On a Singular Perturbation Problem

Abstract

A uniformly valid asymptotic approximation is constructed for the solution to the initial value problem

$$\ddot v + \varepsilon t\dot v + v = 0, v(0) = 0, \dot v(0) = 1$$

, as ε → 0. From this, it is deduced that if εt → 0 then

$$v(t,\varepsilon ) \sim {e^{ - \varepsilon {t^2}/4}}\sin t$$

, and if εt → ∞ then

$$v(t,\varepsilon ) \sim \left( {\sin \frac{\pi }{{2\varepsilon }}} \right){e^{ - 1/2\varepsilon }}\frac{{\sqrt 2 }}{{{{(\varepsilon t)}^{1/\varepsilon }}}}$$

.