Series Solution Example

Some Polynomial Approximations

Truncated series solutions are graphed in red (Order 7) and then maroon (Order 10), blue (Order 12), green (Order 14), magenta (Order 24), brown (Order 36), orange (Order 60)

and black (Order 86).

>	ode:=diff(y(x),x,x)+x*diff(y(x),x)-2*y(x)=sin(x);

>

Order:=7;

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

>

rhs(%);

>	poly:=convert(%,polynom);

>	eval(poly,x=2.0);

>	with(plots):SeriesSoln:=plot(poly,x=0..7,y=-60..60,color=red,thickness=2):

>	display(SeriesSoln);

>	Order:=10;

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

>

rhs(%);

>	poly1:=convert(%,polynom);

>	eval(poly1,x=2.0);

>	with(plots):SeriesSoln1:=plot(poly1,x=0..7,y=-60..20,color=maroon,thickness=2):

>	display(SeriesSoln1);

>	Order:=12;

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

>

rhs(%);

>	poly2:=convert(%,polynom);

>	eval(poly2,x=2.0);

>	SeriesSoln2:=plot(poly2,x=0..7,y=-60..60,color=blue,thickness=2):

>	display(SeriesSoln1,SeriesSoln2);

>	Order:=14;

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

>

rhs(%);

>	poly3:=convert(%,polynom);

>	eval(poly3,x=2.0);

>	SeriesSoln3:=plot(poly3,x=0..7,y=-60..60,color=green,thickness=2):

>	display(SeriesSoln1,SeriesSoln2,SeriesSoln3);

>	Order:=24;

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

>

rhs(%);

>	poly4:=convert(%,polynom);

>	eval(poly4,x=2.0);

>	SeriesSoln4:=plot(poly4,x=0..7,y=-60..60,color=magenta,thickness=2):

>	display(SeriesSoln1,SeriesSoln2,SeriesSoln3,SeriesSoln4);

>	Order:=36;

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

>

rhs(%);

>	poly5:=convert(%,polynom);

>	eval(poly5,x=2.0);

>	SeriesSoln5:=plot(poly5,x=0..7,y=-60..60,color=brown,thickness=2,numpoints=200):

>	display(SeriesSoln1,SeriesSoln2,SeriesSoln3,SeriesSoln4,SeriesSoln5);

>	Order:=60;

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

>

rhs(%);

>	poly6:=convert(%,polynom);

>	eval(poly6,x=2.0);

>	SeriesSoln6:=plot(poly6,x=0..7,y=-60..60,color=orange,thickness=2,numpoints=200):

>	Order:=86;

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

>

rhs(%);

>	poly7:=convert(%,polynom);

>	eval(poly7,x=2.0);

>	SeriesSoln7:=plot(poly7,x=0..7,y=-60..60,color=black,thickness=2,numpoints=200):

>	display(SeriesSoln,SeriesSoln1,SeriesSoln2,SeriesSoln3,SeriesSoln4,SeriesSoln5,SeriesSoln6,SeriesSoln7);

>	display(SeriesSoln,SeriesSoln1,SeriesSoln2,SeriesSoln3,SeriesSoln4,SeriesSoln5,SeriesSoln6,SeriesSoln7);

>