Question

What is the parametric equation for nonlinear initial value problrm?

Answer 1

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The study deals with nonlinear ordinary differential equations defined parametri-

cally by two relations; these arise in fluid dynamics and are a special class of cou-

pled 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 ef-

fective for the numerical integration of problems for implicitly defined equations.