mathieu0.mac and mathieu.mac are from the book "Computer Algebra in
Applied Mathematics: An introduction to MACSYMA", by Richard H Rand,
Pitman (1984).

Mathieu's equation is x''+(delta+e*cos(t))*x=0 

For given values of the parameters delta and e, either all the
solutions are bounded (the equation is stable) or there exist
unbounded solutions (the equation is unstable).  The regions of
stability are separated from those of instability by "transition
curves".

This program computes the transition curves for n=0 (mathieu0.mac) and
n>0 (mathieu.mac) in Mathieu's equation using a perturbation method.

The run below, using maxima-5.9.0cvs, reproduces the result on pages
90-94 of the book.

(C1) load("./mathieu0.mac");
(D1)                            ./mathieu0.mac
(C2) mathieu0();
ENTER DEGREE OF TRUNCATION
8;
              8       6      4    2
       68687 e    29 e    7 e    e
delta= -------- - ----- + ---- - --
        294912     144     32    2


(C1) load("./mathieu.mac");
(D1)                             ./mathieu.mac
(C2) mathieu();
ENTER TRANSITION CURVE NUMBER N
1;
ENTER DEGREE OF TRUNCATION
6;
           6       5    4     3    2
       49 e    11 e    e     e    e    e   1
delta= ----- - ----- - --- + -- - -- - - + -
       36864   4608    384   32   8    2   4

           6       5    4     3    2
       49 e    11 e    e     e    e    e   1
delta= ----- + ----- - --- - -- - -- + - + -
       36864   4608    384   32   8    2   4

(D2)
(C3) mathieu();
ENTER TRANSITION CURVE NUMBER N
2;
ENTER DEGREE OF TRUNCATION
6;
                6        4      2
       1002401 e    763 e    5 e
delta= ---------- - ------ + ---- + 1
        4976640      3456     12

              6       4    2
         289 e     5 e    e
delta= - ------- + ---- - -- + 1
         4976640   3456   12


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