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{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" 
{TEXT -1 80 "Warning, the protected names norm and trace have been red
efined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"A:=matrix(2,2,[2,-4,1,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG
K%'matrixG6#7$7$\"\"#!\"%7$\"\"\"!\"#Q(pprint06\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%\"\"!\"\"#<#K%'vectorG6#7$F%\"\"\"Q(pprint16\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Phi:= matrix(2,2,[1+2*t,-4*t
, t,1-2*t]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiGK%'matrixG6#7$7
$,&\"\"\"F+*&\"\"#F+%\"tGF+F+,$*&\"\"%F+F.F+!\"\"7$F.,&F+F+*&F-F+F.F+F
2Q(pprint26\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ft:=[36*t^
2,6*t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ftG7$,$*&\"#O\"\"\")%\"t
G\"\"#F)F),$*&\"\"'F)F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "Phi1:=inverse(Phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi1GK%'
matrixG6#7$7$,&\"\"\"F+*&\"\"#F+%\"tGF+!\"\",$*&\"\"%F+F.F+F+7$,$F.F/,
&F+F+*&F-F+F.F+F+Q(pprint36\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 110 "xtu:=map(x->simplify(x),evalm(\nPhi&*(map(x->int(x,t),evalm(Phi
1&*ft))+[c1,c2] )\n                         )  );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$xtuGK%'vectorG6#7$,,*&\"\"'\"\"\")%\"tG\"\"%F,F,*&\"
\")F,)F.\"\"$F,F,%#c1GF,*(\"\"#F,F.F,F4F,F,*(F/F,F.F,%#c2GF,!\"\",.*&F
3F,F-F,F,*&F.F,F4F,F,*&F6F,F2F,F9*&F3F,)F.F6F,F,F8F,*(F6F,F.F,F8F,F9Q(
pprint46\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "xt0:=subs(t=0
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%#c1G%#c2GQ(pprint56\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A
:=matrix( 4,4, [ 0,4,8,0, 0,0,0,8, 0,0,2,4, 0,0,0,2]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AGK%'matrixG6#7&7&\"\"!\"\"%\"\")F*7&F*F*F*F,7
&F*F*\"\"#F+7&F*F*F*F/Q(pprint66\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7
%\"\"#F$<#K%'vectorG6#7&\"\"%\"\"!\"\"\"F+Q(pprint76\"7%F+F$<#KF'6#7&F
,F+F+F+Q(pprint8F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "Phi:
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-1+exp(2*t)) , \n0,0, exp(2*t),4*t*exp(2*t),  0,0,0, exp(2*t)]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiGK%'matrixG6#7&7&\"\"\",$*&\"\"%
F*%\"tGF*F*,&F-!\"\"*&F-F*-%$expG6#,$*&\"\"#F*F.F*F*F*F*,$*(\"#;F*F.F*
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t96\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ft:=[0,6*t,0,exp(2
*t)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ftG7&\"\"!,$*&\"\"'\"\"\"%
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{MPLTEXT 1 0 14 "x0:=[4,2,2,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
x0G7&\"\"%\"\"#F'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
Phi1:=inverse(Phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Phi1GK%'matr
ixG6#7&7&\"\"\",$*&\"\"%F*%\"tGF*!\"\",$*(F-F*,&F*F/-%$expG6#,$*&\"\"#
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/,$*(F-F*F.F*F3F/F/7&F=F=F=F?Q)pprint106\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 116 "xtu:=map(x->simplify(x),evalm(\nPhi&*(map(x->int(x
,t),evalm(Phi1&*ft))+[c1,c2,c3,c4] )\n                         )  );" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$xtuGK%'vectorG6#7&,4*&\"\"%\"\"\"
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" {MPLTEXT 1 0 48 "xt0:=map(x->simplify(x),subs(t=0,evalm(xtu))  );" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xt0GK%'vectorG6#7&,&\"\"%!\"\"%#c1
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" 0 "" {MPLTEXT 1 0 35 "eqs:=\{seq(xt0[kk]=x0[kk],kk=1..4)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqsG<&/%#c4G\"\"\"/,&\"\"%!\"\"%#c1
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" {XPPMATH 20 "6#>%%solcG<&/%#c2G\"\"%/%#c4G\"\"\"/%#c3G\"\"#/%#c1G\"
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(xtu));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xtGK%'vectorG6#7&,**&\"
\"%\"\"\")%\"tG\"\"$F,F,*&F+F,-%$expG6#,$*&\"\"#F,F.F,F,F,F,*(\"\")F,F
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{MPLTEXT 1 0 58 "der:= xv -> evalf([1/(xv[1]*xv[2]), 1/(xv[1]^2+xv[2]^
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{MPLTEXT 1 0 96 "x:=[2,3]; der(x);\nh:=.25;\nfor t from h to 1 by h do
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{XPPMATH 20 "6#>%\"xG7$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
$$\"+nmmm;!#5$\"+#p2Bp(!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$
\"#D!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&oilerG7$$\"+=KHg@!\"*$
\"+jz`uIF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now solve the IVP u
sing the Improved Euler's method." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 122 "x:=[2,3];\nh:=.25;\nfor t from h to 1 by
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=xn;\nod:   ei:=x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7$\"\"#\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"#D!\"#" }}{PARA 11 "" 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "x:=[2,3]; der(x);\nh:=.001;\nfor t \+
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7$\"\"#\"\"$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$$\"+nmmm;!#5$\"+#p2Bp(!#6" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
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{MPLTEXT 1 0 126 "x:=[2,3];\nh:=.01;\nfor t from h to 1 by h do\n  un:
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ei1000:=x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7$\"\"#\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"#" }}{PARA 11 "" 1 "" 
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"" {XPPMATH 20 "6#7$$\"+m4Oe@!\"*$\"+(H2Q2$F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$$\"+XTNe@!\"*$\"+5Z!Q2$F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$$\"+G9Ne@!\"*$\"+_?!Q2$F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$$\"+tLNe@!\"*$\"+9W!Q2$F&" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 152 "st:= time(): x:=[2,3];\nh:=.0001;\nfor t from h to
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\n  x:=xn;\nod:   ei1k4:=x;  time()-st;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"xG7$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"
\"\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ei1k4G7$$\"+$Q`$e@!\"*$
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "st:=time(): x:=[2,3]; der(x);\nh:=
.0001;\nfor t from h to 1 by h do\n  xn:= x + h*der(x);\n  x:=xn;\nod:
  e1k4:=x; time()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7$\"\"#
\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+nmmm;!#5$\"+#p2Bp(!#6" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"%" }}{PARA 11 "" 1 "
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"" {XPPMATH 20 "6#$\"$T&!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "S
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0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "bball:= xv -> evalf
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6#%#xvG6\"6$%)operatorG%&arrowGF(-%&evalfG6#7&&9$6#\"\"$&F16#\"\"%,$*(
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35 "x:= [0,3,90*sqrt(2.), 90*sqrt(2.)];" }}{PARA 11 "" 1 "" {XPPMATH 
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{MPLTEXT 1 0 9 "bball(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"+1Az
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1 0 160 "x:= [0,3,90*sqrt(2.), 90*sqrt(2.)];\nh:=.01;\npts:=[ [0,3]]:
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:= [ op(pts), [x[1],x[2]] ]:\nod:  x;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"xG7&\"\"!\"\"$$\"+1Azs7!\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
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{MPLTEXT 1 0 11 "plot( pts);" }}{PARA 13 "" 1 "" {GLPLOT2D 606 142 
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$\"39+++g_D!)[Fjcl$!3)******Hw=(H?F*7$$\"3C+++I//&)[Fjcl$!3')*****z8>L
\"HF*7$$\"3M+++2z\")*)[Fjcl$!3?+++$*)=()z$F*7$$\"3s*****Hp(e%*[Fjcl$!3
Q+++5M\"fo%F*7$$\"3')*****>z\\$**[Fjcl$!37+++v!)*[d&F*7$$\"3K+++2U5/\\
Fjcl$!3s*****fFocY'F*-F]ez6&F_ez$\"#5!\"\"F`ezF`ez-%+AXESLABELSG6%Q!6
\"Fc^gl-%%FONTG6#%(DEFAULTG-%%VIEWG6$Fh^glFh^gl" 1 2 0 1 10 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "Now find the path when the initial angle \+
is 35 degrees." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 193 "x:= [0,3,180*cos(35*evalf(Pi)/180.), 180*sin(35*eval
f(Pi)/180.)];\nh:=.01;\npt35:=[ [0,3]]:\nfor t from h to 5.8 by h do\n
  xn:= x + h*bball(x);\n  x:=xn;\n  pt35:= [ op(pt35), [x[1],x[2]] ]:
\nod:  x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7&\"\"!\"\"$$\"+!ot
WZ\"!\"($\"+&ePC.\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"
!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&$\"+O+X._!\"($!+8s$R@%!\")$
\"+n[8'e&F)$!+8Oud&)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p
35:=plot(pt35,color=violet):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "
Plot both the 35 degree and 45 degree trajectories. \nNote that the ba
ll starting out at a 35 degree angle travels farther than the ball sta
rting at a 45 degree angle. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "display( p35,v2);" }}{PARA 13 "" 1 "" 
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