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[MUG] Re: Airy's Equation
| [MUG] Re: Airy's Equation |
|
Author: Maple Group
Posted: 11/12/2000 14:47:16 GMT
|
>> From: Maple Group "maple_gr"
>>> From: Chuck Baker "geogra4"
| Is it possible to develop a power series solution, in powers of x, for
| Airy's equations using Maple?
| y'' - xy = 0
|
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From: Chris Eilbeck "J.C.Eilbeck"
Date: Fri, 8 Dec 2000 15:34:48 GMT
To: "maple-list"
Subject: Airy's Equation
Use the expansions given in Abramowitz and Stegun, Handbook of
Mathematical Functions, Section 10.4.
----------------------------------------------------------------------
Chris Eilbeck email: "chris"
Department of Mathematics, Fax: +44 (0)131 451 3249
Heriot-Watt University, Phone: +44 (0)131 451 3220
Edinburgh EH14 4AS, Scotland. WWW: http://www.ma.hw.ac.uk/~chris/
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Date: Fri, 08 Dec 2000 10:42:58 -0500
From: Bill Bauldry "BauldryWC"
Subject: Airy's Equation
To: "maple-list"
Chuck,
A naive approach will work here to get one solution:
> de := (D@@2)(y)(x) - x*y(x) =0;
(2)
de := (D )(y)(x) - x y(x) = 0
> Order := 10: # if you want more than 5 terms
> dsolve(de, y(x), series);
3 4
y(x) = y(0) + D(y)(0) x + 1/6 y(0) x + 1/12 D(y)(0) x +
6 7 9 10
1/180 y(0) x + 1/504 D(y)(0) x + 1/12960 y(0) x + O(x )
But it may be more fun to work with the predefined functions:
> Order := 6:
> taylor(AiryAi(x), x);
> taylor(AiryBi(x), x);
(1/3) (1/6) (1/3)
3 3 GAMMA(2/3) 3 3
1/3 ---------- - 1/2 ----------------- x + 1/18 ---------- x -
GAMMA(2/3) Pi GAMMA(2/3)
(1/6)
3 GAMMA(2/3) 4 6
1/24 ----------------- x + O(x )
Pi
(5/6) (2/3) (5/6)
3 3 GAMMA(2/3) 3 3
1/3 ---------- + 1/2 ----------------- x + 1/18 ---------- x +
GAMMA(2/3) Pi GAMMA(2/3)
(2/3)
3 GAMMA(2/3) 4 6
1/24 ----------------- x + O(x )
Pi
Have fun ...
Regards,
Bill
--
===================================
Bill Bauldry
Professor and Chair
Department of Mathematical Sciences
phone: (828) 262-2355
fax: (828) 265-8617
"mailto:BauldryWC"
http://math.appstate.edu/~wmcb/
===================================
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Date: Fri, 8 Dec 2000 08:49:08 -0800 (PST)
From: Robert Israel "israel"
To: "''" "maple-list"
Subject: Airy's Equation
Sure.
> Order:= 10:
dsolve((D@@2)(y)(x)-x*y(x)=0,y(x),series);
3 4
y(x) = y(0) + D(y)(0) x + 1/6 y(0) x + 1/12 D(y)(0) x +
6 7 9
1/180 y(0) x + 1/504 D(y)(0) x + 1/12960 y(0) x +
10
O(x )
Or did you want a symbolic expression for the coefficients?
Robert Israel "israel"
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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Date: Fri, 08 Dec 2000 16:58:01 +0000
From: "Dr Francis J. Wright" "F.J.Wright"
To: "geogra4"
Subject: Airy's Equation
>From the online help for AiryAi in maple6:
series(AiryAi(x),x,4);
1/3 1/6 1/3
3 3 GAMMA(2/3) 3 3 4
1/3 ---------- - 1/2 --------------- x + 1/18 ---------- x + O(x )
GAMMA(2/3) Pi GAMMA(2/3)
There are probably also harder ways to do it! Francis
--
Dr Francis J. Wright | mailto: "F.J.Wright"
School of Mathematical Sciences | tel: (020) 7882 5453 (direct)
Queen Mary, University of London | fax: (020) 8981 9587 (dept.)
Mile End Road, London E1 4NS, UK | http://centaur.maths.qmw.ac.uk/
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To: "maple-list"
Subject: Airy's Equation
Date: Fri, 08 Dec 2000 16:08:10 -0500
From: "Craig B. Watkins" "watko"
It's quite easy. One example I have is:
>Order:=20;
>f1:=dsolve({diff(y(x),x)=exp(x*y),y(0)=0},y(x),'type=series');
>f2:=subs(f1,y(x));
>f3:=convert(f2,polynom);
>plot(f3,x=-1..1);
>g1:=dsolve({diff(y(x),x$2)+x*y=0,y(0)=0,D(y)(0)=1},y(x),'type=series');
>g2:=subs(g1,y(x));
>g3:=convert(g2,polynom);
>plot(g3,x=-1..5,scaling=constrained);
This may be more than you wanted (and I use y'' + x*y = 0 for this
example), but it's what I like to show undergraduates.
An explanation of what this is used for (somewhat MIT-dependent in
terms of subject matter is at
http://web.mit.edu/18.03-esg/www/cws00/maple/seriessol.html
Craig B. Watkins
Experimental Study Group (617) 253-2872
MIT Room 24-611 "watko"
Cambridge, MA 02139 http://web.mit.edu/watko/www/home.html
"... one has to admire a man who for 175 years can get away with
rhyming `eye' and `symmetry'".
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From: "Willard, Daniel Dr DUSA-OR" "Daniel.Willard"
To: "''" "maple-list"
Subject: Airy's Equation
Date: Fri, 8 Dec 2000 16:49:37 -0500
Try this (Release 5.1):
> restart;#?dsolve[series]
> with(ODEtools);
> ode:=diff(y(x),x$2)-x*y(x)=0;
> Order:=20;
> dsolve(ode,y(x),'type=series');# or dsolve({ode,
y(0)=A,D(y)(0)=B},y(x),'type=series');
Daniel Willard
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From: "Bastero de Eleizalde, Carlos" "cbastero"
To: "''" "maple-list"
Subject: Airy's Equation
Date: Mon, 11 Dec 2000 13:12:25 +0100
Chuck
You can write the following sentences
>restart;
>Order:=10; # to change the order of the series expansion (default = 6)
>dsolve({diff(y(x),x$2)- x*y(x)=0},{y(x)},type=series);
That's all
Carlos Bastero
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