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: ***