Solving nonlinear differential equations having periodic solutions.

A technique is proposed that permits solving to within 15 or more decimal places, in explicit form, initial value problems for nonlinear ordinary differential equations having oscillatory periodic solutions. The technique is elementary and relies on employing a numerical solver to generate a solution having a high degree of precision, estimating the period of the numerical solution, and then estimating the Fourier series coefficients of the numerical solution. Using the computer algebra system Mathematica, its routine NDSolve is employed with working precision set to 34, to obtain a (truncated) Fourier series solution which, when substituted back into the equation, yields a residual of less than 4 × 10[sup -16] (in absolute value) and agrees within 1.5 × 10[sup -15] with the numerically generated solution over the first period. The technique is quite suitable for discussion in a second semester beginning course with computer laboratory component and can be applied to any equation expected to have a periodic solution. It is considered that this procedure shows the 'correct use' of mathematical technology, blending strong computing and visualization capability with theoretical considerations that permits one to do new and deeper investigations. [ABSTRACT FROM AUTHOR]