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{SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 48 "Programming and Execu
ting Euler Method in MAPLE." }{TEXT 202 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 203 8 "restart;" }{MPLTEXT 1 204 0 "" }}}{EXCHG 
{PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 205 40 "Exa
mple 2 in Boyce's book (Section 2.7)." }{TEXT 202 0 "" }}{PARA 200 "" 
0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 35 "f
:=(t,y)-> 3+ exp(-t) - (1/2)*y(t);" }{MPLTEXT 1 204 0 "" }}}{EXCHG 
{PARA 200 "" 0 "" {TEXT 205 15 "Exact Solution." }{TEXT 202 0 "" }}
{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 36 "yexact:=t-> 6-2*exp(-t)-3*exp(-t/2);" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 35 " Initial Value and \+
uniform stepsize" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }
}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 28 "tv[1]:=0; yv[1]:=1; h:
= 0.1;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "
" }}{PARA 200 "" 0 "" {TEXT 202 60 "Loop to generate the approximation
s at every partition point" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 50 "for i to 50 \+
do fev:=f(tv[i],yv[i]); tv[i+1]:= i*h;" }{MPLTEXT 1 204 0 "" }}{PARA 
201 "> " 0 "" {MPLTEXT 1 203 23 "yv[i+1]:= yv[i] + fev*h" }{MPLTEXT 1 
204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "
" {TEXT 202 39 "Printing the results at selected points" }{TEXT 202 0 
"" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 20 "for i by 10 to 51 do" }{MPLTEXT 1 204 0 "" }}{PARA 
201 "> " 0 "" {MPLTEXT 1 203 18 "print(tv[i],yv[i])" }{MPLTEXT 1 204 
0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 7 "end do;" }{MPLTEXT 1 204 
0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 206 0 "" }}{PARA 202 "" 0 "" 
{TEXT 207 48 "To create a vector structure that can be plotted" }
{TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 204 35 "A:=[seq( [tv[i],yv[i]], i=1..51 )];" }
{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 72 "Linear I
nterpolation of the set of points obtained by using Euler Method" }
{TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 204 45 "plot(\{yexact(t),A\},t=0..5,color=[brown,
red]);" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 31 
"Relative error at t=5 for h=0.1" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 204 44 "Relerr:=evalf((yv[51]-yexact(5))/yexact(5
));" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }
}{PARA 200 "" 0 "" {TEXT 202 22 "Refining the partition" }{TEXT 202 0 
"" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 29 "tv[1]:=0; yv[1]:=1; h:= 0.05;" }{MPLTEXT 1 204 0 ""
 }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 60 "Loop to generate the approximations at every partition point" 
}{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 203 51 "for i to 100 do fev:=f(tv[i],yv[i]); \+
tv[i+1]:= i*h;" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 
203 23 "yv[i+1]:= yv[i] + fev*h" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> "
 0 "" {MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 39 "Printing \+
the results at selected points" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for i \+
by 10 to 101 do" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 
1 203 18 "print(tv[i],yv[i])" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 
"" {MPLTEXT 1 203 7 "end do;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 204 36 "B:=[seq( [tv[i],yv[i]], i=1..101 )];"
 }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 52 
"plot(\{yexact(t),A,B\},t=0..5,color=[brown,red,blue]);" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 32 "Relative error at t
=5 for h=0.05" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 204 45 "Relerr:=evalf((yv[101]-yexact(5))/yexact(5));" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "
" {TEXT 202 27 "More Refining the partition" }{TEXT 202 0 "" }}{PARA 
200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
203 29 "tv[1]:=0; yv[1]:=1; h:= 0.01;" }{MPLTEXT 1 204 0 "" }}}{EXCHG 
{PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 60 "Loo
p to generate the approximations at every partition point" }{TEXT 202 
0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 51 "for i to 500 do fev:=f(tv[i],yv[i]); tv[i+1]:= i*h;
" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 23 "yv[i+1
]:= yv[i] + fev*h" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" 
{MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 ""
 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 39 "Printing the re
sults at selected points" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for i by 50 \+
to 501 do" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 
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{MPLTEXT 1 203 7 "end do;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 ">
 " 0 "" {MPLTEXT 1 204 36 "C:=[seq( [tv[i],yv[i]], i=1..501 )]:" }
{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 60 "p
lot([yexact(t),A,B,C],t=0..5,color=[brown,red,blue,green]);" }
{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 32 "Relative
 error at t=5 for h=0.01" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 "> " 0 "
" {MPLTEXT 1 204 45 "Relerr:=evalf((yv[501]-yexact(5))/yexact(5));" }
{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 206 0 "" }}}
{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 205 
40 "Example 3 in Boyce's book (Section 2.7)." }{TEXT 202 0 "" }}{PARA 
200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 
203 22 "f:=(t,y)-> 4-t+2*y(t);" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
202 "" 0 "" {TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 207 14 "Exact Sol
ution" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 
36 "yexact:=t->-7/4+1/2*t+11/4*exp(2*t);" }{MPLTEXT 1 204 0 "" }}}
{EXCHG {PARA 200 "" 0 "" {TEXT 202 1 " " }{TEXT 202 0 "" }}{PARA 200 "
" 0 "" {TEXT 202 34 "Initial Value and uniform stepsize" }{TEXT 202 0 
"" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 28 "tv[1]:=0; yv[1]:=1; h:= 0.1;" }{MPLTEXT 1 204 0 "" 
}}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 60 "Loop to generate the approximations at every partition point" 
}{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 203 50 "for i to 50 do fev:=f(tv[i],yv[i]); t
v[i+1]:= i*h;" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 
203 23 "yv[i+1]:= yv[i] + fev*h" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> "
 0 "" {MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 39 "Printing \+
the results at selected points" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 20 "for i \+
by 10 to 51 do" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 
203 18 "print(tv[i],yv[i])" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 ""
 {MPLTEXT 1 203 7 "end do;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "
" 0 "" {TEXT 207 48 "To create a vector structure that can be plotted"
 }{TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 204 35 "A:=[seq( [tv[i],yv[i]], i=1..51 )];" 
}{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 72 "Linear \+
Interpolation of the set of points obtained by using Euler Method" }
{TEXT 206 0 "" }}{PARA 202 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 204 47 "plot([yexact(t),A], t=0..5, color=[brown,
red]);" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 31 
"Relative error at t=5 for h=0.1" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 
"> " 0 "" {MPLTEXT 1 204 44 "Relerr:=evalf((yv[51]-yexact(5))/yexact(5
));" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }
}{PARA 200 "" 0 "" {TEXT 202 22 "Refining the partition" }{TEXT 202 0 
"" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 29 "tv[1]:=0; yv[1]:=1; h:= 0.05;" }{MPLTEXT 1 204 0 ""
 }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 60 "Loop to generate the approximations at every partition point" 
}{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 203 51 "for i to 100 do fev:=f(tv[i],yv[i]); \+
tv[i+1]:= i*h;" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 
203 23 "yv[i+1]:= yv[i] + fev*h" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> "
 0 "" {MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 39 "Printing \+
the results at selected points" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" 
{TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 21 "for i \+
by 10 to 101 do" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 
1 203 18 "print(tv[i],yv[i])" }{MPLTEXT 1 204 0 "" }}{PARA 201 "> " 0 
"" {MPLTEXT 1 203 7 "end do;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 
201 "> " 0 "" {MPLTEXT 1 204 36 "B:=[seq( [tv[i],yv[i]], i=1..101 )];"
 }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 52 
"plot([yexact(t),A,B],t=0..5,color=[brown,red,blue]);" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 207 32 "Relative error at t
=5 for h=0.05" }{TEXT 206 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 
1 204 45 "Relerr:=evalf((yv[101]-yexact(5))/yexact(5));" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "
" {TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 27 "More Refining the p
artition" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 29 "tv[1]:=0; yv[1]:=1; h:= \+
0.01;" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 ""
 }}{PARA 200 "" 0 "" {TEXT 202 60 "Loop to generate the approximations
 at every partition point" }{TEXT 202 0 "" }}{PARA 200 "" 0 "" {TEXT 
202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 203 51 "for i to 500
 do fev:=f(tv[i],yv[i]); tv[i+1]:= i*h;" }{MPLTEXT 1 204 0 "" }}{PARA 
201 "> " 0 "" {MPLTEXT 1 203 23 "yv[i+1]:= yv[i] + fev*h" }{MPLTEXT 1 
204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 7 "end do:" }{MPLTEXT 1 
204 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 202 0 "" }}{PARA 200 "" 0 "
" {TEXT 202 39 "Printing the results at selected points" }{TEXT 202 0 
"" }}{PARA 200 "" 0 "" {TEXT 202 0 "" }}}{EXCHG {PARA 201 "> " 0 "" 
{MPLTEXT 1 203 21 "for i by 50 to 501 do" }{MPLTEXT 1 204 0 "" }}
{PARA 201 "> " 0 "" {MPLTEXT 1 203 18 "print(tv[i],yv[i])" }{MPLTEXT 
1 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 203 7 "end do;" }{MPLTEXT 
1 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 36 "C:=[seq( [
tv[i],yv[i]], i=1..501 )]:" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 201 "
> " 0 "" {MPLTEXT 1 204 60 "plot([yexact(t),A,B,C],t=0..5,color=[brown
,red,blue,green]);" }{MPLTEXT 1 204 0 "" }}}{EXCHG {PARA 202 "" 0 "" 
{TEXT 207 32 "Relative error at t=5 for h=0.01" }{TEXT 206 0 "" }}}
{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 204 45 "Relerr:=evalf((yv[501]-y
exact(5))/yexact(5));" }{MPLTEXT 1 204 0 "" }}}{PARA 203 "" 0 "" 
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