A Singularly Perturbed Turning Point Problem

Abstract

Differential equation problems that depend on small parameter ∈ in such a way that the order of the equation is lower for ∈ = 0 than for ∈≠ 0 but remains positive are now commonly called “singular perturbation problems.” The condition that the order remain positive for ∈ = 0 is not a very distinguishing property of the differential equation as such. The equation

$$\begin{array}{*{20}{c}} { \in u\prime \prime - 2xu\prime + ku = 0,}&{k a constant,} \end{array}$$

(11.1-1)

k a constant, for instance, which will be examined closely in the next section, becomes

$${ \in ^2}v\prime \prime - \left( {{x^2} - \in (1 + k)} \right)v = 0$$

(11.1-2)

under the simple change of variables

$$u = {e^{{x^2}/{2_ \in }}}v.$$

(11.1-3)