Page 208 -

191

Q4.12 Mathieu’s equation for n = 2. See E4.4; consider Mathieu’s equation

with (which is the case n = 2). Introduce T = t and
show that and find the equation for the term in the asymptotic
expansion for x; from this, deduce that the exponent which describes the
amplitude modulation (cf. E4.4) is

What is the nature of the solution of this Mathieu equation, for various
(These results should be compared with those obtained in Q3.15.)
Q4.13 A particular Hill equation. See Q4.12; follow this same procedure for the Hill
equation

where is a fixed constant (independent of and for
Show that and that the leading term is periodic in T and
only if

where are to be identified. (These results should be compared with those
obtained in Q3.17.)
Q4.14 Mathieu’s equation away from critical See Q4.12; consider Mathieu’s equa-
tion, but now with away from the critical values: set
n = 0, 1, 2,..) and fixed independent of Introduce with
and and find the solution correct at (You should
note the singularities, for various that are evident here.)
Q4.15 WKB: higher-order terms. See E4.5; consider the equation

with introduce and write

Show that

and then expand both Determine
and in terms of and the constant (which is independent