A new perturbation technique for differential equations with small parameters

Ordinary linear differential equations containing small parameter $\epsilon$ are investigated in regard to the existence of solutions in power series of the parameter. A new perturbation technique is developed which yields solutions more convenient for computation than comparable solutions obtained by making the usual series expansion in the small parameter. The new method is applied to second and fourth order differential equations in normal form and it is shown that the method yields asymptotic solution for small $\epsilon$. Conditions needed for successful application of the method are discussed and a typical solution is obtained. Comparison of numerical results with an exact solution and with an ordinary perturbation solution indicates the usefulness of the technique.