Resonance and asymptotic solutions of a singular perturbation problem

A singular perturbation problem for a second-order ODE witha pair of singular boundary points and a pair of interior second-orderturning points is studied. Four leading-order uniform approximationsare formally constructed, each is restricted to a region includingone critical point. The neighbouring approximations are formallymatched independently on an overlap domain, yielding an asymptoticapproximation to leading order of the general solution. Twogeneralized formulae for the singular and turning point eigenvaluesthat exhibit the resonance conditions are derived. The resonancecriteria due to the influence of every possible combinationof the critical points are investigated.