Naive singular perturbation theory

The paper demonstrates, via extremely simple examples, the shocks, spikes, and initial layers that arise in solving certain singularly perturbed initi al value problems for first-order ordinary differential equations. As examp les from stability theory, they are basic to many asymptotic techniques. Fi rst, we note that limiting solutions of linear homogeneous equations epsilo n (x) over dot = -a(t)m on t greater than or equal to 0 are specified by th e zeros of A(t) = integral (t)(o) a(s)ds, rather than by the turning points where a(t) becomes zero. Furthermore, solutions to the solvable equations epsilon (x) over dot = -a(t)x - b(t)x(k) for k = 1, 2 or 3 can feature cana rds, where the trivial limit continues to apply after it becomes repulsive. Limiting solutions of the separable equation epsilon (x) over dot = a(t)c( x) may likewise involve switchings between the zeros of c(x) located immedi ately above and below x(0), if they exist, at zeros of A(t). Finally limiti ng solutions of many other problems follow by using asymptotic expansions f or appropriate special functions. For example, solutions of epsilon (x) ove r dot = t(2)(t(2) - x(2)) be given in terms of the Bessel functions K-j(t(4 )/4 epsilon) and I-j (t(4)/4 epsilon) for j = 3/8 and -5/8.