Page 123 -

106 2. Introductory applications

Q2.10 Eigenvalue problems. A standard problem in many branches of applied mathe-
matics and physics is to find the eigenvalues (and eigenfunctions) of appropriate
problems based on ordinary differential equations. In these examples, find the
first two terms in the asymptotic expansions of both the eigenvalues and
the eigenfunctions; for each use the asymptotic sequence
(a)
(b)
(c)
Q2.11 Breakdown of asymptotic solutions of differential equations. These ordinary differ-
ential equations define solutions on the domain with conditions
given on x = 1. In each case, find the first two terms in an asymptotic solution
valid for x = O(1) as which allows the application of the given
boundary condition(s). Show, in each case, that the resulting expansion is not
uniformly valid as find the breakdown, rescale and hence find the
first term in an asymptotic expansion valid near x = 0, matching as necessary.
Finally find, for each problem, the dominant asymptotic behaviour of
as
(a)

(b)
(c)
(d)

(e)
Q2.12 Another breakdown problem. See Q2.11; repeat for the problem

but show that, for a real solution to exist, the domain is where
and then find the dominant asymptotic behaviour of
as
Q2.13 Breakdown as Find the first two terms in an asymptotic solution,
valid for x = O(1) as of

with Now show that this expansion
is not uniformly valid as find the breakdown, rescale and find the first
two terms in an expansion valid for large x, matching as necessary. Show, also,
that this 2-term expansion breaks down for even larger x, but do not take the
analysis further.