We extend the Lindstedt-Poincare small non-linearity based perturbation scheme to strongly non-linear systems. The extended technique starts from a physically non-existent neighboring linear system. This system is created by adding an optimal linear spring term to the system side of each linear differential equation, starting at the zeroth level, and balancing that term by adding an equal one to the forcing side one level down. Unlike the Ritz method, which also works for strongly non-linear systems, results can be obtained beyond the first level without resorting to numerical techniques. Thus, the extended method keeps the intuition-guiding straightforwardness of the perturbation method, while giving accurate, beyond first level, hand-derivable numerical results. Examples are presented for free vibrations with cubic and antisymmetric quadratic non-linearities and for harmonic solutions of the undamped Duffing equation. The forms of the solutions allow the regions of validity of the expansions to be estimated. An internal method of estimating accuracy of the perturbation solutions is developed and checked with the free vibration examples, the exact solutions of which are available. This internal method is then used to estimate the accuracy of the new solutions to the Duffing equation. These new Duffing equation solutions give accurate results in all regions where stable harmonic solutions are known to exist. In particular, they give accurate results at and near the vertical tangent to the resonance curve, a region where Hayashi's amplitude expansion based perturbation solution necessarily breaks down.