An Elementary Approach to a Model Problem of Lagerstrom

The equation studied is u00+ n 1 r u 0+ "u u0 = 0; with boundary conditions u (1) = 0; u (1) = 1. This model equation has been studied by many authors since it was introduced in the 1950s by P. A. Lagerstrom. We use an elementary approach to show that there is an in nite series solution which is uniformly convergent on 1 r < 1: The rst few terms are easily derived, from which one quickly deduces the inner and outer asymptotic expansions. We also give a very short and elementary existence and global uniqueness proof which covers all " > 0 and n 1.