The ''phase function'' method to solve second-order asymptotically polynomial differential equations

The Liouville-Green (WKB) asymptotic theory is used along with the Boruvka's transformation theory, to obtain asymptotic approximations of ''phase functions'' for second-order linear differential equations, whose coefficients are asymptotically polynomial. An efficient numerical method to compute zeros of solutions or even the solutions themselves in such highly oscillatory problems is then derived. Numerical examples, where symbolic manipulations are also used, are provided to illustrate the performance of the method.