0. \] Here, \(f(\dot{x}(t),x(t),t)\) is an \(m\)-dimensional vector-valued function with \({\det( f'_{\dot{x}}(\dot{x}(t),x(t),t))}\equiv{0}\) for all \(t \in T\). The conditions are stated under which (1) a solution to the Cauchy problem is unique; (2) the basic and the modified Newton processes give sequences converging to the exact solution. The convergence rate is calculated for the basic and the modified Newton processes.