[MUG] 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 | -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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/ ---------------------------------------------------------------------- -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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/ =================================== -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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 -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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/ -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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'". -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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 -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- 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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