A Geometric Approach to Boundary Layer Problems Exhibiting Resonance

We consider the boundary value problem for the singularly perturbed equation ε y'' + f(x,ε )y' + g(x,ε )y = 0 under the assumption that f changes sign and fₓ < 0. We show that this problem can be understood by imbedding the equation in a two parameter family ε y'' + f(x,ε ,δ )y' + g(x,ε ,δ )y = 0. Under a transversality condition on f and g, δ = δ (ε ) is computed so that the latter equation displays resonance. The same techniques are used to show that Matkowsky’s criteria provide a simple and effective way to compute the Taylor series of δ (ε ).