A Nonlinear Second Order Equation

Non-Linear Model with a Series Solution

Truncated series solutions to the non-linear model are graphed in blue (SeriesSoln) and green.  The analytical solution is graphed in red (AnalSoln).

The analytical solution satisfying y(0) = 0 and y'(0) = 1 is y = sin(x).

We can observe that the series solution is accurate close to 0 (where the series is centered) and as more terms are added to the series solution the interval over which it is accurate becomes larger.

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

>	Order:=12;

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

>

rhs(%);

>	poly:=convert(%,polynom);

>	with(plots):AnalSoln:=plot(sin(x),x=0..2*Pi,thickness=2,color=red):

>	SeriesSoln:=plot(poly,x=0..2*Pi,thickness=2,color=blue):

>	display(SeriesSoln,AnalSoln);

>	f:=x->x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1/39916800*x^11:

>

f(2.0);

>

sin(2.0);

>	Order:=16;

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

>

rhs(%);

>	poly2:=convert(%,polynom);

>	SeriesSoln2:=plot(poly2,x=0..2*Pi,thickness=2,color=green):

>	display(SeriesSoln,SeriesSoln2,AnalSoln);

>	Order:=20;

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

>

rhs(%);

>	poly3:=convert(%,polynom);

>	with(plots):AnalSoln:=plot(sin(x),x=0..3*Pi,thickness=2,color=red):

>	SeriesSoln3:=plot(poly3,x=0..3*Pi,thickness=2,color=blue):

>	display(SeriesSoln3,AnalSoln);

>	Order:=24;

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

>

rhs(%);

>	poly4:=convert(%,polynom);

>	with(plots):AnalSoln:=plot(sin(x),x=0..3.4*Pi,thickness=2,color=red):

>	SeriesSoln4:=plot(poly4,x=0..3.4*Pi,thickness=2,color=blue):

>	display(SeriesSoln4,AnalSoln);

>	Order:=28;

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

>

rhs(%);

>	poly5:=convert(%,polynom);

>	with(plots):AnalSoln:=plot(sin(x),x=0..3.6*Pi,thickness=2,color=red):

>	SeriesSoln5:=plot(poly5,x=0..3.6*Pi,thickness=2,color=blue):

>	display(SeriesSoln5,AnalSoln);

>	g:=x->x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1/39916800*x^11+1/6227020800*x^13-1/1307674368000*x^15+1/355687428096000*x^17-1/121645100408832000*x^19+1/51090942171709440000*x^21-1/25852016738884976640000*x^23+1/15511210043330985984000000*x^25-1/10888869450418352160768000000*x^27:

>

g(2.0);

>

sin(2.0);

>