1recursiv.mac is from the book "Computer Algebra in Applied
2Mathematics: An introduction to MACSYMA", by Richard H Rand, Pitman
3(1984).
4
5Mathieu's equation is x''+(delta+e*cos(t))*x=0
6
7For given values of the parameters delta and e, either all the
8solutions are bounded (the equation is stable) or there exist
9unbounded solutions (the equation is unstable).  The regions of
10stability are separated from thos of instability by "transition
11curves".
12
13This program computes the transition curves of Mathieu's equation
14using a method due to Levy and Keller (1963) which uses Fourier series
15to solve the perturbation equations.  An improved version of this
16routine is given in newimprv.mac.
17
18The run below, using maxima-5.9.0cvs, reproduces the result on pages
19115-116 of the book.
20
21(C1) load("./recursiv.mac");
22(D1)                            ./recursiv.mac
23(C2) tc();
24ENTER TRANSITION CURVE NUMBER N
250;
26ENTER DEGREE OF TRUNCATION
276;
28             6      4    2
29         29 e    7 e    e
30delta= - ----- + ---- - --
31          144     32    2
32
33(D2)                                 FALSE
34(C3) tc();
35ENTER TRANSITION CURVE NUMBER N
361;
37ENTER DEGREE OF TRUNCATION
386;
39           6       5    4     3    2
40       49 e    11 e    e     e    e    e   1
41delta= ----- - ----- - --- + -- - -- - - + -
42       36864   4608    384   32   8    2   4
43
44           6       5    4     3    2
45       49 e    11 e    e     e    e    e   1
46delta= ----- + ----- - --- - -- - -- + - + -
47       36864   4608    384   32   8    2   4
48
49(D3)
50(C4) tc();
51ENTER TRANSITION CURVE NUMBER N
522;
53ENTER DEGREE OF TRUNCATION
544;
55              4      2
56         763 e    5 e
57delta= - ------ + ---- + 1
58          3456     12
59
60          4    2
61       5 e    e
62delta= ---- - -- + 1
63       3456   12
64
65Reference:
66
67Levy, D.M. and Keller, J.B. "Instability Intervals of Hill's
68Equation", Comm. Pure Appl. Math. 16:469-476 (1963)
69
70Local Variables: ***
71mode: Text ***
72End: ***