A new approach to the approximate solution of the non-linear differential equation x¨ + x + px3+ μ/(x,ẋ,t) = 0 is described, which is not subject to the difficulties of interpretation encountered previously. This solution method is compared with the K-B technique, and numerical results are presented to indicate general solution characteristics for − l<p≤100. The possible mechanism for some apparently anomalous behaviour in the relative error of the two methods is considered in detail. It appears likely that a fortuitous combination of phenomena may, in some instances, yield a K-B solution that exceeds the corresponding elliptic function approximation in accuracy when the system is quasi-linear. When the coefficient p is significantly greater than unity the K-B method fails, whereas the elliptic function approximation continues to improve in accuracy as p increases.