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where We will construct an asymptotic solution, for
as (The actual values, for the Earth and Moon, give
and a trajectory from surface to surface requires
approximately.) Write down a first integral of the equation.
(a) Find the first two terms in an asymptotic expansion valid for x = O(1),
by seeking (cf. Q2.27), and use the data as
and write

(Here, is the non-dimensional initial speed away from the Earth, is small
and the condition on k ensures that the spaceship reaches the
Moon, but not at such a high speed that it can escape to infinity.) Show that
this expansion breaks down as
(b) Seek a scaling of the governing equation in the neighbourhood of x = 1
by writing (which is consistent with the
solution obtained in (a), where the first term, provides the
dominant contribution at x = 1). Find the first term in an asymptotic ex-
pansion of match to your solution from (a) and hence determine
(Be warned that ln terms appear in this problem.)
Q2.29 Eigenvalues for a vibrating beam. The (linearised) problem of an elastic beam
clamped at each end is

for with where is the
eigenvalue (which arises from the time-dependence), and Young’smod-
ulus. Find the first term in an asymptotic expansion of the eigenvalues. (This
problem can be solved exactly, and then the exponents expanded for
this is an alternative that could be explored.)
Q2.30 Heat transfer in 1D. An equation which describes heat transfer in the presence
of a one-dimensional, steady flow (Hanks, 1971) is

with temperature conditions Find the first two
terms in an asymptotic expansion, valid for x = O(1) as and the
leading term valid in the boundary layer, matching as necessary.
Q2.31 Self-gravitating annulus. A particular model for the study of planetary rings is
represented by the equation