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sign and and then find the leading terms in all the relevant
regions for:
(a) (b) (c)
Q2.24 Transition layers and turning points. Consider the equation

introduce and find a choice of which produces
an equation for in the form

and identify F. If F changes sign on the domain of the solution, then the point
where this occurs is called a turning point; find the equation that defines the
turning points in the case
Q2.25 Transition layer at a turning point. Consider the equation

find the position of the turning point and scale in the neighbourhood of the
transition layer. Write down the general solution, to leading order, valid in the
transition layer, as (This solution is best written in terms of Airy func-
tions. A uniformly valid solution is usually expressed using the WKB method;
see Chapter 4.)
Q2.26 Higher-order equations. For these problems, find the first terms only in asymptotic
expansions valid in each region of the solution, for
(a)
(b)
(c)
Q2.27 Vertical motion under gravity. Consider an object that is projected vertically up-
wards from the surface of a planetary body (or, rather, for example, from our
moon, because we will assume no atmosphere in this model). The height above
the surface is z(t), where t is time, and this function is a solution of

where R is the distance from the centre of mass of the body to the point
of projection, and g is the appropriate (constant) acceleration of gravity. (For
our moon, The initial conditions are
find the relevant solution in the form t = t(z) (and you
may assume that