Parametrically defined nonlinear differential equations, differential-algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

The study deals with nonlinear ordinary differential equations defined parametrically by two relations; these arise in fluid dynamics and are a special class of coupled differential-algebraic equations. We propose a few techniques for reducing such equations, first or second order, to systems of standard ordinary differential equations as well as techniques for the exact integration of these systems. Several examples show how to construct general solutions to some classes of nonlinear equations involving arbitrary functions. We specify a procedure for the numerical solution of the Cauchy problem for parametrically defined differential equations and related differential-algebraic equations. The proposed techniques are also effective for the numerical integration of problems for implicitly defined equations.