Approximate solutions of non-linear non-autonomous second-order differential equations†

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Abstract
A method is presented for determining approximate solutions to a class of differential equations characterized by: where resonance phenomena (arising, for example, when ƒ(x, [xdot], t) = x cosω0 t) may be neglected. The approximation is developed from an asymptotic expansion in terms of the amplitude and phase of the solution. Three examples are considered in illustration of the application of the approximation technique, and using an integral error function, solution error is shown graphically for these examples in terms of equation parameters. An expression for the approximate solution is derived which makes it possible to determine solution accuracy for any function f(x, x˙, t) once the approximate amplitude envelope and phase relationships have been derived. Graphical solutions demonstrate the accuracy which can be maintained even up to relatively large values of the parameter µ.

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