MapleDEAiry.mws

Airy's Equation

One Version

Truncated series solutions are produced here and graphed.  The form of Airy's equation being looked at here is

y'' + xy = 0,  y(0) = 0,  y'(0) = 1.

>    ode:=diff(y(x),x,x)+x*y(x)=0;

ode := diff(y(x),`$`(x,2))+x*y(x) = 0

>    Order:=12;

Order := 12

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+O(x^12),x,12)

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+O(x^12),x,12)

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10 

Warning, the name changecoords has been redefined

>    with(plots):SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>    Order:=16;

Order := 16

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13+O(x^16),x,16)

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13+O(x^16),x,16)

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13

>    SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>    Order:=20;

Order := 20

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19+O(x^20),x,20)

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19+O(x^20),x,20)

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19

>    SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>    Order:=24;

Order := 24

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+O(x^24),x,24)

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+O(x^24),x,24)

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22

>    SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>    Order:=32;

Order := 32

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532...
y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532...

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*...
series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*...

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*x...
poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*x...

>    SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>    Order:=40;

Order := 40

>    dsolve({ode,y(0)=0,D(y)(0)=1},y(x),type=series);

y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532...
y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532...
y(x) = series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532...

>    rhs(%);

series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*...
series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*...
series(1*x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*...

>    poly:=convert(%,polynom);

poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*x...
poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*x...
poly := x-1/12*x^4+1/504*x^7-1/45360*x^10+1/7076160*x^13-1/1698278400*x^16+1/580811212800*x^19-1/268334780313600*x^22+1/161000868188160000*x^25-1/121716656350248960000*x^28+1/113196490405731532800000*x...

>    SeriesSoln:=plot(poly,x=0..6,y=-6..6,color=blue):

>    display(SeriesSoln);

[Maple Plot]

>