Multivariable Expansions Using Time Scales Generated from Differential Equations

We deal with the equation (1 + ε t)y" + 2ε y' + y = 0 and show how a set of nonlinear time scales may be derived from the equation itself. Using these time scales multitime expansions of the solution of the equation satisfying the initial conditions y(0) = 0, y'(0) = 1 are found. Proof of the uniform asymptotic convergence of the expansions is given. The technique used is an extension of the method used by Reiss [1] and has been successfully applied to a broad class of second order linear differential equations with slowly varying coefficients.