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where and are constants to be determined; also find the recurrence relation
for the coefficients (This is to be compared with the familiar Frobenius
method for second order ODEs.) Write down the most general solution avail-
able (by using both values of and retain terms as far as Show that this
is consistent with the solution

although it is not possible, here, to find values for C and
Q1.19 Expansion of a function with a parameter. For these functions (all with the domain
expand as for each of x = O(l), and
and find the first two terms in each asymptotic expansion. Show that your
expansions satisfy the matching principle (and you may wish to note wherever
any breakdowns are evident in your two-term expansions). Remember that the
matching principle applies only to adjacent regions.

(a) (b) (c)

(d) (e)

for this one, also include (f)

Q1.20 Expansion with exponentially small terms. For these two functions (both with the
domain expand as for x = O(1) and retain terms O(1),
and for expand and now retain the first two terms
only. Show that your expansions satisfy the matching principle.

(a) (b)

Q1.21 Matching with logarithms. The domain of these functions is given as x > 0 with
in each case, find the first two terms in each of the asymptotic expan-
sions valid for x = O(1) and for as Show that, with the
interpretation your expansions satisfy the matching principle.

(a) (this example was introduced by Eckhaus);
(b)
(c)
Q1.22 Composite expansions I. For these functions, given that find the
first two terms in asymptotic expansions valid for x = O(1) and for