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[MUG] Re: Airy
| [MUG] Re: Airy |
|
Author: Maple Group
Posted: 15/12/2000 19:50:47 GMT
|
>> From: Maple Group
>>> From: Chuck Baker
| Is it possible to develop a power series solution, in powers of x, for
| Airy's equations using Maple?
| y'' - xy = 0
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
Date: Mon, 11 Dec 2000 11:03:17 -0800 (PST)
From: Robert Israel
To:
Subject: Airy's Equation
There were lots of replies (including mine) that used the "series" option
for dsolve, but Chuck later confirmed to me that he was interested in
a symbolic expression for the coefficients. In Maple 6, this can be
obtained using the new "formal_series" option:
> dsolve((D@@2)(y)(x)-x*y(x)=0,y(x),'formal_series',coeffs=mhypergeom);
4
_C[1] x + 1/12 _C[1] x
/infinity \
| ----- n (3 n + 1) |
| \ (1/3) x |
2/9 _C[1] Pi sqrt(3) | ) ------------------------------|
| / n |
| ----- 3 GAMMA(n + 4/3) GAMMA(n + 1)|
\ n = 2 /
+ --------------------------------------------------------------,
GAMMA(2/3)
3
_C[1] + 1/6 _C[1] x
/infinity \
| ----- n (3 n) |
| \ (1/3) x |
+ _C[1] GAMMA(2/3) | ) ------------------------------|{}
| / n |
| ----- GAMMA(n + 1) 3 GAMMA(n + 2/3)|
\ n = 2 /
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
Date: Mon, 11 Dec 2000 22:58:00 +0100
From: boucherf
To:
Subject: Airy's Equation
two ideas (with mapleV4) :
> > with(powseries):
> > edo:=diff(y(x),x,x)-x*y(x);
>
> / 2 \
> |d |
> edo := |--- y(x)| - x y(x)
> | 2 |
> \dx /
>
> > sol:=powsolve(edo,y(0)=1,D(y)(0)=0{});
>
> sol := proc(powparm) ... end
>
> > sol(_k);
>
> a(_k - 3)
> -----------
> _k (_k - 1)
>
> > ak:=rsolve(a(k)=a(k-1)/3/k/(3*k-1),a(0)=1{},a(k));
>
> (-k)
> 9 GAMMA(2/3)
> ak := ---------------------------
> GAMMA(k + 1) GAMMA(k + 2/3)
>
> > sum(ak*x^(3 *k),k=0..infinity);
>
> 3
> hypergeom([], [2/3], 1/9 x )
>
>
> > restart:
> > with(share):
> See ?share and ?share,contents for information about the share library
> > readshare(gfun,analysis):
> > with(gfun):
> > edo:=diff(y(x),x,x)-x*y(x);
>
> / 2 \
> |d |
> edo := |--- y(x)| - x y(x)
> | 2 |
> \dx /
>
> > rec:=diffeqtorec(edo,y(0)=1,D(y)(0)=0{},y(x),u(m));
>
> rec :=
>
> 2
> -u(m) + (m + 5 m + 6) u(m + 3), u(2) = 0, u(0) = 1, u(1) = 0{}
> rsolve don't work on this but :
> > rec2:=op(1,rec);
>
> 2
> rec2 := -u(m) + (m + 5 m + 6) u(m + 3)
>
> > rec2:=subs([u(m)=v(n),u(m+3)=v(n+1),m=3*n],rec2);
>
> 2
> rec2 := -v(n) + (9 n + 15 n + 6) v(n + 1)
>
> > vn:=rsolve(rec2=0,v(0)=1{},v(n));
>
> (-n)
> 9 GAMMA(2/3)
> vn := ---------------------------
> GAMMA(n + 1) GAMMA(n + 2/3)
>
> > sol:=sum(vn*x^(3*n),n=0..infinity);
>
> 3
> sol := hypergeom([], [2/3], 1/9 x )
>
bye
Francois Boucher
|
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