ABSTRACT
The singular and non-linear differential equations can be solved by analytical numerical approximate methods. The forms of a differential equation may indicate what type of solutions are possible for example periodic solutions and guide the construction of methods of solution such as power series or parametric integrals or continued fractions. The proof of the theorems of existence, unicity, robustness, and uniformity is based on replacing the differential equation by an equivalent integral equation which is solved by successive approximations. A sufficient condition for the convergence to a fixed point is that the form of the differential equation is specified by a Lipschitz function. It is sufficient to prove the existence, unicity, robustness, and uniformity for a system of simultaneous first-order differential equations because a differential equation of any order or a simultaneous system can always be reduced to a set of first-order simultaneous equations.
