MATH 134 -- Intensive Calculus for Science 2

An Euler's Method Example

April 5, 2006

We study the approximate solution of   y ' =

with initial condition   y (0) = 1 .   First, we load the plotting

packages.  Then we do 10 steps of Euler's method with

  to approximate the solution:

>	with(DEtools):

>	with(plots):

>	xl[0]:=0; yl[0]:=1;

>	for i to 10 do xl[i]:=xl[i-1]+.1; yl[i]:= eval(yl[i-1]+(xl[i-1]^2+yl[i-1]^2)*(.1)); end do;

>	SField:=DEplot(diff(y(x),x)= x^2 + y(x)^2,[y(x)],x=-.1..1,y=0..10):

>

>	Pts:=plot([seq([xl[i],yl[i]],i=0..10)],style=point,symbol=circle,color=blue):

>	Lines:=plot([seq([xl[i],yl[i]],i=0..10)],color=black):

>	display(SField,Pts,Lines);

We would get a better approximation with smaller .  For instance, with  .01:

>	xl[0]:=0: yl[0]:=1:

>	for i to 100 do xl[i]:=xl[i-1]+.01; yl[i]:= eval(yl[i-1]+(xl[i-1]^2+yl[i-1]^2)*(.01)); end do:

>	Pts:=plot([seq([xl[i],yl[i]],i=0..100)],style=point,symbol=circle,color=blue):

>	Lines:=plot([seq([xl[i],yl[i]],i=0..100)],color=black):

>	display(SField,Pts,Lines);

>