{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 262 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Grey Emphasis" -1 269 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 273 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 274 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "The pet rock problem" }}{PARA 0 " " 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 " " 0 "" {TEXT -1 19 "Version: 18.8.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT 261 5 "Note:" }{TEXT -1 15 " The procedure " }{TEXT 0 9 "de solveRK" }{TEXT -1 29 " is needed in this worksheet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 " load " }{TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 269 7 "DEsol.m" }{TEXT -1 32 " is req uired by this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be r ead into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The pet roc k problem" }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 2 ": " } }{PARA 0 "" 0 "" {TEXT -1 61 "A man takes his pet rock for a walk. Ini tially the rock is at" }{XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\" \"F'" }{TEXT -1 18 " and the man is at" }{XPPEDIT 18 0 " ``(0,0)" "6#- %!G6$\"\"!F&" }{TEXT -1 241 " (units are metres). The man walks to the right at 1 m/sec and the rock follows along behind on his nonelastic \+ leash, with his nose always pointing toward his owner. Find the path f ollowed by the rock. Where will the rock be after 2 seconds?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "This problem c omes from a Maple worksheet by: Carl Eberhart, Dept. of Mathematics, \+ University of Kentucky: " }{URLLINK 17 "http://www.ms.uky.edu/~carl" 4 "http://www.ms.uky.edu/~carl" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Runge Kutta solu tion using " }{TEXT 0 9 "desolveRK" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "R(x,y)" "6#-%\"RG6$%\"xG%\"yG" }{TEXT -1 68 " be the position of th e rock at time t. The man will be located at " }{XPPEDIT 18 0 "M(t,0) " "6#-%\"MG6$%\"tG\"\"!" }{TEXT -1 9 " at time " }{TEXT 264 1 "t" } {TEXT -1 46 ". Since the leash is inelastic, we have that " } {XPPEDIT 18 0 " (t-x)^2 + y^2 = 2" "6#/,&*$,&%\"tG\"\"\"%\"xG!\"\"\"\" #F(*$%\"yGF+F(F+" }{TEXT -1 29 ", the square of the distance " } {XPPEDIT 18 0 "d = sqrt(2);" "6#/%\"dG-%%sqrtG6#\"\"#" }{TEXT -1 5 " f rom" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" }{TEXT -1 3 " to" } {XPPEDIT 18 0 " ``(-1,1)" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 2 ". " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 " 60-%'CURVESG6$7S7$$!\"\"\"\"!$\"25++!**********!#<7$$!3%>#=wg0YM**!#=$ \"3[D0,K+kM**F17$$!3Gg\"=^/Nu()*F1$\"3Q)\\%o*F1$ \"3O(3p!pDx)o*F17$$!3kA?+W2$\\i*F1$\"3%)e#**HM55j*F17$$!3=i7F*4_Kc*F1$ \"3)fr]B;]:d*F17$$!3'f\\em&RY*\\*F1$\"3;,5B**pV5&*F17$$!3yv!GMQ!)eV*F1 $\"3oI-6.@\"*\\%*F17$$!3E#)f%H'oZq$*F1$\"3+u@sm42)Q*F17$$!37va]1%pGJ*F 1$\"3Eet8ep&RL*F17$$!3W8C%[08%)*G=*F1$ \"3RP:'Q-?J@*F17$$!3z1?xNX9?\"*F1$\"3?o)*R@;Tb\"*F17$$!3Km_9.!eJ1*F1$ \"3$[9;b=tL5*F17$$!3YIm*R]&R&**)F1$\"3iHsX5c&>/*F17$$!3J)3'zy<*z$*)F1$ \"3QT@!=!\\K!**)F17$$!31SQ&)*f<7())F1$\"3d\\-:TGtI*)F17$$!3W`2/P667))F 1$\"3m,0\"z))3%y))F17$$!3+)Qk%z>EZ()F1$\"3M\\%fEhn9#))F17$$!3U*zhfC5bo )F1$\"3k3'fpP1xw)F17$$!3H\"4M-zy5i)F1$\"3tup:!3)47()F17$$!3]\"R_0g5>c) F1$\"3+*4q+nv9m)F17$$!37**ej9(*3)\\)F1$\"3oMJzY_N2')F17$$!3v$)=q]xzJ%) F1$\"3k0'o!=tn^&)F17$$!3#y8?7S!4u$)F1$\"3P=&za0iO])F17$$!3w]`$G)[w6$)F 1$\"3E6;(e$QG_%)F17$$!33/vyBlPZ#)F1$\"31Kz?5^t*R)F17$$!3a4YAw]Q%=)F1$ \"33l4Y)F17$$!3f>rxmi.IzF1$\"3c fPj7`')[\")F17$$!3e+U=Tf?ryF1$\"3e52xjf)Q5)F17$$!3#=&e=c)))o!yF1$\"3wA N<5[Fb!)F17$$!3!f/]:\\rju(F1$\"3r8vqx!z+,)F17$$!3%Gi>#3M6$o(F1$\"3/w>/ 7nSjzF17$$!3Q+#R#pqE@wF1$\"3pr'3t0Y$=zF17$$!3_!3ZRo>lb(F1$\"3SFN-,Hyry F17$$!3w*[,lDgT\\(F1$\"3o,/`1]`FyF17$$!3t*pm^t)QIuF1$\"3-Dw(>5(p*p(F17$$! 3]704-dUUsF1$\"3wrF1$\"3-d(y,D6K*p F1$\"3/#GNO5!>%\\(F1-%'COLOURG6&%$RGBG$\"#5F)$F*F*F_[l-F$6%7%7$FezF_[l 7$Fez$\"3Q+++0,>%\\(F17$$\"3++++++++]F1F_[l-Fjz6&F\\[lF*F*F*-%*LINESTY LEG6#\"\"#-F$6%7$7$F_[lF_[lFg[l-Fjz6&F\\[lF_[lF_[l$\"*++++\"!\")-%*THI CKNESSGF^\\l-F$6&7$Fd[lFg[l-%'SYMBOLG6#%'CIRCLEGFj[l-%&STYLEG6#%&POINT G-F$6&F]]l-F_]l6#%(DIAMONDGFj[lFb]l-F$6&F]]l-F_]l6#%&CROSSGFj[lFb]l-%% TEXTG6$7$$!#>!\"#$\"#^Ff^lQ\"d6\"-Fa^l6$7$$!#fFf^l$\"\")F)Q'R(x,y)Fj^l -Fa^l6$7$$\"#bFf^l$Fa_lFf^lQ'M(t,0)Fj^l-Fa^l6$7$$!\"&Ff^l$\"$Z\"Ff^lQ \"yFj^l-Fa^l6$7$$\"\"\"F*$\"\"(Ff^lQ\"xFj^l-%+AXESLABELSG6%Q!Fj^lF]al- %%FONTG6#%(DEFAULTG-%*AXESTICKSG6$\"\"$F_\\l-%%VIEWG6$;F(Fe`l;F_[l$\"# :F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" "Curve 11" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The \+ slope of RM is " }{XPPEDIT 18 0 "(0-y)/(t-x) = -y/(t-x);" "6#/*&,&\" \"!\"\"\"%\"yG!\"\"F',&%\"tGF'%\"xGF)F),$*&F(F',&F+F'F,F)F)F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 113 "Since the rock's nose alway s points towards his master and his velocity is tangent to his path of motion, we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " dy/dx = -y/(x-t)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"yGF&,&%\"xGF&%\"tG F(F(F(" }{TEXT -1 16 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 13 "The variable " }{TEXT 266 1 "t" }{TEXT -1 75 " can be eliminated from the right hand side of (i) by solving the equation " }{XPPEDIT 18 0 " (t-x)^2+y^2 = 2;" "6#/,&*$,&%\"tG\"\"\"%\"xG!\"\"\"\"#F(*$%\"yGF+F(F+ " }{TEXT -1 5 " for " }{TEXT 265 1 "t" }{TEXT -1 9 " to give " } {XPPEDIT 18 0 "t=x+sqrt(2-y^2)" "6#/%\"tG,&%\"xG\"\"\"-%%sqrtG6#,&\"\" #F'*$%\"yGF,!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "S ubstituting in (i) gives the differential equation" }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = -y/sqrt(2-y^2)" "6#/*&%#dyG \"\"\"%#dxG!\"\",$*&%\"yGF&-%%sqrtG6#,&\"\"#F&*$F+F0F(F(F(" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 25 "The initial condit ion is " }{XPPEDIT 18 0 "y(-1)=1" "6#/-%\"yG6#,$\"\"\"!\"\"F(" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Maple's " }{TEXT 0 6 "dsolve" }{TEXT -1 111 " cannot find an ana lytical solution in closed form, as we shall see later, so we look for a numerical solution." }}{PARA 0 "" 0 "" {TEXT -1 61 "We use the comb ined 7th and 8th order Runge-Kutta method via " }{TEXT 0 9 "desolveRK " }{TEXT -1 38 " to find a solution over the interval " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=10" "6#1%!G\" #5" }{TEXT -1 51 ", and set up the solution as a numerical procedure \+ " }{TEXT 269 3 "rk1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "de := diff(y(x),x)=-y(x)/ sqrt(2-y(x)^2);\nic := y(-1)=1;\nrk1 := desolveRK(\{de,ic\},x=-1..10,m ethod=rk78);\nplot('rk1'(x),x=-1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F)\"\"\",&\"\"#F/*$)F)F1F/!\"\" #F4F1F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"\"\"\" " }}{PARA 13 "" 1 "" {GLPLOT2D 597 216 216 {PLOTDATA 2 "6%-%'CURVESG6$ 7U7$$!\"\"\"\"!$\"\"\"F*7$$!+v`3Y$*!#5$\"+oMy%Q*F07$$!+]2<#p)F0$\"+LZH M))F07$$!+JJ?B\")F0$\"+!G$G(R)F07$$!+7bBavF0$\"+(yCG*zF07$$!+D$3XF'F0$ \"+@TB!=(F07$$!+v#)H')\\F0$\"+BK.rkF07$$!+i3@/PF0$\"+*Qy5&eF07$$!+7q0]F]o$\"+i(\\51$F07$$\"*DM^I'F]o$\"+Azm'y#F07$$\"*0ytb(F ]o$\"+CrSYDF07$$\"*RNXp)F]o$\"+#RznM#F07$$\"+XDn/5F]o$\"+Zr=I@F07$$\"+ !y?#>6F]o$\"+w%eF'>F07$$\"+4wY_7F]o$\"+)QxZy\"F07$$\"+IOTq8F]o$\"+(e\\ 4k\"F07$$\"+4\">)*\\\"F]o$\"+1=h'\\\"F07$$\"+EP/B;F]o$\"+(y36P\"F07$$ \"+)o:;v\"F]o$\"+?@Y^7F07$$\"+%)[op=F]o$\"+&*[(3:\"F07$$\"+i%Qq*>F]o$ \"+bpZ^5F07$$\"+RIKH@F]o$\"+h!eNd*!#67$$\"+^rZWAF]o$\"+B&)QB))F`t7$$\" +[n%)oBF]o$\"+MlIz!)F`t7$$\"+5FL(\\#F]o$\"+p*HmP(F`t7$$\"+e6.BEF]o$\"+ &>B&[nF`t7$$\"+p3lWFF]o$\"+G(f=>'F`t7$$\"+A))ozGF]o$\"+HH]FcF`t7$$\"+I k-,IF]o$\"+n8[k^F`t7$$\"+D-eIJF]o$\"+w>87ZF`t7$$\"+>_(zC$F]o$\"+!*\\eO VF`t7$$\"+b*=jP$F]o$\"+_M?gRF`t7$$\"+4/3(\\$F]o$\"+*ogfj$F`t7$$\"+C4JB OF]o$\"+nIRDLF`t7$$\"+DVsYPF]o$\"+[?VZIF`t7$$\"+>n#f(QF]o$\"+PVI\"y#F` t7$$\"+!)RO+SF]o$\"+>V*pa#F`t7$$\"+_!>w7%F]o$\"+*='yFBF`t7$$\"+*Q?QD%F ]o$\"+@\\.H@F`t7$$\"+5jypVF]o$\"+#z-9'>F`t7$$\"+Ujp-XF]o$\"+;$\\ay\"F` t7$$\"+gEd@YF]o$\"+k@\\T;F`t7$$\"+4'>$[ZF]o$\"+Gtw+:F`t7$$\"+6Ejp[F]o$ \"+11Rx8F`t7$$\"\"&F*$\"+F$*3c7F`t-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fj[l-% +AXESLABELSG6$Q\"x6\"Q!F_\\l-%%VIEWG6$;F(F`[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "To find out where the \+ rock is when " }{XPPEDIT 18 0 "t = 2;" "6#/%\"tG\"\"#" }{TEXT -1 33 ", we need to solve the equation " }{XPPEDIT 18 0 "(t-x)^2+y(x)^2 = 2; " "6#/,&*$,&%\"tG\"\"\"%\"xG!\"\"\"\"#F(*$-%\"yG6#F)F+F(F+" }{TEXT -1 5 " for " }{TEXT 270 1 "x" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "t = 2 " "6#/%\"tG\"\"#" }{TEXT -1 6 ".\nThe " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 57 " coordinates for the location of the rock when t = 2 are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x=fsolve((2-x)^2+'rk1'(x)^2=2,x=0.. 2);\ny=rk1(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"+QS\"= 9'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$\"+Y`p>G!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "t=2" "6#/%\"tG\"\"#" }{TEXT -1 29 " the pet rock is at \+ the point" }{XPPEDIT 18 0 " ``( .6141814038 ,.2819695346 )" "6#-%!G6 $-%&FloatG6$\"+QS\"=9'!#5-F'6$\"+Y`p>GF*" }{TEXT -1 18 ", approximatel y. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Solution v ia a numerical inverse function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 60 "We can separate the variable s in the differential equation " }{XPPEDIT 18 0 "dy/dx = -y/sqrt(2-y^ 2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"yGF&-%%sqrtG6#,&\"\"#F&*$F+F0F( F(F(" }{TEXT -1 9 " to give" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(sqrt(2-y^2)/y,y) = -int(1,x);" "6#/-%$IntG6$*&-%%sq rtG6#,&\"\"#\"\"\"*$%\"yGF,!\"\"F-F/F0F/,$-%$intG6$F-%\"xGF0" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x = -Int(sqrt(2-y^2)/y,y)" "6#/%\"xG,$-%$IntG6$*&-%%sqr tG6#,&\"\"#\"\"\"*$%\"yGF.!\"\"F/F1F2F1F2" }{TEXT -1 16 " ------- (iii ). " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "u=s qrt(2-y^2)" "6#/%\"uG-%%sqrtG6#,&\"\"#\"\"\"*$%\"yGF)!\"\"" }{TEXT -1 52 " in the integral on the right side of (iii) we have " }{XPPEDIT 18 0 "du/dx=-y/sqrt(2-y^2)" "6#/*&%#duG\"\"\"%#dxG!\"\",$*&%\"yGF&-%%s qrtG6#,&\"\"#F&*$F+F0F(F(F(" }{TEXT -1 5 ", or " }{XPPEDIT 18 0 "du=`` (-y/sqrt(2-y^2))*dy" "6#/%#duG*&-%!G6#,$*&%\"yG\"\"\"-%%sqrtG6#,&\"\"# F,*$F+F1!\"\"F3F3F,%#dyGF," }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y^2=2 -u^2" "6#/*$%\"yG\"\"#,&F&\"\"\"*$%\"uGF&!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-Int(sqrt(2-y^2)/y,y)=-Int((2-y^2)*y/(y^2*sqrt(2-y^2 )),y)" "6#/,$-%$IntG6$*&-%%sqrtG6#,&\"\"#\"\"\"*$%\"yGF-!\"\"F.F0F1F0F 1,$-F&6$*(,&F-F.*$F0F-F1F.F0F.*&F0F--F*6#,&F-F.*$F0F-F1F.F1F0F1" } {XPPEDIT 18 0 "``=Int(u^2/(2-u^2),u)" "6#/%!G-%$IntG6$*&%\"uG\"\"#,&F* \"\"\"*$F)F*!\"\"F.F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "student[changevar](sqrt(2-y^2)=u,-Int(sqrt(2-y^2)/y,y),u);\nco mbine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,$*&%\"uG\"\"# ,&F*\"\"\"*$)F)F*F,!\"\"F/F/F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$*&%\"uG\"\"#,&F(\"\"\"*$)F'F(F*!\"\"F-F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " } {XPPEDIT 18 0 "u^2/(2-u^2)" "6#*&%\"uG\"\"#,&F%\"\"\"*$F$F%!\"\"F)" } {TEXT -1 36 " has the partial fraction expansion " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "u^2/(2-u^2)=sqrt(2)/2" "6#/*&%\"uG\" \"#,&F&\"\"\"*$F%F&!\"\"F**&-%%sqrtG6#F&F(F&F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(1/(sqrt(2)+u)+1/(sqrt(2)-u))-1;" "6#,&-%!G6#,&*&\"\" \"F),&-%%sqrtG6#\"\"#F)%\"uGF)!\"\"F)*&F)F),&-F,6#F.F)F/F0F0F)F)F)F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 64 "u^2/(2-u^2);\n``=factor(%,sqrt(2));\n``=conver t(rhs(%),parfrac,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"uG\"\"#,& F%\"\"\"*$)F$F%F'!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(%\"u G\"\"#,&F&\"\"\"*$-%%sqrtG6#F'F)F)!\"\",&F&F.F*F)F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,(!\"\"\"\"\"*(#F'\"\"#F'F*F),&%\"uGF'*$-%%sqrtG6 #F*F'F'F&F'*(F)F'F*F),&F,F&F-F'F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Int(u^2/(2-u^2),u) = sqrt(2)/2" "6#/-%$IntG6$*&%\"uG\" \"#,&F)\"\"\"*$F(F)!\"\"F-F(*&-%%sqrtG6#F)F+F)F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(``(1/(sqrt(2)+u)+1/(sqrt(2)-u)),u)-Int(1,u);" "6#,& -%$IntG6$-%!G6#,&*&\"\"\"F,,&-%%sqrtG6#\"\"#F,%\"uGF,!\"\"F,*&F,F,,&-F /6#F1F,F2F3F3F,F2F,-F%6$F,F2F3" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(2)/2" "6#/%!G*&-%%sqrtG6#\"\"# \"\"\"F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(sqrt(2)+u)-ln(sqrt( 2)-u)]-u+c;" "6#,(7#,&-%#lnG6#,&-%%sqrtG6#\"\"#\"\"\"%\"uGF.F.-F'6#,&- F+6#F-F.F/!\"\"F5F.F/F5%\"cGF." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(2)/2" "6#/%!G*&-%%sqrtG6#\"\"# \"\"\"F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((sqrt(2)+u)/(sqrt(2) -u))-u+c" "6#,(-%#lnG6#*&,&-%%sqrtG6#\"\"#\"\"\"%\"uGF-F-,&-F*6#F,F-F. !\"\"F2F-F.F2%\"cGF-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=sqrt(2)/2" "6#/%!G*&-%%sqrtG6#\"\"#\"\"\"F)!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((sqrt(2)+sqrt(2-y^2))/(sqrt(2)-sq rt(2-y^2)))-sqrt(2-y^2)+c;" "6#,(-%#lnG6#*&,&-%%sqrtG6#\"\"#\"\"\"-F*6 #,&F,F-*$%\"yGF,!\"\"F-F-,&-F*6#F,F--F*6#,&F,F-*$F2F,F3F3F3F--F*6#,&F, F-*$F2F,F3F3%\"cGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Si nce " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 10 ", we have \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c=-sqrt(2)/2" "6# /%\"cG,$*&-%%sqrtG6#\"\"#\"\"\"F*!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((sqrt(2)+1)/(sqrt(2)-1)) = sqrt(2)/2" "6#/-%#lnG6#*&,&-%%sqrt G6#\"\"#\"\"\"F-F-F-,&-F*6#F,F-F-!\"\"F1*&-F*6#F,F-F,F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((sqrt(2)-1)/(sqrt(2)+1))" "6#-%#lnG6#*&,&-%%sqrt G6#\"\"#\"\"\"F,!\"\"F,,&-F)6#F+F,F,F,F-" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "so that the soluti on of the differential equation " }{XPPEDIT 18 0 "dy/dx = -y/sqrt(2-y^ 2)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"yGF&-%%sqrtG6#,&\"\"#F&*$F+F0F(F (F(" }{TEXT -1 35 ", subject to the initial condition " }{XPPEDIT 18 0 "y(-1)=1" "6#/-%\"yG6#,$\"\"\"!\"\"F(" }{TEXT -1 39 ", is given impl icitly by the equation " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x= sqrt(2)/2" "6#/%\"xG*&-% %sqrtG6#\"\"#\"\"\"F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(ln((sqr t(2)+sqrt(2-y^2))/(sqrt(2)-sqrt(2-y^2)))+ln((sqrt(2)-1)/(sqrt(2)+1)))- sqrt(2-y^2);" "6#,&-%!G6#,&-%#lnG6#*&,&-%%sqrtG6#\"\"#\"\"\"-F.6#,&F0F 1*$%\"yGF0!\"\"F1F1,&-F.6#F0F1-F.6#,&F0F1*$F6F0F7F7F7F1-F)6#*&,&-F.6#F 0F1F1F7F1,&-F.6#F0F1F1F1F7F1F1-F.6#,&F0F1*$F6F0F7F7" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 30 "____________________ __________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Alternative ly, since " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=sqrt (2)/2" "6#/%\"xG*&-%%sqrtG6#\"\"#\"\"\"F)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "``(ln( (1+sqrt(2-y^2)/sqrt(2))/(1+sqrt(2-y^2)/sqrt(2)) \+ ) - ln( (1+1/sqrt(2))/(1-1/sqrt(2)) ) ) - sqrt(2-y^2)" "6#,&-%!G6#,&-% #lnG6#*&,&\"\"\"F-*&-%%sqrtG6#,&\"\"#F-*$%\"yGF3!\"\"F--F06#F3F6F-F-,& F-F-*&-F06#,&F3F-*$F5F3F6F--F06#F3F6F-F6F--F)6#*&,&F-F-*&F-F--F06#F3F6 F-F-,&F-F-*&F-F--F06#F3F6F6F6F6F--F06#,&F3F-*$F5F3F6F6" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 12 "the formula " }{XPPEDIT 18 0 "arctan( z)=1/2" "6#/-%'arctanG6#%\"zG*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln( (1+z)/(1-z) )" "6#-%#lnG6#*&,&\"\"\"F(%\"zGF(F(,&F( F(F)!\"\"F+" }{TEXT -1 15 ", means that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = sqrt(2)*(arctanh(sqrt(2-y^2)/sqrt(2))-arc tanh(1/sqrt(2)))-sqrt(2-y^2);" "6#/%\"xG,&*&-%%sqrtG6#\"\"#\"\"\",&-%( arctanhG6#*&-F(6#,&F*F+*$%\"yGF*!\"\"F+-F(6#F*F6F+-F.6#*&F+F+-F(6#F*F6 F6F+F+-F(6#,&F*F+*$F5F*F6F6" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 31 "_______________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "g := y -> sqrt(2)*(arctanh(sqrt(2-y^2)/sqrt(2))-arcta nh(1/sqrt(2)))-sqrt(2-y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf *6#%\"yG6\"6$%)operatorG%&arrowGF(,&*&-%%sqrtG6#\"\"#\"\"\",&-%(arctan hG6#*&-F/6#,&F1F2*$)9$F1F2!\"\"F2F.F>F2-F56#*&F2F2F.F>F>F2F2F8F>F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The \+ graph of g plotted using " }{TEXT 0 12 "implicitplot" }{TEXT -1 90 " c ertainly looks like the graph for the numerical solution given in the \+ previous section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "plots[implicitplot](x=g(y),x=-1..4,y=0..1,gr id=[20,40]);" }}{PARA 13 "" 1 "" {GLPLOT2D 530 217 217 {PLOTDATA 2 "6$ -%'CURVESG6bp7$7$$!3mz:j_5Uot!#=$\"312RY\"=.$oyF*7$$!3S;pasEf*[(F*$\"3 o_zr[zr[zF*7$7$$!3_+3$*QoLzF*$\"3smWF(40-E)F *7$7$$!37')oU%yi(4#)F*$\"3!fYQ:YQ:Y)F*FB7$7$$!3B()oU%yi(4#)F*$\"3+n%Q: YQ:Y)F*7$$!3b1iy%*>lK$)F*$\"3+jFso\"*[b&)F*7$7$$!3;8.i-?:X&)F*$\"3]A([ zr[zr)F*FS7$FY7$$!3x\"y^c&>'zq)F*$\"3e*e')z3o%[))F*7$7$$!3o'oORL)*\\') )F*$\"35z*eV(*eV(*)F*Fin7$F_o7$$!31UsNI*>61*F*$\"3d:*)f3')GR\"*F*7$7$$ !3mXY>I;!*p\"*F*$\"3sN#p2Bp2B*F*Feo7$F[p7$$!3]yVx!RRMR*F*$\"3#*G.()4(y !G%*F*7$7$$!3/X W)********!#<$\"2yS%[)*********F\\r7$7$$!2))****f)********F\\r$\"3W+++ ++++5F\\rFiq7$7$$!3ueJE0@%ot%F*$\"3/v^sS.\\XjF*7$$!3+p*Gck)[n[F*$\"3\" >Tc-Tc-T'F*7$7$F\\s$\"3-8kD5kD5kF*7$$!33@fU@#=v*[F*$\"3#oCJ=.7fU'F*7$7 $$!3'441q!yJf`F*$\"3ipmmmmmmmF*Fds7$Fjs7$$!3%=yapk'*R\\&F*$\"3**y^]aF*7$7$$!3 kI#on[qwEt0QF*$\"3!)**eV(*eV(*eF*Fiy7$7$F`z$\"3!4!f V(*eV(*eF*7$$!3[Gg1RRR\\UF*$\"3$pD>V?^j5'F*7$7$$!3IW(\\c>c-N%F*$\"3Uch %Q:YQ:'F*Fhz7$7$$!3'[u\\c>c-N%F*$\"3_dh%Q:YQ:'F*Ffr7$7$$\"3sIot%*y:j_! #>$\"3gB?KCfZ]UF*7$$\"3f/')>)z$Q,=F]\\l$\"39gV(*eV(*eVF*7$7$$\"3,2')>) z$Q,=F]\\lFc\\l7$$\"3/;(f!QMj%y$!#?$\"31z8q=)olS%F*7$7$$!3ueb8%z`W'eF] \\l$\"3v;YQ:YQ:YF*Fi\\l7$7$$!3Mdb8%z`W'eF]\\lFc]l7$$!3N7ka#*4%e+\"F*$ \"3CTKX'orYw%F*7$7$$!3Q;KpGYy18F*$\"3Ot[zr[zr[F*Fi]l7$7$F`^l$\"3#R([zr [zr[F*7$$!3,z4\\/WnV>F*$\"3r'o,IegC6&F*7$7$$!3!4m9![-M&)>F*$\"3(*H^?G^ ?G^F*Fh^l7$7$$!3igY,[-M&)>F*Fa_lF_x7$7$$\"3)R5Uot%*y:$F*$\"3@%[niKpL]$ F*7$$\"3^8W=2]e:GF*$\"3I!fV(*eV(*e$F*7$F]`l7$$\"3:)z8.LD\\i#F*$\"3l`X< URnTOF*7$7$$\"3!RsZKl(=u=F*$\"3#p%Q:YQ:YQF*Fc`l7$7$$\"3=CxC`w=u=F*F\\a l7$$\"3y*e.<)pqL7F*$\"3-L4\"p!)QO.%F*7$7$$\"3%y)3&z!RY#)**F]\\l$\"3`.T c-Tc-TF*Fbal7$7$$\"3Y')3&z!RY#)**F]\\lF[bl7$F[\\l$\"3/B?KCfZ]UF*7$7$$ \"3MC0@%ot%*y&F*$\"3mkX)ebCZ*GF*7$$\"3())pl]H*HM\\F*$\"35xI#p2Bp2$F*7$ 7$$\"3U*pl]H*HM\\F*F]cl7$$\"3_<-BycmF*7$$\"3CMD%fJlu0\"F\\r$\"3 m]?G^?G^?F*7$7$Fafl$\"3%40#G^?G^?F*7$$\"3')\\y.sPhw(*F*$\"39lCW\"*>hv@ F*7$7$$\"3Np62ro#)G*)F*$\"3F2Bp2Bp2BF*Fifl7$7$F`gl$\"3*pI#p2Bp2BF*7$Fa dl$\"3S?\"[g5(R'R#F*7$7$$\"3ay:j_5Uo8F\\r$\"3;>Gt[I;[;F*7$$\"3yVav.dbW 7F\\r$\"3g%zr[zr[z\"F*7$7$$\"3+Wav.dbW7F\\rFdhlF[fl7$7$$\"3e?%ot%*y:j \"F\\r$\"3#y*4iBN-o8F*7$$\"3Ae*QbST5Y\"F\\r$\"3FQ:YQ:YQ:F*7$7$Fail$\"3 *z`h%Q:YQ:F*F\\hl7$7$$\"3gi_5Uot%*=F\\r$\"3qTtY6]kP6F*7$$\"30)R#G#R,wr \"F\\r$\"3#>G^?G^?G\"F*7$7$$\"3$yR#G#R,wr\"F\\rFbjlF[il7$7$$\"3'[5Uot% *y:#F\\r$\"3UkYqcN9E@.#F\\r$\"3hD5kD5kD5F*7$F^[mFj il7$Fijl7$$\"3'*f!>.=]&*Q#F\\r$\"3-j9I`YD**zF]\\l7$7$$\"3!p%*y:j_5U#F \\r$\"3yI(zRm9.!yF]\\lFe[m7$7$$\"3#*)y:j_5Uo#F\\r$\"33op#p)3#4f'F]\\l7 $$\"3sV=c+d:QCF\\r$\"3u#p2Bp2Bp(F]\\l7$Ff\\mF[\\m7$Fa\\m7$$\"3k*=q>G0L *GF\\r$\"3Wq]nvT(\\l&F]\\l7$7$$\"3)4j_5Uot%HF\\r$\"3nQKp)exHT&F]\\lF]] m7$7$$\"3,t%*y:j_5KF\\r$\"3K]>9Ow21YF]\\l7$$\"3/6!*)\\)y)4,$F\\r$\"3.G ^?G^?G^F]\\l7$F^^m7$Fd]m$\"3ORKp)exHT&F]\\l7$7$Fj]m$\"3,^>9Ow21YF]\\l7 $$\"3yx`=$p(y$G$F\\r$\"3RN\"4GKwVT%F]\\l7$7$$\"3/:j_5UotMF\\r$\"3bQ!Hw fyu\"RF]\\lF[_m7$7$Fb_m$\"3DR!Hwfyu\"RF]\\l7$$\"3+d!G,7lNk$F\\r$\"3k'* yxR_&HZ$F]\\l7$7$$\"33dJE0@%ot$F\\r$\"3[Fh6f&z)GKF]\\lFj_m7$7$$\"37*** ***********RF\\r$\"3us.kGP]hDF]\\l7$$\"3-.\"4!og*3*RF\\r$\"3-kD5kD5kDF ]\\l7$F[am7$$\"3_dJE0@%ot$F\\rFc`m-%'COLOURG6&%$RGBG\"\"\"\"\"!Fiam-%+ AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Solution using " }{TEXT 0 6 "dsolve" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "We can use " }{TEXT 0 6 "dsolve" }{TEXT -1 74 " to obtain an implicit form for the solution of the differential equat ion " }{XPPEDIT 18 0 "dy/dx = -y/sqrt(2-y^2)" "6#/*&%#dyG\"\"\"%#dxG! \"\",$*&%\"yGF&-%%sqrtG6#,&\"\"#F&*$F+F0F(F(F(" }{TEXT -1 35 ", subjec t to the initial condition " }{XPPEDIT 18 0 "y(-1)=1" "6#/-%\"yG6#,$\" \"\"!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "The soluti on involves the inverse hyperbolic tangent function " }{TEXT 269 7 "ar ctanh" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "de := diff(y(x),x)=-y(x)/sqrt(2-y( x)^2);\nic := y(-1)=1;\ndsolve(\{de,ic\},y(x));\nisolate(subs(y(x)=y,% ),x);\nh := unapply(rhs(%),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d eG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F)\"\"\",&\"\"#F/*$)F)F1F/!\"\"#F4F1F4 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,*F'\"\"\"*$ -%%sqrtG6#,&*$)%#_ZG\"\"#F,!\"\"F5F,F,F,*&-F/6#F5F,-%(arctanhG6#*$-F/6 #,$*&F,F,,&F2F,F5F6F6!\"#F,F,F6*&F8F,-F;6#*$F8F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,(*$-%%sqrtG6#,&\"\"#\"\"\"*$)%\"yGF+F,!\"\"F, F0*&-F(6#F+F,-%(arctanhG6#*$-F(6#,$*&F,F,,&!\"#F,F-F,F0F=F,F,F,*&F2F,- F56#*$F2F,F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"yG6\"6$ %)operatorG%&arrowGF(,(*$-%%sqrtG6#,&\"\"#\"\"\"*$)9$F2F3!\"\"F3F7*&-F /6#F2F3-%(arctanhG6#*$-F/6#,$*&F3F3,&!\"#F3F4F3F7FDF3F3F3*&F9F3-F<6#*$ F9F3F3F7F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "desolve" }{TEXT -1 60 " (from another worksheet) gives essentially the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "de := \+ diff(y(x),x)=-y(x)/sqrt(2-y(x)^2);\nic := y(-1)=1;\ndesolve(\{de,ic\}, y(x),info=true);\nisolate(subs(y(x)=y,%),x);\nh := unapply(rhs(%),y); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now \+ " }{XPPEDIT 18 0 "arctanh(z)" "6#-%(arctanhG6#%\"zG" }{TEXT -1 14 " is real when " }{TEXT 275 1 "z" }{TEXT -1 22 " is a real number and " } {XPPEDIT 18 0 "abs(z)<1" "6#2-%$absG6#%\"zG\"\"\"" }{TEXT -1 25 ", but it is complex when " }{XPPEDIT 18 0 "abs(z)>1" "6#2\"\"\"-%$absG6#%\" zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Hence the expressi on " }{XPPEDIT 18 0 "arctanh(sqrt(2))" "6#-%(arctanhG6#-%%sqrtG6#\"\"# " }{TEXT -1 13 " is complex. " }}{PARA 0 "" 0 "" {TEXT -1 14 "In fact, when " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 152 "p := y -> arctanh(sqrt( 2)/sqrt(2-y^2))+Pi*I/2;\nq := y -> arctanh(sqrt(2-y^2)/sqrt(2));\nplot ([p(y),q(y)],y=0..sqrt(2),color=[red,green],thickness=[1,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"yG6\"6$%)operatorG%&arrow GF(,&-%(arctanhG6#*&-%%sqrtG6#\"\"#\"\"\"-F26#,&F4F5*$)9$F4F5!\"\"F%\"qGf* 6#%\"yG6\"6$%)operatorG%&arrowGF(-%(arctanhG6#*&-%%sqrtG6#,&\"\"#\"\" \"*$)9$F4F5!\"\"F5-F16#F4F9F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7jn7$$\"++A2L'*!#8$\"+\")y$[)z!\"*7 $$\"+SWhE>!#7$\"+u3p\"H(F-7$$\"+g;#**)GF1$\"+cgA')oF-7$$\"+!))GK&QF1$ \"+!RW&)f'F-7$$\"+?L%)zdF1$\"+U/3$>'F-7$$\"+gxX1xF1$\"+pOV0fF-7$$\"+k' of:\"!#6$\"+D&R**\\&F-7$$\"+_:HT:FK$\"+%pQA@&F-7$$\"+Gt$>J#FK$\"+P)Qn! 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However, since complex numbers are not involved in \+ the evaluation of the function " }{TEXT 269 1 "g" }{TEXT -1 57 ", this earlier formulation will be used in what follows. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "If we wan t to make a direct comparison with the earlier numerical solution we s hould construct a " }{TEXT 261 17 "numerical inverse" }{TEXT -1 5 " fo r " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g(y) = sqrt(2) *(arctanh(sqrt(2-y^2)/sqrt(2))-arctanh(1/sqrt(2)))-sqrt(2-y^2);" "6#/- %\"gG6#%\"yG,&*&-%%sqrtG6#\"\"#\"\"\",&-%(arctanhG6#*&-F+6#,&F-F.*$F'F -!\"\"F.-F+6#F-F8F.-F16#*&F.F.-F+6#F-F8F8F.F.-F+6#,&F-F.*$F'F-F8F8" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "First we write a procedu re which provides improvements in efficiency for numerical evaluation. Maple can help with this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "with(codegen,optimize,makeproc):\nm akeproc(g(x),x):\noptimize(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#f*6# %\"xG6'%#t2G%#t3G%#t5G%$t10G%$t15G6\"F,C(>8$*$-%%sqrtG6#\"\"#\"\"\">8% *$)9$F4F5>8&*$-F26#,&F4F5F7!\"\"F5>8'-%#lnG6#*&,&F/F5F8(-FE6#*&,&F/F5F5FAF5,&F/F5F5F5FA,&*&F/F5,&FCF5FKF5F5#F5F4F " 0 "" {MPLTEXT 1 0 132 "newg := proc(y)\n local s,s2;\n s := sqrt(2-y^2) ;\n s2 := sqrt(2);\n s2/2*(ln((s2+s)/(s2-s))+ln((s2-1)/(s2+1)))-s; \nend:\nnewg(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%sqrtG6#\"\" #\"\"\",&-%#lnG6#*&,&*$F%F)F)*$-F&6#,&F(F)*$)%\"yGF(F)!\"\"F)F)F),&F0F )F1F8F8F)-F,6#*&,&F0F)F)F8F),&F0F)F)F)F8F)F)#F)F(F1F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "\nFor a given " }{TEXT 282 1 "y" } {TEXT -1 23 ", we want to calculate " }{TEXT 277 1 "x" }{TEXT -1 9 " s o that " }{XPPEDIT 18 0 "x = g(y)" "6#/%\"xG-%\"gG6#%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "The solution curve looks a bit li ke " }{XPPEDIT 18 0 "y = exp(-x-1);" "6#/%\"yG-%$expG6#,&%\"xG!\"\"\" \"\"F*" }{TEXT -1 11 " so, given " }{TEXT 283 1 "x" }{TEXT -1 23 ", we could try taking " }{XPPEDIT 18 0 "y = exp(-x-1);" "6#/%\"yG-%$expG6 #,&%\"xG!\"\"\"\"\"F*" }{TEXT -1 32 "as a starting approximation for \+ " }{TEXT 0 6 "fsolve" }{TEXT -1 28 " applied to the solution of " } {XPPEDIT 18 0 "g(y) = x " "6#/-%\"gG6#%\"yG%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 138 "It isn't absolutely necessary to incorpo rate g inside the definition of the inverse, except that this makes th e definition self-contained." }}{PARA 0 "" 0 "" {TEXT -1 78 "The next \+ procedure gives a new numerical solution to the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = -y/sqrt( 2-y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&%\"yGF&-%%sqrtG6#,&\"\"#F&*$F+ F0F(F(F(" }}{PARA 0 "" 0 "" {TEXT -1 26 "in the form of a function " } {XPPEDIT 18 0 "y = y(x)" "6#/%\"yG-F$6#%\"xG" }{TEXT -1 17 " of the va riable " }{TEXT 278 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "rk2 := proc(x::realc ons)\n local y1,y,g,s2,xx;\n xx := evalf(x);\n s2 := evalf(sqrt( 2));\n g := proc(y)\n local s;\n s := sqrt(2-y^2);\n \+ s2/2*(ln((s2+s)/(s2-s))+ln((s2-1)/(s2+1)))-s;\n end:\n y1 := evalf (exp(-xx-1));\n fsolve(g(y)=x,y=y1);\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Before we check the numer ical inverse let's redefine g by the new procedure." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "g := 'g': g := proc(x)\n local s,s2;\n s := sqrt(2-x^2);\n s2 := sqrt(2); \n s2/2*(ln((s2+s)/(s2-s))+ln((s2-1)/(s2+1)))-s;\nend:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Here are a couple \+ of numerical examples to check that " }{TEXT 269 3 "rk2" }{TEXT -1 39 " provides an (approximate) inverse for " }{TEXT 269 1 "g" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rk2(0);\nevalf(g(%));\nrk2(1);\nevalf(g(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZQx " 0 "" {MPLTEXT 1 0 16 "plot(rk2,-1..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 480 219 219 {PLOTDATA 2 "6%-%'CURVESG6$7U7$!\"\"$\"+++++5!\"*7$$!+Y62b%*!# 5$\"+'\\kB[*F/7$$!+\"HU,\"*)F/$\"+>]b6!*F/7$$!+4E+O%)F/$\"+4L;L')F/7$$ !+EH'='zF/$\"+3l]z#)F/7$$!+.OU&*oF/$\"+w!f#fvF/7$$!+G_\">#eF/$\"+RJ**> pF/7$$!+]!4Nv%F/$\"+FTi_jF/7$$!+e(fHw$F/$\"+\\2&y(eF/7$$!+TLIPFF/$\"+# =62V&F/7$$!+3!oln\"F/$\"+^;m4]F/7$$!*mWB>'F/$\"+')4sEYF/7$$\")0j$o%F+$ \"+tOomUF/7$$\"*_>jU\"F+$\"+y5,vRF/7$$\"*j^Z]#F+$\"+n`JsOF/7$$\"*)=h(e $F+$\"+0+6$R$F/7$$\"*Q[6j%F+$\"+%p%GXJF/7$$\"*\\z(ybF+$\"+mrrOHF/7$$\" *b/cq'F+$\"+>VP2FF/7$$\"*F+$\"+Fq!\\6\"F/7$$\"+KE >>?F+$\"+z#*3N5F/7$$\"+#RU07#F+$\"+JFKL'*!#67$$\"+?S2LAF+$\"+wG'\\*))F gu7$$\"+$p)=MBF+$\"+iE4!G)Fgu7$$\"+*=]@W#F+$\"+#3!eqwFgu7$$\"+]$z*RDF+ $\"+M\"=s:(Fgu7$$\"+kC$pk#F+$\"+c8INmFgu7$$\"+3qcZFF+$\"+yP4zhFgu7$$\" +/\"fF&GF+$\"+Q)ydt&Fgu7$$\"+0OgbHF+$\"+b*pJL&Fgu7$$\"+nAFjIF+$\"+W:'> %\\Fgu7$$\"+&)*pp;$F+$\"+QeO#f%Fgu7$$\"+ye,tKF+$\"+i#[Z$F+$\"+-7m$p$Fgu7$$\"+(G!e&e$F+$\"+>ZL:MFgu 7$$\"+&)Qk%o$F+$\"+4>A%=$Fgu7$$\"+UjE!z$F+$\"+&4;]&HFgu7$$\"+60O\"*QF+ $\"+ " 0 "" {MPLTEXT 1 0 103 "de := diff(y(x),x)=-y(x)/sqrt(2-y(x)^2);\nic := y(-1)=1;\nrk1 := desolveRK(\{de,ic\},x=-1..10,method=rk78):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&F)\"\"\",&\"\"#F/*$ )F)F1F/!\"\"#F4F1F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6# !\"\"\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rk1(-0.5);\nrk2(-0.5);\nrk1(0);\nrk2(0);\nrk1(1); \nrk2(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M!*3yk!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+T!*3yk!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+OQx " 0 "" {MPLTEXT 1 0 53 "x=fsolve((2-x)^2+'rk2'(x)^2=2,x=0..2);\ny=rk2(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"+QS\"=9'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$\"+k`p>G!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "There is no difficulty in obtaining a mor e accurate result with this method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Digits := 20:\nx=fsolve((2- x)^2+'rk2'(x)^2=2,x=0..2);\ny=rk2(rhs(%));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"54D7m8QS\"=9'!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$\"5w$*[FQY`p>G!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "For comparison, the previous resul t was:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x=fsolve((2-x)^2 +'rk1'(x)^2=2,x=0..2);\ny=rk1(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"+QS\"=9'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$ \"+Y`p>G!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "To obtain a more accurate answer with the Runge-Kutta solution we need to reconstruct the numerical solution for the differential eq uation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "de := diff(y(x),x)=-y(x)/sqrt(2-y(x)^2):\nic := y(-1 )=1:\nDigits := 21:\nrk3 := desolveRK(\{de,ic\},x=-1..1,method=rk78): \nx=fsolve((2-x)^2+'rk3'(x)^2=2,x=0..1);\ny=rk3(rhs(%));\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"6*3D7m8QS\"=9'!#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG$\"6QP*[FQY`p>G!#@" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 " Task using " }{TEXT 0 9 "desolveRK" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 100 "Suppose that inste ad of walking in a straight line, the master follows a circular arc wi th centre at" }{XPPEDIT 18 0 " ``(0,4)" "6#-%!G6$\"\"!\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 60 "Then the position of the maste r at time t secs after leaving" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\" \"!F&" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "M(4*sin(t/4),4-4*cos(t/4)); " "6#-%\"MG6$*&\"\"%\"\"\"-%$sinG6#*&%\"tGF(F'!\"\"F(,&F'F(*&F'F(-%$co sG6#*&F-F(F'F.F(F." }{TEXT -1 2 ". 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Although it takes a few seconds to obtain initial ly, evaluating this solution requires no further evaluations of the sl ope function, so it should be fairly speedy.\n" }}{PARA 0 "" 0 "" {TEXT -1 33 "Plot the graph of this solution. " }{TEXT 260 23 "Where i s the rock when " }{TEXT 267 1 "t" }{TEXT 268 5 " = 4?" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Hint " }{TEXT -1 7 ": When " }{XPPEDIT 18 0 "t = 4" "6#/%\"tG\"\"%" }{TEXT -1 29 ", the master is at the point " }{XPPEDIT 18 0 "``(4*sin(1),4-4* cos(1));" "6#-%!G6$*&\"\"%\"\"\"-%$sinG6#F(F(,&F'F(*&F'F(-%$cosG6#F(F( !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "g := (x,y) -> x+sqrt(2-y^2);\nh := (x,y, t) -> (4-4*cos(t/4)-y)/(4*sin(t/4)-x);\nf := (x,y) -> h(x,y,g(x,y));\n de := diff(y(x),x)=f(x,y(x));\nic := y(-1)=1;\nrk := desolveRK(\{de,ic \},x=-1..3,output=rkinterp);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "Code for drawing the pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "Code for drawing 1st picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 612 "x := 'x':y := 'y':\ncrv := x ->.4634422253*x^3+1.805111854*x^2+1.219897032*x+.8782274032:\nh := \+ evalf(1/2-sqrt(2)*cos(8*Pi/45)):\ns := evalf(sqrt(2)*sin(8*Pi/45)):\nc urve := plot(crv,-1..h):\nextra := plot([[h,0],[h,s],[0.5,0]],linestyl e=2,color=black):\npts := plot([[[h,s],[.5,0]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nwalk := plot([[0,0] ,[.5,0]],thickness=2,color=blue):\nt1 := plots[textplot]([[-0.19,0.51, `d`],[-0.59,0.8,`R(x,y)`],\n[0.55,0.08,`M(t,0)`],[-0.05,1.47,`y`],[1,0 .07,`x`]]):\nplots[display]([curve,extra,walk,pts,t1],\n \+ view=[-1..1,0..1.5],tickmarks=[3,2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 28 "Code for drawing 2nd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 46 ": The procedure rungk45 is required for this.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "read \"D:\\\\Maple7/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1041 "x :='x': y := 'y': t := 't':\ng : = (x,y) -> x+sqrt(2-y^2):\nh := (x,y,t) -> (4-4*cos(t/4)-y)/(4*sin(t/4 )-x):\nf := (x,y) -> h(x,y,g(x,y)):\nrk := rungk45(f(x,y),x=-1..3,y=1, output=rkinterp):\nxM := evalf(4*sin(1)):\nyM := evalf(4-4*cos(1)):\nx R :=2.278907739:\nyR :=.9340904170: \nrockpath := plot(rk,-1..xR,color =red,thickness=2):\nmasterpath := plot(4-sqrt(16-x^2),x=0..xM,y=0..yM, \n color=blue,thickness=2):\nlines := plot([[[xR,yR], [xM,yM],[0,4]],[[-1,1],[0,0]]],\n linestyle=2,color=bl ack):\npts := plot([[[xR,yR],[xM,yM],[-1,1],[0,0]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nd := evalf(Pi/40 ):\nangle := plot([seq([.5*cos(d*i),4+.4*sin(d*i)],i=-20..-8)],\n \+ color=black):\nt1 := plots[textplot]([[xR-0.25,yR+0.1,`R(x,y)`] ,\n [xM-0.85,yM,`M(4sin(t/4),4-4cos(t/4))`],[-0.1,3.75,`y`],\n [3. 5,0.2,`x`],[1.9,3,`r = 4`],[0.15,3.75,`t/4`],\n [2.37,0.56,`arc = \+ t`]]):\nplots[display]([masterpath,rockpath,lines,pts,angle,t1],\n \+ view=[-1..3.5,0..4],tickmarks=[5,5],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }