Page 160 -

143

(see equation (1.7)), but the use of the ‘strained’ coordinate (see (1.8))
immediately removes the non-uniformity that is otherwise present as Further,
the leading term above (~ sin t) is essentially correct, but ‘in the wrong place’; if t is
replaced by then sin becomes a uniformly valid first approximation as We
will show how this method develops for a particular type of equation (first examined by
Lighthill, 1949 & 1961; similar examples have been discussed by Carrier, 1953 & 1954).

E3.7 An ordinary differential equation with a strained coordinate
asymptotic structure
We consider the problem

(and possesses the property that it can be expanded uniformly for
as this is essentially the problem discussed by Lighthill (1949). The ideas are
satisfactorily presented in a special case (which leads to a more transparent calculation);
we choose to examine the problem with

where and are constants independent of Thus (3.84) and (3.85) become

We will start by seeking the conventional type of asymptotic solution, as
in the form

and then satisfies

which produces the general solution

where A is an arbitrary constant. It is immediately clear that, with this solution
is not defined on x = 0, although we may use the boundary condition on x = 1,