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154 3. Further applications

Q3.17 A particular Hill equation. (Hill’s equation is a generalisation of the Mathieu
equation.) Consider the equation

where is a constant independent of seek a solution

where the are independent of Impose the condition that and are
to be periodic; what condition(s) must and satisfy?
Q3.18 Matched expansion applied to E3.7. Consider

with (and constants independent of Find the equation for the
second term of the expansion, and from this deduce that

this can be done by first approximating the equation for before integrating
it—see the method leading to equation (3.96). Hence show that the asymptotic
expansion for breaks down where rescale x and y in
the neighbourhood of x = 0, and then find and solve the equation describing
the dominant term in this region (matching as necessary). What is the behaviour
of as based on your solution?
Q3.19 A strained-coordinate problem I. (This is a problem introduced by Carrier, 1953.)
Find an asymptotic solution of

as in the form

Find and use your solution to find the dominant behaviour of
where (You may assume that the asymptotic expansion of
the coordinate is uniformly valid on
Q3.20 A strained-coordinate problem II. See Q3.19; follow the same procedure for the
problem

as Use your results to show that