Page 157 -

140 3. Further applications

E3.6 Mathieu’s equation for
We consider the equation

where we use the over-dot to denote the derivative with respect to t; special curves
in the separate stable (oscillatory) from unstable (exponentially growing)
solutions: on these curves there exist both oscillatory and linearly growing solutions.
We will seek these curves in the case
First, with we obtain

which has periodic solutions, with period or only if (n = 0, 1, 2 ...)
although n = 0 might be thought an unimportant exceptional case. The form of the
equation suggests that we should seek a solution in the form

and invoke the requirement that each be periodic; each is a constant inde-
pendent of We will explore the cases n = 0 and n = 1 ( and the case n = 2 is left
as an exercise).

(a) Case n = 0
Equation (3.79), with (3.80a,b), becomes

where, as is our convention, ‘= 0’ means zero to all orders in Thus we have the
sequence of equations

and so on. The only periodic solution for —and trivially so—is
which we will normalise to = 1. Then equation (3.81b) becomes

and the solution for which is periodic requires thus