{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 "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 } {PSTYLE "Normal" -1 0 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 }{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 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 } } {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Chapter 8" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "dsolve(diff (p(t),t)=0.012*p(t)+0.3, p(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"pG6#%\"tG,&\"#D!\"\"*&-%$expG6#,$*(\"\"$\"\"\"\"$]#F*F'F2F2F2%$_C1GF 2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " It means: " }{XPPEDIT 18 0 "p(t) = -25+C*exp(3*t/250); " "6#/-%\"pG6#%\"tG,&\"#D!\"\"*&%\"CG\"\"\"-%$expG6#*(\"\"$F-F'F-\"$]# F*F-F-" }{TEXT -1 35 " ---- the general sol. to the ODE" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now consider " } {XPPEDIT 18 0 "p(0) = 25;" "6#/-%\"pG6#\"\"!\"#D" }{TEXT -1 14 ": \+ When " }{XPPEDIT 18 0 "t = 0;" "6#/%\"tG\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "p = 25;" "6#/%\"pG\"#D" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " " } {XPPEDIT 18 0 "-25+C = 25;" "6#/,&\"#D!\"\"%\"CG\"\"\"F%" }{TEXT -1 6 ": " }{XPPEDIT 18 0 "C = 50;" "6#/%\"CG\"#]" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Finally: \+ We have the exact solution: " }{XPPEDIT 18 0 "p(t) = -25+50*exp(3 *t/250);" "6#/-%\"pG6#%\"tG,&\"#D!\"\"*&\"#]\"\"\"-%$expG6#*(\"\"$F-F' F-\"$]#F*F-F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p := t-> -25 + 50*exp(3*t/25 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"tG6\"6$%)operatorG %&arrowGF(,&\"#D!\"\"*&\"#]\"\"\"-%$expG6#,$*&#\"\"$\"$]#F19$F1F1F1F1F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plot(p(t),t=0..5); " }}{PARA 13 "" 1 "" {GLPLOT2D 378 187 187 {PLOTDATA 2 "6%-%'CURVESG6$ 7S7$$\"\"!F)$\"#DF)7$$\"3GLLL3x&)*3\"!#=$\"3%))3eTUVl]#!#;7$$\"3umm\"H 2P\"Q?F/$\"3JLk/*yVA^#F27$$\"3MLL$eRwX5$F/$\"3]B]s*>i'=DF27$$\"33ML$3x %3yTF/$\"3g9Yzc98DDF27$$\"3emm\"z%4\\Y_F/$\"3WRBPY#y:`#F27$$\"3`LLeR-/ PiF/$\"3%>q()Qjiv`#F27$$\"3]***\\il'pisF/$\"3Z%*y7@mwVDF27$$\"3>MLe*)> VB$)F/$\"37jQlI3>]DF27$$\"3Y++DJbw!Q*F/$\"3'GF(z!e-mb#F27$$\"3%ommTIOo /\"!#<$\"3_KvDZj?jDF27$$\"3YLL3_>jU6Fhn$\"3S5Ka\"4I!pDF27$$\"37++]i^Z] 7Fhn$\"3M)z!yjUfvDF27$$\"33++](=h(e8Fhn$\"3Mjw*=%R>#e#F27$$\"3/++]P[6j 9Fhn$\"3%HqYT2i&)e#F27$$\"3UL$e*[z(yb\"Fhn$\"3kJ@Fhn$\"3>^$o>cO*>EF27$$\"3K+]i!f #=$3#Fhn$\"3=UQ5HjcEEF27$$\"3?+](=xpe=#Fhn$\"3AX1i9u)Gj#F27$$\"37nm\"H 28IH#Fhn$\"3=D4)H6\"\\REF27$$\"3um;zpSS\"R#Fhn$\"3Yl5b_GcXEF27$$\"3GLL 3_?`(\\#Fhn$\"3'yzRb3?@l#F27$$\"3fL$e*)>pxg#Fhn$\"31$f_c0S*eEF27$$\"33 +]Pf4t.FFhn$\"3/3)*44U)[m#F27$$\"3uLLe*Gst!GFhn$\"3I>IFhn$\"3=(HQ\"RJZ%o#F27$$\"3F +]i!RU07$Fhn$\"3')\\PA;By!p#F27$$\"3+++v=S2LKFhn$\"3#ekQtd'z(p#F27$$\" 3Jmmm\"p)=MLFhn$\"3rGi?(G2Tq#F27$$\"3B++](=]@W$Fhn$\"3T%[9\\x`3r#F27$$ \"35L$e*[$z*RNFhn$\"3PrWZ[Y(pr#F27$$\"3e++]iC$pk$Fhn$\"3bZlA2YnBFF27$$ \"3[m;H2qcZPFhn$\"3Y*H$3.m)*HFF27$$\"3O+]7.\"fF&QFhn$\"3%)*HG1h#fOFF27 $$\"3Ymm;/OgbRFhn$\"3He@@d#fIu#F27$$\"3w**\\ilAFjSFhn$\"3)yr]LzP)\\FF2 7$$\"3yLLL$)*pp;%Fhn$\"3u@w*Rhujv#F27$$\"3)RL$3xe,tUFhn$\"3oru#[Z%Fhn$\"3Mg>SZ(Gex#F27$ $\"3SnmT&G!e&e%Fhn$\"3IPf1:b%Gy#F27$$\"3#RLLL)Qk%o%Fhn$\"3Y>/],$H\"*y# F27$$\"37+]iSjE!z%Fhn$\"3_>$Q " 0 "" {MPLTEXT 1 0 528 " Euler := proc( f :: operator, # RHS of the ODE\n a :: \+ numeric, # left end\n b :: numeric, # right end\n \+ y0 :: numeric, # ini. cond\n n :: integer \+ # no. of subintervals\n )\n local t, y, h, k;\n\n h : = evalf((b-a)/n): # stepsize\n t := Vector([seq( evalf(a+(k-1)*h), \+ k=1..n+1)]):\n y := Vector(n+1): # space for y\n y[1] := evalf (y0):\n for k from 1 to n do\n y[k+1] := y[k] + evalf(h*f(t[k], y[k])):\n end do:\n\n return t, y;\nend proc: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Now, we s olve the population problem using the Euler method" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f := (t,p) \+ -> 0.012*p+0.3; # RHS of the ODE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGf*6$%\"tG%\"pG6\"6$%)operatorG%&arrowGF),&*&$\"#7!\"$\"\"\"9%F2F 2$\"\"$!\"\"F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a, \+ b := 0, 5; # the interval" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\" aG%\"bG6$\"\"!\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "y0 : = 25; # initial condition" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #y0G\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n := 9; # no. \+ of subintervals" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "t, y := Euler(f, a, b, y0, n );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"tG%\"yG6$-%'RTABLEG6%\"*W \\>Y\"-%'MATRIXG6#7,7#$\"\"!F27#$\"+cbbbb!#57#$\"+66666!\"*7#$\"+nmmm; F:7#$\"+AAAAAF:7#$\"+yxxxFF:7#$\"+MLLLLF:7#$\"+*)))))))QF:7#$\"+XWWWWF :7#$\"+++++]F:&%'VectorG6#%'columnG-F)6%\"*%)\\>Y\"-F-6#7,7#$\"#DF27#$ \"+LLLLD!\")7#$\"+*))))oc#Fjn7#$\"+[\"o1g#Fjn7#$\"+-EnMEFjn7#$\"+`P!*o EFjn7#$\"+6JO.FFjn7#$\"+)>_!QFFjn7#$\"+YD(Hx#Fjn7#$\"+'pD\"3GFjnFP" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "pts := [seq([t[k],y[k]],k=1 ..n+1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(pts,style= point,symbolsize=20);" }}{PARA 13 "" 1 "" {GLPLOT2D 420 190 190 {PLOTDATA 2 "6'-%'CURVESG6$7,7$$\"\"!F)$\"#DF)7$$\"3G+++cbbbb!#=$\"3') *****HLLL`#!#;7$$\"3/+++66666!#<$\"3;+++*))))oc#F27$$\"35+++nmmm;F6$\" 35+++[\"o1g#F27$$\"33+++AAAAAF6$\"3%)*****>gsYj#F27$$\"3#******zxxxx#F 6$\"3-+++`P!*oEF27$$\"3?+++MLLLLF6$\"3<+++6JO.FF27$$\"3'*******))))))) )QF6$\"36+++)>_!QFF27$$\"3C+++XWWWWF6$\"35+++YD(Hx#F27$$\"\"&F)$\"36++ +'pD\"3GF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&STYLEG6#%&POINTG-%'SYMBOL G6$%(DEFAULTG\"#?-%+AXESLABELSG6$Q!6\"Fio-%%VIEWG6$FdoFdo" 1 5 0 1 20 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot1 := plot(p(s),s=0..5):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot2 := plot(pts,style=poin t,symbolsize=20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plots[ display](\{plot1,plot2\});" }}{PARA 13 "" 1 "" {GLPLOT2D 420 254 254 {PLOTDATA 2 "6&-%'CURVESG6&7,7$$\"\"!F)$\"#DF)7$$\"3G+++cbbbb!#=$\"3') *****HLLL`#!#;7$$\"3/+++66666!#<$\"3;+++*))))oc#F27$$\"35+++nmmm;F6$\" 35+++[\"o1g#F27$$\"33+++AAAAAF6$\"3%)*****>gsYj#F27$$\"3#******zxxxx#F 6$\"3-+++`P!*oEF27$$\"3?+++MLLLLF6$\"3<+++6JO.FF27$$\"3'*******))))))) )QF6$\"36+++)>_!QFF27$$\"3C+++XWWWWF6$\"35+++YD(Hx#F27$$\"\"&F)$\"36++ +'pD\"3GF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%&STYLEG6#%&POINTG-%'SYMBOL G6$%(DEFAULTG\"#?-F$6$7SF'7$$\"3GLLL3x&)*3\"F/$\"3%))3eTUVl]#F27$$\"3u mm\"H2P\"Q?F/$\"3JLk/*yVA^#F27$$\"3MLL$eRwX5$F/$\"3]B]s*>i'=DF27$$\"33 ML$3x%3yTF/$\"3g9Yzc98DDF27$$\"3emm\"z%4\\Y_F/$\"3WRBPY#y:`#F27$$\"3`L LeR-/PiF/$\"3%>q()Qjiv`#F27$$\"3]***\\il'pisF/$\"3Z%*y7@mwVDF27$$\"3>M Le*)>VB$)F/$\"37jQlI3>]DF27$$\"3Y++DJbw!Q*F/$\"3'GF(z!e-mb#F27$$\"3%om mTIOo/\"F6$\"3_KvDZj?jDF27$$\"3YLL3_>jU6F6$\"3S5Ka\"4I!pDF27$$\"37++]i ^Z]7F6$\"3M)z!yjUfvDF27$$\"33++](=h(e8F6$\"3Mjw*=%R>#e#F27$$\"3/++]P[6 j9F6$\"3%HqYT2i&)e#F27$$\"3UL$e*[z(yb\"F6$\"3kJ@F6$\"3>^$o>cO*>EF27$$\"3K+]i!f#=$3# F6$\"3=UQ5HjcEEF27$$\"3?+](=xpe=#F6$\"3AX1i9u)Gj#F27$$\"37nm\"H28IH#F6 $\"3=D4)H6\"\\REF27$$\"3um;zpSS\"R#F6$\"3Yl5b_GcXEF27$$\"3GLL3_?`(\\#F 6$\"3'yzRb3?@l#F27$$\"3fL$e*)>pxg#F6$\"31$f_c0S*eEF27$$\"33+]Pf4t.FF6$ \"3/3)*44U)[m#F27$$\"3uLLe*Gst!GF6$\"3I>IF6$\"3=(HQ\"RJZ%o#F27$$\"3F+]i!RU07$F6$\"3 ')\\PA;By!p#F27$$\"3+++v=S2LKF6$\"3#ekQtd'z(p#F27$$\"3Jmmm\"p)=MLF6$\" 3rGi?(G2Tq#F27$$\"3B++](=]@W$F6$\"3T%[9\\x`3r#F27$$\"35L$e*[$z*RNF6$\" 3PrWZ[Y(pr#F27$$\"3e++]iC$pk$F6$\"3bZlA2YnBFF27$$\"3[m;H2qcZPF6$\"3Y*H $3.m)*HFF27$$\"3O+]7.\"fF&QF6$\"3%)*HG1h#fOFF27$$\"3Ymm;/OgbRF6$\"3He@ @d#fIu#F27$$\"3w**\\ilAFjSF6$\"3)yr]LzP)\\FF27$$\"3yLLL$)*pp;%F6$\"3u@ w*Rhujv#F27$$\"3)RL$3xe,tUF6$\"3oru#[Z%F6$\"3Mg>SZ(Gex#F27$$\"3SnmT&G!e&e%F6$\"3IPf1:b%G y#F27$$\"3#RLLL)Qk%o%F6$\"3Y>/],$H\"*y#F27$$\"37+]iSjE!z%F6$\"3_>$Q " 0 "" {MPLTEXT 1 0 94 "# plot the error, if we know the ex act solution\nerrpts := [seq([t[k],y[k]-p(t[k])],k=1..n+1)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "plot(errpts);" }}{PARA 13 " " 1 "" {GLPLOT2D 420 189 189 {PLOTDATA 2 "6%-%'CURVESG6$7,7$$\"\"!F)F( 7$$\"3G+++cbbbb!#=$!3$**********pN6\"!#?7$$\"3/+++66666!#<$!3:+++++1UA F07$$\"35+++nmmm;F4$!3%**********>bQ$F07$$\"33+++AAAAAF4$!35+++++=WXF0 7$$\"3#******zxxxx#F4$!3&)*********p\"=dF07$$\"3?+++MLLLLF4$!3\"****** *****e2pF07$$\"3'*******))))))))QF4$!3l+++++n7\")F07$$\"3C+++XWWWWF4$! 3b*********RNL*F07$$\"\"&F)$!33++++!Rq0\"!#>-%'COLOURG6&%$RGBG$\"#5!\" \"F(F(-%+AXESLABELSG6$Q!6\"F_o-%%VIEWG6$%(DEFAULTGFdo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "If the accuracy i s not good enough, increase n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "nn := 20:\ntt, yy := Euler(f ,a,b,y0,nn):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "errpt2 := [ seq([tt[k],yy[k]-p(tt[k])],k=1..n+1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "plot(errpt2);" }}{PARA 13 "" 1 "" {GLPLOT2D 420 219 219 {PLOTDATA 2 "6%-%'CURVESG6$7,7$$\"\"!F)F(7$$\"3++++++++D!#=$!3++++ ++]_A!#@7$$\"3++++++++]F-$!3y***********z^%F07$$\"3++++++++vF-$!3;++++ +](z'F07$$\"\"\"F)$!31+++++]!4*F07$$\"3+++++++]7!#<$!30+++++tR6!#?7$$ \"3++++++++:FC$!3++++++ur8FF7$$\"3+++++++]+%=FF7$$\"3+++++++]AFC$!3/+++++Cw?FF-%'COLOURG6& %$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q!6\"F_o-%%VIEWG6$%(DEFAULTGFdo" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In class project: P186, Problem 1(a)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}}{MARK "24 1 0" 39 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 146194944 146194984 } {RTABLE M7R0 I6RTABLE_SAVE/146194944X*%)anythingG6"6"[gl!#%!!!"+"+$""!F($"+cbbbb!#5$"+66666! "*$"+nmmm;F.$"+AAAAAF.$"+yxxxFF.$"+MLLLLF.$"+*)))))))QF.$"+XWWWWF.$"+++++]F.F& } {RTABLE M7R0 I6RTABLE_SAVE/146194984X*%)anythingG6"6"[gl!#%!!!"+"+$"#D""!$"+LLLLD!")$"+*)))) oc#F,$"+["o1g#F,$"+-EnMEF,$"+`P!*oEF,$"+6JO.FF,$"+)>_!QFF,$"+YD(Hx#F,$"+'pD"3GF ,F& }