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}{CSTYLE "" -1 239 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 240 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 241 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 242 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 243 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 244 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 245 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 208 "" 0 "" {TEXT 217 51 "Math 321 Differential Equatio ns: Maple Introduction" }{TEXT 217 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 387 "Maple V is a comprehensive com puter system for advanced mathematics. It includes facilities for int eractive algebra, calculus, differential equations, graphics, and nume rical computation. In this worksheet, Maple will facilitate validatio n and solution of differential equation models of United States popula tion change. US population data is given in table 1.1 on page 7 of th e text." }{TEXT 218 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 219 4 "Data " }{TEXT 219 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 204 "" 0 "" {TEXT 221 297 "Place US population data in a list and store the lis t in a variable. For the storage of data, I prefer to use a word for \+ the variable name (\"population\") rather than a single letter (\"p\") because the single letters are often useful in defining formulas. Th e assignment operator is colon-equals (" }{MPLTEXT 1 222 2 ":=" }{TEXT 221 14 ") not equals (" }{MPLTEXT 1 222 1 "=" }{TEXT 221 2 ")." } {TEXT 221 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 95 "populatio n := [3.9,5.3,7.2,9.6,12,17,23,31,38,50,62,75,91,105,122,131,151,179,2 03,226,249,281];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7 8$\"#R!\"\"$\"#`F%$\"#sF%$\"#'*F%\"#7\"#<\"#B\"#J\"#Q\"#]\"#i\"#v\"#\" *\"$0\"\"$A\"\"$J\"\"$^\"\"$z\"\"$.#\"$E#\"$\\#\"$\"G" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 28 "Access an eleme nt of a list." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 14 "population[2];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"#`!\"\"" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 216 "Create and store a list of the times. I wo uld have prefered to use \"time\" but that word is already used by Map le for timing computations. The call seq(f(i), i = m..n) generates th e sequence f(m), f(m+1), ..., f(n)." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 30 "ttime := [seq(10*i, i=0..21)];" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "78\"\"!\"#5\"#?\"#I \"#S\"#]\"#g\"#q\"#!)\"#!*\"$+\"\"$5\"\"$?\"\"$I\"\"$S\"\"$]\"\"$g\"\" $q\"\"$!=\"$!>\"$+#\"$5#" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 121 "Create and store a list of the years. The \+ call seq(f(i), i = x) generates a sequence by applying f to each eleme nt of x." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 31 "year := [seq(1790+i, i=ttime)];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "78\"%!z\"\"%+=\"%5=\"%?=\"%I=\"%S=\"%]=\"%g=\"%q=\"% !)=\"%!*=\"%+>\"%5>\"%?>\"%I>\"%S>\"%]>\"%g>\"%q>\"%!)>\"%!*>\"%+?" }} }{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 201 "C reate and store a list of points to be plotted.. Maple's zip creates \+ a list by applying the binary function in the first argument to the el ements of the lists given in the second and third arguments." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 47 "points := zip(( a,b)->[a,b], ttime, population);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "787$\"\"!$\"#R!\"\"7$\"#5$\"#`F'7$\"#?$\"#sF'7$\"#I$\"# '*F'7$\"#S\"#77$\"#]\"#<7$\"#g\"#B7$\"#q\"#J7$\"#!)\"#Q7$\"#!*F87$\"$+ \"\"#i7$\"$5\"\"#v7$\"$?\"\"#\"*7$\"$I\"\"$0\"7$\"$S\"\"$A\"7$\"$]\"\" $J\"7$\"$g\"\"$^\"7$\"$q\"\"$z\"7$\"$!=\"$.#7$\"$!>\"$E#7$\"$+#\"$\\#7 $\"$5#\"$\"G" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 151 "Plot the points. Without the style option, the points \+ would be connected with lines. The color option controls the color of the points, not the axes." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 38 "plot(points, style=point, color=blue);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6#787$$\"\"!!\"\"$\"#R!\"\"7$$\"$+\"!\"\"$\"# `!\"\"7$$\"$+#!\"\"$\"#s!\"\"7$$\"$+$!\"\"$\"#'*!\"\"7$$\"$+%!\"\"$\"$ ?\"!\"\"7$$\"$+&!\"\"$\"$q\"!\"\"7$$\"$+'!\"\"$\"$I#!\"\"7$$\"$+(!\"\" $\"$5$!\"\"7$$\"$+)!\"\"$\"$!Q!\"\"7$$\"$+*!\"\"$\"$+&!\"\"7$$\"%+5!\" \"$\"$?'!\"\"7$$\"%+6!\"\"$\"$](!\"\"7$$\"%+7!\"\"$\"$5*!\"\"7$$\"%+8! \"\"$\"%]5!\"\"7$$\"%+9!\"\"$\"%?7!\"\"7$$\"%+:!\"\"$\"%58!\"\"7$$\"%+ ;!\"\"$\"%5:!\"\"7$$\"%+!\"\"$\"%gA!\"\"7$$\"%+?!\"\"$\"%!\\#!\"\"7$$\"%+@!\"\"$\"%5G!\" \"-%+AXESLABELSG6'Q!6\"Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZO NTALG%+HORIZONTALG-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"!!\"\"$\"#5!\"\"-%& STYLEG6#%&POINTG" 1 5 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" } }{TEXT 246 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 207 "" 0 "" {TEXT 218 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 219 11 "First Model" }{TEXT 219 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 " " }}{PARA 207 "" 0 "" {TEXT 218 80 "Notation: Let P(t) be the model \+ population, in millions, t years after 1790." }{TEXT 218 0 "" }} {PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 77 "Ass umption: the rate of population change is proportional to the populati on. " }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 " " 0 "" {TEXT 218 48 "Model: P'(t) = k P(t) where k is a constant." }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 " " {TEXT 218 49 "Create and store the differential equation model." } {TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 31 "ode := di ff(P(t),t) = k * P(t);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I%diffG%*protectedG6$-I\"PG6\"6#I\"tGF)F+*&I\"kGF)\"\" \"F'F." }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 56 "Find the analytic solution to the differential equation." } {TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(od e);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"PG6\"6#I \"tGF%*&I$_C1GF%\"\"\"-I$expG6$%*protectedGI(_syslibGF%6#*&I\"kGF%F*F' F*F*" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 120 "The model solution has two undetermined constants. One appro ach to choosing these constants is to use two data points: " }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 25 " (0, 3.9) => P(0) = 3.9" }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 27 " (10, 5.3) => P(10 ) = 5.3" }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 44 "The first e quality implies that _C1 = 3.9. " }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 47 "The second equality can be used to solve for k." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 35 "solve(3.9 * ex p(k * 210) = 281, k);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+An%o.#!#6" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 94 "Define the solution function. A copy and paste wer e used to insert the calculated constant k." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 46 "solution := t -> 3.9 * exp(0.0203 6846722 * t);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6# I\"tG6\"F%6$I)operatorGF%I&arrowGF%F%*&$\"#R!\"\"\"\"\"-I$expG6$%*prot ectedGI(_syslibGF%6#*&$\"+An%o.#!#6F-9$F-F-F%F%F%" }}}{PARA 207 "" 0 " " {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 56 "Use the solution fu nction to find predicted populations." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 13 "solution(10);" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"+++++`!\"*" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 27 "Plot the solution fun ction." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 28 " plot(solution(t), t=0..210);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$ \"\"!!\"\"$\"#R!\"\"7$$\"+vBSxX!\"*$\"1y-j(H06G%!#:7$$\"-D1dEN?#!#6$\"1&4E\"o>B4h!#:7$$\"2(**\\i+pb>E!#:$\"0ha'y )*[\\m!#97$$\".DcfK.0$!#6$\"14iIvPHfs!#:7$$\".DcVTe\\$!#6$\"1)>5&=*Q)[ z!#:7$$\".DJ_@*RR!#6$\"17]&>#*\\8q)!#:7$$\",vZ7nR%!\"*$\"0WrP8$z\\&*!# 97$$\"-v)>a!*z%!#5$\"2pAI#))o`O5!#:7$$\",Do&*>D&!\"*$\"26%QJ\\\\rO6!#: 7$$\"2'****\\()pz1d!#:$\"1tdB@s/Z7!#97$$\"2/++vJ#3Xh!#:$\"2Y]Q,-%\\j8! #:7$$\".DcQ(3Vl!#6$\"1_[^%3L'y9!#97$$\",v!RN;q!\"*$\"1'>#3&Gl#G;!#97$$ \"*tssT(!\"($\"17$\\m*4\"ow\"!#97$$\".D1jOO)y!#6$\"2x(=9=A(G%>!#:7$$\" +0xW'H)!\")$\"2;-)Rn**H8@!#:7$$\".D1)oO\\()!#6$\"1Le6&HLvJ#!#97$$\".v= /`1=*!#6$\"/CY#4I.`#!#77$$\"1,+D1\\lI'*!#9$\"2MM$zOw@tF!#:7$$\"2,]7$4( *Q/5!#9$\"1D'*3]xu;I!#97$$\".v=Yj*[5!#5$\"1lXA%=cMI$!#97$$\"/DcjIE&4\" !#6$\"29cV.r^,j$!#:7$$\"/v$H+nb8\"!#6$\"1d))GEbtSR!#97$$\"2,+D;O'4z6!# 9$\"2;-;>z)31V!#:7$$\"2*****\\[k1C7!#9$\"2Yx1g*G9>Z!#:7$$\"2***\\7041o 7!#9$\"2E+H%=7bh^!#:7$$\"/D1/yi58!#6$\"2mAu7yU!Hc!#:7$$\".vy3\"*yN\"!# 5$\"0x*et5)y>'!#87$$\"2(****\\]#f.S\"!#9$\"1MV'[zvyv'!#97$$\"-vyIqX9! \"*$\"1%Q)*oR`'*\\6!#97$$\"/ Dc^Vd1P%f#>!#5$\"1()4Z*eJ7(>!#87$$\"+ J/bn>!\"($\"1n%*R)[pb9#!#87$$\"2-]iI'=\">,#!#9$\"1&fezBm%[B!#87$$\"/v$ RTrV0#!#6$\"1\\()es8hgD!#87$$\"%+@!\"\"$\"16_++++5G!#8-%&COLORG6&%$RGB G$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"%+@!\"\"%(DE FAULTG-%+AXESLABELSG6'-I#miG6#/I+modulenameG6\"I,TypesettingGI(_syslib G6\"65Q\"t6\"/%'familyGQ!6\"/%%sizeGQ#106\"/%%boldGQ&false6\"/%'italic GQ%true6\"/%*underlineGQ&false6\"/%*subscriptGQ&false6\"/%,superscript GQ&false6\"/%+foregroundGQ([0,0,0]6\"/%+backgroundGQ.[255,255,255]6\"/ %'opaqueGQ&false6\"/%+executableGQ&false6\"/%)readonlyGQ&false6\"/%)co mposedGQ&false6\"/%*convertedGQ&false6\"/%+imselectedGQ&false6\"/%,pla ceholderGQ&false6\"/%6selection-placeholderGQ&false6\"/%,mathvariantGQ 'italic6\"Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZONTALG%+HORIZO NTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 246 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 308 "We can compare the theoretical model with the actual data by \+ overlaying the two plots. This requires a function that is in a separ ate library of functions which can be included with the following call . Note the use of a colon instead of semicolon at the end of the comm and. This suppresses printed output." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{MPLTEXT 1 0 0 "" }}} {PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 246 "Di splay a plot of the model and data simultaneously. The colons suppres s the two individual plots. The assignments stored the two plots so t hat they can be referred to in the last command. Shift-Enter places a nother line in an execution block." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 45 "p1 := plot(points, style=point, color=blac k):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "p2 := p lot(solution(t), t=0..210):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "di splay([p1,p2]);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6%787$$\"\"!!\"\"$\"#R !\"\"7$$\"$+\"!\"\"$\"#`!\"\"7$$\"$+#!\"\"$\"#s!\"\"7$$\"$+$!\"\"$\"#' *!\"\"7$$\"$+%!\"\"$\"$?\"!\"\"7$$\"$+&!\"\"$\"$q\"!\"\"7$$\"$+'!\"\"$ \"$I#!\"\"7$$\"$+(!\"\"$\"$5$!\"\"7$$\"$+)!\"\"$\"$!Q!\"\"7$$\"$+*!\" \"$\"$+&!\"\"7$$\"%+5!\"\"$\"$?'!\"\"7$$\"%+6!\"\"$\"$](!\"\"7$$\"%+7! \"\"$\"$5*!\"\"7$$\"%+8!\"\"$\"%]5!\"\"7$$\"%+9!\"\"$\"%?7!\"\"7$$\"%+ :!\"\"$\"%58!\"\"7$$\"%+;!\"\"$\"%5:!\"\"7$$\"%+!\"\"$\"%gA!\"\"7$$\"%+?!\"\"$\"%!\\#!\" \"7$$\"%+@!\"\"$\"%5G!\"\"-%&COLORG6&%$RGBG$\"\"!!\"\"$\"\"!!\"\"$\"\" !!\"\"-%&STYLEG6#%&POINTG-%'CURVESG6$7S7$$\"\"!!\"\"$\"#R!\"\"7$$\"+vB SxX!\"*$\"1y-j(H06G%!#:7$$\"-D1dEN?#!#6$\" 1&4E\"o>B4h!#:7$$\"2(**\\i+pb>E!#:$\"0ha'y)*[\\m!#97$$\".DcfK.0$!#6$\" 14iIvPHfs!#:7$$\".DcVTe\\$!#6$\"1)>5&=*Q)[z!#:7$$\".DJ_@*RR!#6$\"17]&> #*\\8q)!#:7$$\",vZ7nR%!\"*$\"0WrP8$z\\&*!#97$$\"-v)>a!*z%!#5$\"2pAI#)) o`O5!#:7$$\",Do&*>D&!\"*$\"26%QJ\\\\rO6!#:7$$\"2'****\\()pz1d!#:$\"1td B@s/Z7!#97$$\"2/++vJ#3Xh!#:$\"2Y]Q,-%\\j8!#:7$$\".DcQ(3Vl!#6$\"1_[^%3L 'y9!#97$$\",v!RN;q!\"*$\"1'>#3&Gl#G;!#97$$\"*tssT(!\"($\"17$\\m*4\"ow \"!#97$$\".D1jOO)y!#6$\"2x(=9=A(G%>!#:7$$\"+0xW'H)!\")$\"2;-)Rn**H8@!# :7$$\".D1)oO\\()!#6$\"1Le6&HLvJ#!#97$$\".v=/`1=*!#6$\"/CY#4I.`#!#77$$ \"1,+D1\\lI'*!#9$\"2MM$zOw@tF!#:7$$\"2,]7$4(*Q/5!#9$\"1D'*3]xu;I!#97$$ \".v=Yj*[5!#5$\"1lXA%=cMI$!#97$$\"/DcjIE&4\"!#6$\"29cV.r^,j$!#:7$$\"/v $H+nb8\"!#6$\"1d))GEbtSR!#97$$\"2,+D;O'4z6!#9$\"2;-;>z)31V!#:7$$\"2*** **\\[k1C7!#9$\"2Yx1g*G9>Z!#:7$$\"2***\\7041o7!#9$\"2E+H%=7bh^!#:7$$\"/ D1/yi58!#6$\"2mAu7yU!Hc!#:7$$\".vy3\"*yN\"!#5$\"0x*et5)y>'!#87$$\"2(** **\\]#f.S\"!#9$\"1MV'[zvyv'!#97$$\"-vyIqX9!\"*$\"1%Q)*oR`'*\\6!#97$$\"/Dc^Vd1P%f#>!#5$\"1()4Z*eJ7(>!#87$$\"+J/bn>!\"($\"1n%*R)[pb9#!#87 $$\"2-]iI'=\">,#!#9$\"1&fezBm%[B!#87$$\"/v$RTrV0#!#6$\"1\\()es8hgD!#87 $$\"%+@!\"\"$\"16_++++5G!#8-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\" !!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"%+@!\"\"%(DEFAULTG-%+AXESLABELSG6'Q!6\" Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\"#5%+HORIZONTALG%+HORIZONTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" }}{TEXT 246 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 219 12 "Second M odel" }{TEXT 219 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 " " 0 "" {TEXT 218 80 "Notation: Let P(t) be the model population, in \+ millions, t years after 1790." }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 187 "Assumption: the rate of population change is due to two additive factors (1) proportional \+ to the population (as in the first model), and (2) proportional to the square of the population. " }{TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 27 "Model: P'(t) = k P(t) - a " }{XPPEDIT 205 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetti ng:-mrow(Typesetting:-msup(Typesetting:-mrow(Typesetting:-mi(\"P\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFu nction;\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\" , rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesett ing:-mi(\"t\", italic = \"true\", mathvariant = \"italic\")), mathvari ant = \"normal\")), Typesetting:-mn(\"2\", mathvariant = \"normal\"), \+ superscriptshift = \"0\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+mod ulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6#-I%msupGF$6% -F#6%-F,6%Q\"PF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q 0⁡F'/F=Q'normalF'/%&fenceGQ&falseF'/%*separatorGFG/%)str etchyGFG/%*symmetricGFG/%(largeopGFG/%.movablelimitsGFG/%'accentGFG/%' lspaceGQ&0.0emF'/%'rspaceGFV-I(mfencedGF$6$-F#6#-F,6%Q\"tF'F9F " 0 "" {MPLTEXT 1 0 44 "ode := diff(P(t),t) = k * P(t) - a * P(t)^2;" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I%diffG%*protected G6$-I\"PG6\"6#I\"tGF)F+,&*&I\"kGF)\"\"\"F'F/F/*&I\"aGF)F/F'\"\"#!\"\"" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 56 "Find the analytic solution to the differential equation." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(ode);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"PG6\"6#I\"tGF%* &I\"kGF%\"\"\",&I\"aGF%F**(-I$expG6$%*protectedGI(_syslibGF%6#,$*&F)F* F'F*!\"\"F*I$_C1GF%F*F)F*F*F6" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }} {PARA 207 "" 0 "" {TEXT 218 124 "The model solution has three undeterm ined constants. One approach to choosing these constants is to use th ree data points. " }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 108 "solve(\{k/(a+exp(-k*0)*_C1*k) = 3.9, k/(a+exp(-k*10) *_C1*k) = 5.3, k/(a+exp(-k*20)*_C1*k) = 7.2\}, \{k,a,_C1\});" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/I$_C1G6\"$\"+Ef.d D!#5/I\"aGF%$\"+?fcu@!#9/I\"kGF%$\"+L#Gs2$!#6" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 29 "Define the solution f unction." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 103 "solution := t -> 0.3077228233e-1/(0.2174565920e-4+exp(-0.30772282 33e-1*t)*.2557035926*0.3077228233e-1);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"tG6\"F%6$I)operatorGF%I&arrowGF%F%*&$\"+L# Gs2$!#6\"\"\",&$\"+?fcu@!#9F-*(-I$expG6$%*protectedGI(_syslibGF%6#,$*& F*F-9$F-!\"\"F-$\"+Ef.dD!#5F-F*F-F-F " 0 "" {MPLTEXT 1 0 45 "p1 := plot(points, style=point, color=black):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "p2 := plot( solution(t), t=0..210):" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 17 "display([p1,p2]);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6%7 87$$\"\"!!\"\"$\"#R!\"\"7$$\"$+\"!\"\"$\"#`!\"\"7$$\"$+#!\"\"$\"#s!\" \"7$$\"$+$!\"\"$\"#'*!\"\"7$$\"$+%!\"\"$\"$?\"!\"\"7$$\"$+&!\"\"$\"$q \"!\"\"7$$\"$+'!\"\"$\"$I#!\"\"7$$\"$+(!\"\"$\"$5$!\"\"7$$\"$+)!\"\"$ \"$!Q!\"\"7$$\"$+*!\"\"$\"$+&!\"\"7$$\"%+5!\"\"$\"$?'!\"\"7$$\"%+6!\" \"$\"$](!\"\"7$$\"%+7!\"\"$\"$5*!\"\"7$$\"%+8!\"\"$\"%]5!\"\"7$$\"%+9! \"\"$\"%?7!\"\"7$$\"%+:!\"\"$\"%58!\"\"7$$\"%+;!\"\"$\"%5:!\"\"7$$\"%+ !\"\"$\"%gA!\"\"7$$ \"%+?!\"\"$\"%!\\#!\"\"7$$\"%+@!\"\"$\"%5G!\"\"-%&COLORG6&%$RGBG$\"\"! !\"\"$\"\"!!\"\"$\"\"!!\"\"-%&STYLEG6#%&POINTG-%'CURVESG6$7S7$$\"\"!! \"\"$\"28p&o********Q!#;7$$\"+vBSxX!\"*$\"/=Y\"*G#R7r]!#:7$$\"-D'3ARI\"!#5$\"1QKuO%eu\"e!#:7$$\"-v.czaEN?#!#6$\"2D#=]')R$Gm(!#;7$$\"2(**\\i+pb>E!#:$ \"1Zrt*HtHq)!#:7$$\".DcfK.0$!#6$\"16(pmgQz#**!#:7$$\".DcVTe\\$!#6$\"29 jyG^(\\P6!#:7$$\".DJ_@*RR!#6$\"2'**=a'=HDI\"!#:7$$\",vZ7nR%!\"*$\"2aYl 8JKq\\\"!#:7$$\"-v)>a!*z%!#5$\"2lq?`w#)>p\"!#:7$$\",Do&*>D&!\"*$\"2$He $=)fcT>!#:7$$\"2'****\\()pz1d!#:$\"1VoN$eL'GA!#97$$\"2/++vJ#3Xh!#:$\"1 abzRhjWD!#97$$\".DcQ(3Vl!#6$\"1<7?JoXpG!#97$$\",v!RN;q!\"*$\"1u`Y#*Qz3 L!#97$$\"*tssT(!\"($\"1KiiAzzJP!#97$$\".D1jOO)y!#6$\"1dX6koA!H%!#97$$ \"+0xW'H)!\")$\"2a[P&3nU^[!#:7$$\".D1)oO\\()!#6$\"10\"3..E&[b!#97$$\". v=/`1=*!#6$\"1$3c48W4I'!#97$$\"1,+D1\\lI'*!#9$\"1Mx*H.H#*=(!#97$$\"2,] 7$4(*Q/5!#9$\"0xUIF`#3\")!#87$$\".v=Yj*[5!#5$\"1KF:*)egA#*!#97$$\"/Dcj IE&4\"!#6$\"2`NVNKsH0\"!#97$$\"/v$H+nb8\"!#6$\"2hB&GaDT!=\"!#97$$\"2,+ D;O'4z6!#9$\"2OU!#97$$\".v y3\"*yN\"!#5$\"1r)z,N3D;#!#87$$\"2(****\\]#f.S\"!#9$\"2-J;(RI$HT#!#97$ $\"-vyIqX9!\"*$\"1*3L$4)p^q#!#87$$\"2)*\\ilK\"z'[\"!#9$\"1V\"*))*)Rm#* H!#87$$\"-DM;rJ:!\"*$\"1$f'GmF%=L$!#87$$\"2)*\\iI9yRd\"!#9$\"1mP\">`oW n$!#87$$\"/DJB)e\"=;!#6$\"1BtC\"Q'>cS!#87$$\"-v8NNh;!\"*$\"2:X&\\]HV^W !#97$$\"/Dc^Vd1P%f#>!#5$\"0ymb\"Qa+s!#77 $$\"+J/bn>!\"($\"1rYNfWA_w!#87$$\"2-]iI'=\">,#!#9$\"1%y?CSl&G\")!#87$$ \"/v$RTrV0#!#6$\"1$4khmybd)!#87$$\"%+@!\"\"$\"1#p-XWsA/*!#8-%&COLORG6& %$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%%VIEWG6$;$\"\"!!\"\"$\"%+@!\" \"%(DEFAULTG-%+AXESLABELSG6'Q!6\"Q!6\"-%%FONTG6%%(DEFAULTG%(DEFAULTG\" #5%+HORIZONTALG%+HORIZONTALG" 1 2 2 1 10 1 2 6 0 4 2 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" }}{TEXT 246 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 74 "In an attempt to obtain a bette r fit, use a different set of three points." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 247 116 "par := solve(\{k/(a+exp(-k*0)* _C1*k) = 3.9, k/(a+exp(-k*100)*_C1*k) = 62, k/(a+exp(-k*200)*_C1*k) = \+ 249\}, \{k,a,_C1\});" }{MPLTEXT 1 247 1 "\n" }{MPLTEXT 1 247 32 "subs( par,k/(a+exp(-k*t)*_C1*k));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/I$_C1G6\"$\"+RCPID!#5/I\"aGF%$\"+`()p25!#8/I\"kGF%$\"+ !RLv)H!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*$,&$\"+`()p25!#8\"\"\"-I $expG6$%*protectedGI(_syslibG6\"6#,$I\"tGF.$!+!RLv)H!#6$\"+^@dfv!#7!\" \"$\"+!RLv)HF4" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 247 90 "solut ion := t -> 0.2987533390e-1/(0.1007698753e-3+0.7559572151e-2*exp(-0.29 87533390e-1*t))" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 247 1 "\n" }{MPLTEXT 1 247 45 "p1 := plot(points, style=point, color=black):" }{MPLTEXT 1 247 1 "\n" }{MPLTEXT 1 247 34 "p2 := plot(solution(t), t=0..210):" } {MPLTEXT 1 247 1 "\n" }{MPLTEXT 1 247 17 "display([p1,p2]);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"tG6\"F%6$I)operatorGF %I&arrowGF%F%*&$\"+!RLv)H!#6\"\"\",&$\"+`()p25!#8F-*&$\"+^@dfv!#7F--I$ expG6$%*protectedGI(_syslibGF%6#,$*&F*F-9$F-!\"\"F-F-F?F%F%F%" }} {PARA 13 "" 1 "" {TEXT 246 0 "" }{GLPLOT2D 400 400 400 {PLOTDATA 2 "6& -%'CURVESG6%787$$\"\"!!\"\"$\"#R!\"\"7$$\"$+\"!\"\"$\"#`!\"\"7$$\"$+#! \"\"$\"#s!\"\"7$$\"$+$!\"\"$\"#'*!\"\"7$$\"$+%!\"\"$\"$?\"!\"\"7$$\"$+ &!\"\"$\"$q\"!\"\"7$$\"$+'!\"\"$\"$I#!\"\"7$$\"$+(!\"\"$\"$5$!\"\"7$$ \"$+)!\"\"$\"$!Q!\"\"7$$\"$+*!\"\"$\"$+&!\"\"7$$\"%+5!\"\"$\"$?'!\"\"7 $$\"%+6!\"\"$\"$](!\"\"7$$\"%+7!\"\"$\"$5*!\"\"7$$\"%+8!\"\"$\"%]5!\" \"7$$\"%+9!\"\"$\"%?7!\"\"7$$\"%+:!\"\"$\"%58!\"\"7$$\"%+;!\"\"$\"%5:! \"\"7$$\"%+!\"\"$\" %gA!\"\"7$$\"%+?!\"\"$\"%!\\#!\"\"7$$\"%+@!\"\"$\"%5G!\"\"-%&COLORG6&% $RGBG$\"\"!!\"\"$\"\"!!\"\"$\"\"!!\"\"-%&STYLEG6#%&POINTG-%'CURVESG6$7 S7$$\"\"!!\"\"$\"11Xm********Q!#:7$$\"+vBSxX!\"*$\"1ota+[\"HY%!#:7$$\" -D1dEN?#!#6$\"1&Q)oqLwTu!#:7$$\"2(**\\i+pb>E !#:$\"1/Gn\\;u)R)!#:7$$\".DcfK.0$!#6$\"1/;=**eB:&*!#:7$$\".DcVTe\\$!#6 $\"2QFI1gR?3\"!#:7$$\".DJ_@*RR!#6$\"2lS!3&p(=H7!#:7$$\",vZ7nR%!\"*$\"2 m>8SDF/S\"!#:7$$\"-v)>a!*z%!#5$\"2'[s8JN#)p:!#:7$$\",Do&*>D&!\"*$\"2#) f$3Wqg$y\"!#:7$$\"2'****\\()pz1d!#:$\"2jhReR`a-#!#:7$$\"2/++vJ#3Xh!#:$ \"28xCPC`pG#!#:7$$\".DcQ(3Vl!#6$\"1XX4ak'3b#!#97$$\",v!RN;q!\"*$\"29yg )ydP+H!#:7$$\"*tssT(!\"($\"0(p>KJBHK!#87$$\".D1jOO)y!#6$\"1^T]k$GDl$!# 97$$\"+0xW'H)!\")$\"1bwbo<>mS!#97$$\".D1)oO\\()!#6$\"17r%*3KkkX!#97$$ \".v=/`1=*!#6$\"14)eiq-Z3&!#97$$\"1,+D1\\lI'*!#9$\"1UZf@EGwc!#97$$\"2, ]7$4(*Q/5!#9$\"2OFFml]XE'!#:7$$\".v=Yj*[5!#5$\"0n^Lbax%p!#87$$\"/DcjIE &4\"!#6$\"1s7Z*Rr.r(!#97$$\"/v$H+nb8\"!#6$\"1W**y[(yoT)!#97$$\"2,+D;O' 4z6!#9$\"1$\\<1D*>A#*!#97$$\"2*****\\[k1C7!#9$\"2BL^())pm45!#97$$\"2** *\\7041o7!#9$\"165'y%*>*)4\"!#87$$\"/D1/yi58!#6$\"2&**G=8u@)=\"!#97$$ \".vy3\"*yN\"!#5$\"2E&*\\_[3+H\"!#97$$\"2(****\\]#f.S\"!#9$\"2Kp,`b.JQ \"!#97$$\"-vyIqX9!\"*$\"1Nv,.$fL[\"!#87$$\"2)*\\ilK\"z'[\"!#9$\"2%=N$z xAUd\"!#97$$\"-DM;rJ:!\"*$\"1=`e[)\\Fn\"!#87$$\"2)*\\iI9yRd\"!#9$\"1 \\/&)oc$Rw\"!#87$$\"/DJB)e\"=;!#6$\"2l\\#*>zLp&=!#97$$\"-v8NNh;!\"*$\" 2_nz()e:\\%>!#97$$\"/Dc^Vd1V-IL?!#87$$\"2-++IRF,v\"!#9$\"2 j[\"\\6NM9@!#97$$\"2,+v$omm%z\"!#9$\"0d)HS=q#>#!#77$$\"/D1Or$)Q=!#6$\" 2LFX]cOcE#!#97$$\"2-++&3_Uz=!#9$\"1A++68LGB!#87$$\".v)>P%f#>!#5$\"2nc' 4Md.&R#!#97$$\"+J/bn>!\"($\"17vB/R1]C!#87$$\"2-]iI'=\">,#!#9$\"2MLf= \\ " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 207 "" 0 "" {TEXT 218 0 "" }} {SECT 0 {PARA 206 "" 0 "" {TEXT 219 27 "Important Notes About Maple" } {TEXT 219 0 "" }}{PARA 207 "" 0 "" {TEXT 218 66 "Maple attempts to car ry out manipulation symbolically and exactly." }{TEXT 218 0 "" }} {EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 28 "solve(x^2 + 3*x + 1 = 0, x );" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 52 "Maple can be forced to use numerical approximat ions." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 29 "f solve(x^2 + 3*x + 1 = 0, x);" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 111 "Sometimes exact answ ers are impossible to express in notation seen in high school or under graduate mathematics." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 23 "solve(exp(x) = 4*x, x);" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 149 "We can p erhaps understand the numeric approximation better, but here only one \+ of the two solutions is reported (remember how Newton's method works?) ." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 24 "fsolv e(exp(x) = 4*x, x);" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 40 "The following works better \+ in this case." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 30 "evalf(solve(exp(x) = 4*x, x));" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 53 "Sometimes you may assign a number to a variable . . ." }{TEXT 218 0 "" }} {EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 7 "x := 9;" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 75 ". . . and then want to use the variable as if it were not assigned a \+ number" }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 28 " solve(x^2 + 3*x + 1 = 0, x);" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 35 "Here is how to unassi gn a variable." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 9 "x := 'x';" }{MPLTEXT 1 0 0 "" }}}{PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 49 "Now the command works the way we \+ had intended it." }{TEXT 218 0 "" }}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 28 "solve(x^2 + 3*x + 1 = 0, x);" }{MPLTEXT 1 0 0 "" }}} {PARA 207 "" 0 "" {TEXT 218 0 "" }}{PARA 207 "" 0 "" {TEXT 218 227 "Cl ick on Help | New User's Tour to obtain more of an overview of the typ es of computations that Maple can perform. Click on Help | Topic Sear ch when you want to find details about the syntax and capabilities of \+ Maple commands." }{TEXT 218 0 "" }}}{EXCHG {PARA 215 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 217 "" 0 "" {TEXT 223 0 "" }}{PARA 216 "" 0 "" {TEXT 248 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }