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Gold Medal Software 2
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Gold Medal Software Volume 2 (Gold Medal) (1994).iso
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win31
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macsyma.arj
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MACSDEMO.EXE
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ODE1.OUT
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1993-09-14
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(c1) /*
USE PACKAGE `ODE' TO SOLVE BESSEL AND LEGENDRE EQUATIONS.
*/
(kill(x,y), tr_warn_bad_function_calls:false, depends(y,x))$
(c2) /* 1. A Bessel Equation */
eq1: x^2*diff(y,x,2)+x*diff(y,x)+(x^2-n^2)*y=0;
|$label(0,15,Times New Roman,$(d2$))$sup(x,2)$in( )$q($sup(d,2)y,d$sup(x,2))$hinge()$in( + )x$in( )$q(dy,dx)$hinge()$in( + )$paren($sup(x,2)$in( - )$sup(n,2),$(,$))$in( )y$hinge()$in( = )0
(c3) /* Let's assume we don't know it's a Bessel, so we will try a SERIES
solution ... */
/* If it asks about N being an integer, say NO; otherwise only one
solution will be returned. */
ode(eq1,y,x,series);
C:\MACSD2B\ode\odeaux.fas being loaded.
C:\MACSD2B\ode\series.fas being loaded.
C:\MACSD2B\ode\sings.fas being loaded.
|$label(-1,15,Times New Roman,)Is $in() $in()n$in() $in() positive$, negative$, or zero?
P;
|$label(-1,15,Times New Roman,)Is $in()2$in( )n$in() $in()an integer$in()? $(Answer e.g. yes; no; unknown; or help; .$)
N;
|$label(0,15,Times New Roman,$(d3$))y$hinge()$in( = )%k2$in( )$greektext(G)$paren(n$in( + )1)$in( )$sup(x,n)$in( )$sum($q($sup($paren($in( - )1,$(,$)),%%n)$in( )$sup(x,2$in( )%%n),$sup(4,%%n)$in( )%%n!$in( )$greektext(G)$paren(n$in( + )%%n$in( + )1)),%%n = 0,$greektext(Ñ))$hinge()$in( + )$q(%k1$in( )$sum($q($sup($paren($in( - )1,$(,$)),%%n)$in( )$sup(x,2$in( )%%n),$sup(4,%%n)$in( )%%n!$in( )pochhammer$paren(1$in( - )n$ina($, )$hinge()%%n)),%%n = 0,$greektext(Ñ)),$sup(x,n))
(c4) /* But that is a cumbersome answer, maybe we can get a closed form ... */
closedform:true$
(c5) /* And for kicks, maybe ODE will tell us something interesting about the
equation ... */
odetutor:true$
(c6) ode(eq1,y,x,series);
|$label(-1,15,Times New Roman,)Trying $in() $in()series
|$label(-1,15,Times New Roman,)Recursion relation: $in() $in()$sub(%a,%%n)$in( = )$in( - )$q($sub(%a,%%n$in( - )2),$paren(r11$in( - )n$in( + )%%n,$(,$))$in( )$paren(r11$in( + )n$in( + )%%n,$(,$)))
|$label(-1,15,Times New Roman,)Is $in() $in()n$in() $in() positive$, negative$, or zero?
P;
|$label(-1,15,Times New Roman,)Is $in()2$in( )n$in() $in()an integer$in()?
N;
|$label(-1,15,Times New Roman,)DIAGNOSIS:$in() $in()TYPE:$in() $in()REGULAR SINGULAR$in() $in() $in() $in()singular$in() $in() ROOTS: R1=$in() $in()$in( - )n$in() $in() R2=$in() $in()n$in() $in()SINGULARITIES:$in() $in()$paren(0$ina($, )$hinge()$greektext(Ñ),[,])
|$label(0,15,Times New Roman,$(d6$))y$hinge()$in( = )%k2$in( )$sup(4,n$in(/)2)$in( )$greektext(G)$paren(n$in( + )1)$in( )$sub(bessel_j,n)$paren(x)$hinge()$in( + )$q(%k1$in( )$greektext(G)$paren(1$in( - )n)$in( )$sub(bessel_j,- n)$paren(x),$sup(4,n$in(/)2))
(c7) /* Normally all those GAMMA functions,etc are absorbed into the arbitrary
constant, but since we solved it the "hard way" we must accept them.
We can also solve the Legendre equation by series */
/* 2. The Legendre Equation */
eq2: (1-x^2)*diff(y,x,2)-2*x*diff(y,x)+n*(n+1)*y=0;
|$label(0,15,Times New Roman,$(d7$))$paren(1$in( - )$sup(x,2),$(,$))$in( )$q($sup(d,2)y,d$sup(x,2))$hinge()$in( - )2$in( )x$in( )$q(dy,dx)$hinge()$in( + )n$in( )$paren(n$in( + )1,$(,$))$in( )y$hinge()$in( = )0
(c8) ode(eq2,y,x,series);
|$label(-1,15,Times New Roman,)Trying $in() $in()series
|$label(-1,15,Times New Roman,)DIAGNOSIS: ORDINARY POINT
|$label(-1,15,Times New Roman,)Recursion relation: $in() $in()$sub(%a,%%n)$in( = )$q($sub(%a,%%n$in( - )2)$in( )$paren(r11$in( - )n$in( + )%%n$in( - )2,$(,$))$in( )$paren(r11$in( + )n$in( + )%%n$in( - )1,$(,$)),$paren(r11$in( + )%%n$in( - )1,$(,$))$in( )$paren(r11$in( + )%%n,$(,$)))
|$label(0,15,Times New Roman,$(d8$))y$hinge()$in( = )%k2$in( )$sub(hyper_f,2$ina($, )$hinge()1)$paren($paren($in( - )$q(n$in( - )1,2)$ina($, )$hinge()$q(n$in( + )2,2),[,])$ina($, )$hinge()$paren($q(3,2),[,])$ina($, )$hinge()$sup(x,2))$in( )x$hinge()$in( + )%k1$in( )$sub(hyper_f,2$ina($, )$hinge()1)$paren($paren($in( - )$q(n,2)$ina($, )$hinge()$q(n$in( + )1,2),[,])$ina($, )$hinge()$paren($q(1,2),[,])$ina($, )$hinge()$sup(x,2))
(c9) /* The reason we didn't get Legendre functions is that SERIES doesn't
transform the equation for solution at the singularity as does the
SOLVEHYPER routine. The user can do this himself by using the
TRANSFORM routine. */
transform(eq2,y,x,v,z,[y=v,z=x-1]);
C:\MACSD2B\ode\tran.fas being loaded.
|$label(0,15,Times New Roman,$(d9$))$q($sup(d,2)v,d$sup(z,2))$in( )$sup(z,2)$hinge()$in( + )2$in( )$q($sup(d,2)v,d$sup(z,2))$in( )z$hinge()$in( + )2$in( )$q(dv,dz)$in( )z$hinge()$in( + )2$in( )$q(dv,dz)$hinge()$in( - )$sup(n,2)$in( )v$hinge()$in( - )n$in( )v$hinge()$in( = )0
(c10) /* Let's just get the first solution ... */
ode(%,v,z,series), bothsol:false;
|$label(-1,15,Times New Roman,)Trying $in() $in()series
|$label(-1,15,Times New Roman,)Recursion relation: $in() $in()$sub(%a,%%n)$in( = )$in( - )$q($sub(%a,%%n$in( - )1)$in( )$paren(r11$in( - )n$in( + )%%n$in( - )1,$(,$))$in( )$paren(r11$in( + )n$in( + )%%n,$(,$)),2$in( )$sup($paren(r11$in( + )%%n,$(,$)),2))
|$label(-1,15,Times New Roman,)DIAGNOSIS:$in() $in()TYPE:$in() $in()REGULAR SINGULAR$in() $in() $in() $in()ser_equal$in() $in() ROOTS: R1=$in() $in()0$in() $in() R2=$in() $in()0$in() $in()SINGULARITIES:$in() $in()$paren($in( - )2$ina($, )$hinge()0$ina($, )$hinge()$greektext(Ñ),[,])
|$label(0,15,Times New Roman,$(d10$))v$hinge()$in( = )%k1$hinge()$in( )$sub(alegendre_p,n$ina($, )$hinge()0)$paren(z$in( + )1)
(c11) /* So there is the Legendre function, obtained by SERIES where z=x-1 */
subst([v=y,z=x-1],%);
|$label(0,15,Times New Roman,$(d11$))y$hinge()$in( = )%k1$hinge()$in( )$sub(alegendre_p,n$ina($, )$hinge()0)$paren(x)
(c12) /* Clean Up */
(remove(y,dependency), remvalue(eq1,eq2),
reset(tr_warn_bad_function_calls), odetutor:false)$