MapleSeriesTHno6.mws

Series Solution Take-Home Number 6

With and Without Initial Conditions

Below are examples with type = series and unspecified type.

>    ode:=diff(y(x),x,x,x)-2*(x^2)*diff(y(x),x,x)+3*x*diff(y(x),x)-2*y(x)=0;

ode := diff(y(x),`$`(x,3))-2*x^2*diff(y(x),`$`(x,2))+3*x*diff(y(x),x)-2*y(x) = 0

>    Order:=20;

Order := 20

>    dsolve({ode},y(x),type=series);

y(x) = series(y(0)+D(y)(0)*x+1/2*`@@`(D,2)(y)(0)*x^2+1/3*y(0)*x^3+(-1/24*D(y)(0))*x^4+1/72*y(0)*x^6+(-1/360*D(y)(0))*x^7+11/9072*y(0)*x^9+(-13/51840*D(y)(0))*x^10+17/155520*y(0)*x^12+(-19/855360*D(y)(0...
y(x) = series(y(0)+D(y)(0)*x+1/2*`@@`(D,2)(y)(0)*x^2+1/3*y(0)*x^3+(-1/24*D(y)(0))*x^4+1/72*y(0)*x^6+(-1/360*D(y)(0))*x^7+11/9072*y(0)*x^9+(-13/51840*D(y)(0))*x^10+17/155520*y(0)*x^12+(-19/855360*D(y)(0...

>    dsolve({ode},y(x));

{y(x) = _C1*x^2+_C2*hypergeom([-2/3, -1/6],[1/3, 2/3],2/3*x^3)+_C3*x*hypergeom([-1/3, 1/6],[2/3, 4/3],2/3*x^3)}

>    dsolve({ode,y(0)=0,D(y)(0)=0,D(D(y))(0)=2},y(x),type=series);

y(x) = x^2

>    dsolve({ode,y(0)=0,D(y)(0)=0,D(D(y))(0)=2},y(x));

y(x) = x^2

Notice what a difference a change in one of the initial conditions makes.

>    dsolve({ode,y(0)=0,D(y)(0)=1,D(D(y))(0)=2},y(x),type=series);

y(x) = series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19+O(x^20),x,20)

>    rhs(%);

series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19+O(x^20),x,20)

>    poly1:=convert(%,polynom);

poly1 := x+x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19

>    with(plots):

>    SeriesSoln1:=plot(poly1,x=0..4,y=-4..16,color=red):

>    Soln:=plot(x^2,x=0..4,y=-4..16,color=blue):

>    display(Soln,SeriesSoln1);

[Maple Plot]

>    Order:=32;

Order := 32

>    dsolve({ode,y(0)=0,D(y)(0)=1,D(D(y))(0)=2},y(x),type=series);

y(x) = series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513...
y(x) = series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513...

>    rhs(%);

series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513600*x^3...
series(1*x+1*x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513600*x^3...

>    poly2:=convert(%,polynom);

poly2 := x+x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513600*x^31
poly2 := x+x^2-1/24*x^4-1/360*x^7-13/51840*x^10-19/855360*x^13-19/10450944*x^16-31/228427776*x^19-1147/124157214720*x^22-49321/85668478156800*x^25-49321/1494143395430400*x^28-1591/909024303513600*x^31

>    SeriesSoln2:=plot(poly2,x=0..4,y=-4..16,color=green):

>    display(Soln,SeriesSoln1,SeriesSoln2);

[Maple Plot]

>    dsolve({ode,y(0)=0,D(y)(0)=1,D(D(y))(0)=2},y(x));

y(x) = x^2+x*hypergeom([-1/3, 1/6],[2/3, 4/3],2/3*x^3)

>