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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 55 "Using Maple for Runge-Kutta Order 4 (or Euler's Method)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Use this \+
as a template for the homework:  Page 438, numbers 4 and 6.  " }}
{PARA 0 "" 0 "" {TEXT -1 97 "Below is the solution to Problem 5.  We h
ave also included an approximation using Euler's Method." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "STEP ONE: \+
 Define the differential equation and try to get an exact solution (St
ored as Soln)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "de:=diff(y(t),t)=(y(t)^2+2
*t*y(t))/(3+t^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 256 85 "Ignore the next line when doing the HW problems- \+
there is no exact solution for those" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#Y:=dsolve(\{de,y(0)=1/2
\},y(t));  Soln:=rhs(Y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "STEP \+
TWO:  Get an approximation using Euler, step size 0.1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Soln1:=dsolve(\{de,y(0)=1/2\},numer
ic,method=classical[foreuler],stepsize=0.1);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 66 "STEP THREE:  Get an approximation using Runge-Kutta, st
ep size 0.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Soln2:=dsolv
e(\{de,y(0)=1/2\},numeric,method=classical[rk4],stepsize=0.1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "STEP FOUR:  Get an approximation u
sing Runge-Kutta, step size 0.05" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "Soln3:=dsolve(\{de,y(0)=1/2\},numeric,method=classica
l[rk4],stepsize=0.05);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 268 "If you've left the names of the four sol
utions the same, you do not have to change the following code.  It wil
l print out the values of the solutions at each desired time, and then
 plot the difference between the actual solution and the three numeric
al approximations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from \+
1 to 4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "  t:=j/10;  #This will
 give times 0.1,0.2,0.3,0.4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#  y
1:=Soln(t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  y2:=rhs(Soln1(t)[2
]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  y3:=rhs(Soln2(t)[2]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  y4:=rhs(Soln3(t)[2]);" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 99 "  printf(\"The differences between RK4, ste
psize 0.05, and others:  %g %g\\n\",abs(y4-y2),abs(y4-y3));" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13 6
 0" 64 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }