Series Solution Example Number 4

With initial conditions

Truncated series solutions are graphed in blue (Order = 8), red (Order = 12), green (Order = 16), and black (Order = 32) .

>	ode:=diff(y(x),x,x)-4*x*diff(y(x),x)+x*y(x)=-sin(x)-4*x*cos(x)+x*sin(x);

>

Order:=8;

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

>

rhs(%);

>	poly1:=convert(%,polynom);

>	with(plots):

>	SeriesSoln1:=plot(poly1,x=0..7,color=blue):

>	display(SeriesSoln1);

>	Order:=12;

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

>

rhs(%);

>	poly2:=convert(%,polynom);

>	SeriesSoln2:=plot(poly2,x=0..7,color=red):

>	display(SeriesSoln1,SeriesSoln2);

>	Order:=16;

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

>

rhs(%);

>	poly3:=convert(%,polynom);

>	SeriesSoln3:=plot(poly3,x=0..7,color=green):

>	display(SeriesSoln1,SeriesSoln2,SeriesSoln3);

>	soln1:=plot(poly1,x=0..4,y=-100..2,color=blue):

>	soln2:=plot(poly2,x=0..4,y=-100..2,color=red):

>	soln3:=plot(poly3,x=0..4,y=-100..2,color=green):

>	display(soln1,soln2,soln3);

>	Order:=32;

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

>

rhs(%);

>	poly4:=convert(%,polynom);

>	with(plots):

>	SeriesSoln4:=plot(poly4,x=0..7,color=black):

>	display(SeriesSoln4);

>	soln4:=plot(poly4,x=0..4,y=-100..2,color=black):

>	display(soln1,soln2,soln3,soln4);

>	soln1:=plot(poly1,x=0..2,y=-1..2,color=blue):

>	soln2:=plot(poly2,x=0..2,y=-1..2,color=red):

>	soln3:=plot(poly3,x=0..2,y=-1..2,color=green):

>	display(soln1,soln2,soln3);

Changing an initial condition in finding soln5 (orange) produces what?

>	Order:=16;

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

>

rhs(%);

>	poly5:=convert(%,polynom);

>	soln1:=plot(poly1,x=0..3.14,y=-1..2,color=blue):

>	soln2:=plot(poly2,x=0..3.14,y=-1..2,color=red):

>	soln3:=plot(poly3,x=0..3.14,y=-1..2,color=green):

>	soln5:=plot(poly5,x=0..3.14,y=-1..2,color=orange):

>	display(soln1,soln2,soln3,soln5);

We can see below that maybe the computer does not yet know everything (and it took a long time on this one).

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

>

>