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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 550 "In the previous worksheet, we examined the classical methods o
f Euler and Runge-Kutta.  In this worksheet, we look at the \"Workhors
e\" of the numerical solvers, Runge-Kutta-Fehlberg orders 4 and 5.  Th
e idea is that we get two solutions, Runge-Kutta order 4 and Runge-Kut
ta order 5.  We use these two algorithms to give us an estimate of the
 current error.  If this value is too large, we cut the stepsize.  If \+
the error is small, we increase the stepsize.  This way, we do not hav
e to give the algorithm a stepsize, it tries to compute it on its own.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"de:=diff(y(t),t)=t+y(t)-3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 148 "Here, we'll use dsolve, give the range \+
of possible t values, and output a generic procedure to be used later \+
in plotting or evaluating the solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Soln1:=dsolve(\{de,y(0
)=1\},numeric,method=rkf45,range=0..1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Here we will have dsolve \+
output the value of the solution at some predefined values of time:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 82 "Soln2:=dsolve(\{de,y(0)=1\},numeric,method=rkf45,output=array([0
,0.1,0.2,0.3,0.4]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 35 "We can plot the numerical solution:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wit
h(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "odeplot(Soln1,
0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0
" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }