The Oscillating Pendulum

Linear and Non-Linear Models

Truncated series solutions to the non-linear model are graphed in blue (SeriesSoln).  The analytical solution to the linear model is graphed in red (AnalSoln).  Note the different initial condition in this example.

>	ode:=diff(y(x),x,x)+sin(y(x))=0;

>	Order:=12;

>	dsolve({ode,y(0)=Pi/3,D(y)(0)=0},y(x),type=series);

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot((Pi/3)*cos(x),x=0..Pi,color=red):

Warning, the name changecoords has been redefined

>	SeriesSoln:=plot(poly,x=0..Pi,y=-1.5..1.2,color=blue):

>	display(SeriesSoln,AnalSoln);

>	Order:=16;

>	dsolve({ode,y(0)=Pi/3,D(y)(0)=0},y(x),type=series);

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot((Pi/3)*cos(x),x=0..Pi,color=red):

>	SeriesSoln:=plot(poly,x=0..Pi,y=-1.5..1.2,color=blue):

>	display(SeriesSoln,AnalSoln);

>	Order:=20;

>	dsolve({ode,y(0)=Pi/3,D(y)(0)=0},y(x),type=series);

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot((Pi/3)*cos(x),x=0..9*Pi/8,color=red):

>	SeriesSoln:=plot(poly,x=0..9*Pi/8,y=-1.5..1.2,color=blue):

>	display(SeriesSoln,AnalSoln);

>	Order:=24;

>	dsolve({ode,y(0)=Pi/3,D(y)(0)=0},y(x),type=series);

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot((Pi/3)*cos(x),x=0..9*Pi/8,color=red):

>	SeriesSoln:=plot(poly,x=0..9*Pi/8,y=-1.5..1.2,color=blue):

>	display(SeriesSoln,AnalSoln);

>	Order:=32;

>	dsolve({ode,y(0)=Pi/3,D(y)(0)=0},y(x),type=series);

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot((Pi/3)*cos(x),x=0..9*Pi/8,color=red):

>	SeriesSoln:=plot(poly,x=0..9*Pi/8,y=-1.5..1.2,color=blue):

>	display(SeriesSoln,AnalSoln);

>