Surprises in a classic boundary-layer problem

We revisit a textbook example of singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows wealth phenomena that seem to have been overlooked previously, including pitchfork bifurcation in the number solutions as one varies small parameter, and transcendentally terms initial conditions can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students courses perturbation methods, applied dynamical systems, or numerical analysis.