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We begin by demonstrating how Maple can be used to analytical ly find a solution and how Maple can draw direction fields to graphica lly analyze the solution. We then show how Maple can analyze populatio n models." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Finding the Analytic Solution of a Differ ential Equation" }}{PARA 0 "" 0 "" {TEXT -1 159 "Maple can quickly fin d the analytic solution for many different types of differential equat ions. We demonstrate how this can be done in the following examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Suppose we want to find an analytic solut ion to the differential equation " }{TEXT 257 12 "y' = x - y. " } {TEXT -1 48 "That, is we want to find a function of the form " }{TEXT 258 1 "y" }{TEXT -1 3 " = " }{TEXT 259 1 "y" }{TEXT -1 1 "(" }{TEXT 261 1 "x" }{TEXT -1 1 ")" }{TEXT 260 1 " " }{TEXT -1 147 "that satifie s this differential equation. We first enter and store this differenti al equation by entering the following command. After issuing the " } {TEXT 272 7 "restart" }{TEXT -1 82 " command to unassign all variables , we enter and store this differential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "de1 := diff(y(x), x) = x - y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"yG6#%\"xGF,,&F,\"\"\"F)!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Note how in the previous command h ow the solution variable, " }{TEXT 262 1 "y" }{TEXT -1 63 ", must be e xpressed as a function of its independent variable, " }{TEXT 263 1 "x " }{TEXT -1 24 ", that is, we must type " }{TEXT 264 4 "y(x)" }{TEXT -1 18 ". Recall that the " }{TEXT 265 5 "diff " }{TEXT -1 87 "command \+ is used to take the derivative of a function with respect to a given v ariable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "To find the solution of the differential equation, we use the " } {TEXT 266 7 "dsolve " }{TEXT -1 105 "command. The following command wi ll find a general solution of the differential equation we stored abov e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(de1, y(x));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(\"\"\"!\"\"F'F)*&-%$e xpG6#,$F'F*F)%$_C1GF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "The _ C1 in the previous answer is the arbitrary constant used in expressing the general solution. Maple can also quickly find the particular solu tion. Suppose we want to solve the differential equation " }{TEXT 267 11 "y' = x - y " }{TEXT -1 23 "with initial condition " }{TEXT 268 1 " y" }{TEXT -1 121 "(0) = 5. The following command finds this particular solution (note how the stored differential equation in the variable \+ " }{TEXT 269 3 "de1" }{TEXT -1 61 " and the initial condition are grou ped together in brackets)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dsolve([de1, y(0) = 5], y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,(\"\"\"!\"\"F'F)*&\"\"'F)-%$expG6#,$F'F*F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 51 "Suppose we want to solve the initia l value problem " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dy/dt;" "6#*&%#dyG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "(y^2-y)*sin(t);" "6#*&,&*$%\"yG\"\"#\"\"\"F&!\"\"F(-%$s inG6#%\"tGF(" }{TEXT -1 3 ", " }{TEXT 270 1 "y" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "pi/2;" "6#*&%#piG\"\"\"\"\"#!\"\"" }{TEXT -1 5 ") = 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "The fo llowing command stores this differential equation. Note this time the \+ solution variable is expressed as a function of " }{TEXT 271 1 "t" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "de2 := diff(y(t), t) = (y(t )^2 - y(t))*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/-%%diff G6$-%\"yG6#%\"tGF,*&,&*$)F)\"\"#\"\"\"F2F)!\"\"F2-%$sinGF+F2" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The following command gives the ge neral solution of this differential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(de2, y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&\"\"\"F),&F)F)*&-%$expG6#,$-%$cosGF&!\"\"F)%$_C1 GF)F)F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "This would be the part icular solution." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "dsolve( [de2, y(Pi/2) = 4], y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6 #%\"tG,$*&\"\"%\"\"\",&F*!\"\"*&\"\"$F+-%$expG6#,$-%$cosGF&F-F+F+F-F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 50 "Examining Solutions Graphically - Direction Fields " }}{PARA 0 "" 0 "" {TEXT -1 255 "Maple can quickly generate direction fields. Direction fields can be displayed, with or without solution c urves. To start, we must read in a special package known as the DETOOL S package, which has routines specifically designed for differential e quations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ir%+AreSimilarG%)DEnormalG%'DEpl otG%)DEplot3dG%/DEplot_polygonG%(DFactorG%,DFactorLCLMG%,DFactorsolsG% +DchangevarG%6FunctionDecompositionG%%GCRDG%'GosperG%)HeunsolsG%.Homom orphismsG%3IsHyperexponentialG%%LCLMG%,MeijerGsolsG% " 0 "" {MPLTEXT 1 0 32 "de1 := diff(y (x), x) = x - y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de1G/-%%diff G6$-%\"yG6#%\"xGF,,&F,\"\"\"F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "The command that is used to plot directions fields is the " } {TEXT 274 10 "dfieldplot" }{TEXT -1 106 " command. The following comma nd will generate a basic direction field plot for this differential eq uation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dfieldplot(de1, \+ y(x), x = -2..2, y = -2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6%-%'CURVESG6^dl7%7$$!3sOZ*y:j_5#!#<$!\"#\"\"!7$$!3Ej _5Uot%*=F*F+7$$!3dq\\^ErJS>F*$!3s:j_5Uot>F*7%7$$!3?ZatC*yC*=F*$!3;?Zz8 `o@?F*7$$!3W0coV%oko\"F*$!3%)z_?'o9$y>F*7$$!3MRIk\\B\\O'>F*7%7$$!3)eeN%=;'fn\"F*$!3Hx#p_5[3/#F*7$$!3U>lS=J$>[\"F*$!3%HsIZ*=: f>F*7$$!3,zYCZN:M:F*$!33F*7%7$$!3/w&HX))>uX\"F*$!3)f:n&)y4i0# F*7$$!3#>eL2AA%z7F*$!3.WGV6-zV>F*7$$!3-u2a#H7?L\"F*$!3O*fye0!)e%>F*7%7 $$!3!G&\\10'oxPx5F*$!3IBgGPi>K>F*7$$!3? 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The second parameter " }{TEXT 276 4 " y(x)" }{TEXT -1 102 " is the solution variable, which must be expresse d in function notation. The third variable gives the " }{TEXT 277 2 "x -" }{TEXT -1 74 "axis range of the graph (here from -2..2). The fourth parameter gives the " }{TEXT 278 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 279 2 "y-" }{TEXT -1 36 "axis range of the graph (from -2..2) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "By s pecifying additional parameters, we can provide added details to the g raph, as can be seen with the following command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "dfieldplot(de1, y(x), x = -2..2, y = -2..2, \+ color = blue, arrows = MEDIUM, dirgrid = [30,30], title = \"Direction \+ Field Plot of y' = x - y\");" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6&-%'CURVESG6bcn7'7$$!3Wz8CF*F+7$$!3%[q;xU(*3'>F*$!3vW.Jz8CF*F.7'7$ $!3!pv&3kwQI>F*$!3Jr!o(oKU4?F*7$$!3uD([pE]Pz\"F*$!3!*G>BJnd!*>F*7$$!3& 4i8SJx4#=F*$!3*o(R%4!ot6?F*7$$!3hcwR[*)oD=F*$!35>(f;Xxv(>F*F?7'7$$!33O fc!=?1z\"F*$!3cG_!4$*R$=?F*7$$!3AHI]\"obwl\"F*$!3WrZ4p+m\")>F*7$$!3S4U Fd$e=o\"F*$!3WC9^a?A1?F*7$$!3oBosA$G5p\"F*$!3s(pX(H4)H(>F*FT7'7$$!3eS% 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The " }{TEXT 281 7 "arrows " }{TEXT -1 131 "parameter specifies the thickness and l ook of the direction field lines (this parameter can be LINE, THIN, ME DIUM, and LARGE). The " }{TEXT 282 7 "dirgrid" }{TEXT -1 134 " paramet ers specifies the spacing between and the number of direction field li nes to be drawn (the default is [20, 20]). Finally, the " }{TEXT 283 6 "title " }{TEXT -1 44 "option allows one to give the graph a title. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 254 "We can also plot the \+ direction field of a differential equation together with several diffe rent selected solutions. This is especially important when an analytic solutions does not exist for the differential equation. To demonstrat e, consider the example" }{XPPEDIT 18 0 "dy/dt;" "6#*&%#dyG\"\"\"%#dtG !\"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "t^2+y^2;" "6#,&*$%\"tG\"\"# \"\"\"*$%\"yGF&F'" }{TEXT -1 48 ". We enter and store this differentia l equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "de2 := diff(y (t), t) = t^2 + y(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de2G/-%% diffG6$-%\"yG6#%\"tGF,,&*$)F,\"\"#\"\"\"F1*$)F)F0F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "As the next " }{TEXT 309 7 "dsolve " } {TEXT -1 92 "command illustrates, a clean general solution for this di fferential equation does not exist." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(de2,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"tG,$*(F'\"\"\",&*&%$_C1GF*-%(BesselJG6$#!\"$\"\"%,$*&\"\"# !\"\"F'F6F*F*F*-%(BesselYGF0F*F*,&*&F-F*-F/6$#F*F3F4F*F*-F9F=F*F7F7" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "However, the " }{TEXT 310 7 "DEp lot " }{TEXT -1 92 "command can be used to graph the direction field w ith several specified initial conditions. 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iebmFiebmFiebm-%*THICKNESSG6#\"\"\"-F$6'777$%*undefinedGFbfbmFafbm7$$! +++++;!\"*$!3i)=&p-%)ow?F*7$$!+++++9Fffbm$!3p0o;h**GL6F*7$$!+++++7Fffb m$!3Kt[Y6GV6kFes7$$!+++++5Fffbm$!3uF\"[9\\vD]$Fes7$$!+++++!)!#5$!3r'QL (4=!4u\"Fes7$$!+++++gF[hbm$!3#p%HTN+7XsFfy7$$!+++++SF[hbm$!3x2L:Q!4g8# Ffy7$$!+++++?F[hbm$!3)pmMLpmom#F^hy7$$FiebmFiebmF^ibm7$$\"+++++?F[hbm$ \"3)pmMLpmom#F^hy7$$\"+++++SF[hbm$\"3x2L:Q!4g8#Ffy7$$\"+++++gF[hbm$\"3 #p%HTN+7XsFfy7$$\"+++++!)F[hbm$\"3r'QL(4=!4u\"Fes7$$\"+++++5Fffbm$\"3u F\"[9\\vD]$Fes7$$\"+++++7Fffbm$\"3Kt[Y6GV6kFes7$$\"+++++9Fffbm$\"3p0o; h**GL6F*7$$\"+++++;Fffbm$\"3i)=&p-%)ow?F*FafbmFafbm777$$FjhbmFffbm$!3q BSSDLf[TF*7$$!+++++=Fffbm$!3q!>R;0#)=#=F*7$Fdfbm$!3S;\\g+pY!*))Fes7$Fj fbm$!36s9tQ@)pg$Fes7$F_gbm$!3gO\"**HOn)y8Ffy7$Fdgbm$\"3vM6CzepDBFes7$F igbm$\"3')[T.B<^uTFes7$F_hbm$\"3%RNR\"4)H?l&Fes7$Fdhbm$\"3Iii`&*f5epFe s7$Fihbm$\"3(f)4tB?&)3$)Fes7$F^ibm$F]fbmFiebm7$F`ibm$\"3wus!)33*HD\"F* 7$Feibm$\"3_!R.1#)>fp\"F*7$Fjibm$\"3w>HeA(4@k#F*FafbmFafbmFafbmFafbmFa fbmFafbmFafbm77FafbmFafbmFafbmFafbmFafbmFafbmFafbm7$F_hbm$!3w>HeA(4@k# F*7$Fdhbm$!3_!R.1#)>fp\"F*7$Fihbm$!3wus!)33*HD\"F*7$F^ibm$!\"\"Fiebm7$ F`ibm$!3(f)4tB?&)3$)Fes7$Feibm$!3Iii`&*f5epFes7$Fjibm$!3%RNR\"4)H?l&Fe s7$F_jbm$!3')[T.B<^uTFes7$Fdjbm$!3vM6CzepDBFes7$Fijbm$\"3gO\"**HOn)y8F fy7$F^[cm$\"36s9tQ@)pg$Fes7$Fc[cm$\"3S;\\g+pY!*))Fes7$$\"+++++=Fffbm$ \"3q!>R;0#)=#=F*7$$FaibmFffbm$\"3qBSSDLf[TF*-%&STYLEG6#%%LINEG-F[fbm6# \"\"$-%'COLOURG6&%$RGBGF^ibmF^ibm$\"*++++\"!\")-%%VIEWG6$;$!+LLLL@Fffb m$\"+LLLL@FffbmFcbcm-%+AXESLABELSG6$Q\"t6\"Q%y(t)F\\ccm" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "In the previous command, the first parameter " }{TEXT 311 3 "de2" }{TEXT -1 87 " is the differential equation to generate th e direction field of. The second parameter " }{TEXT 312 4 "y(t)" } {TEXT -1 102 " is the solution variable, which must be expressed in fu nction notation. The third variable gives the " }{TEXT 313 2 "t-" } {TEXT -1 74 "axis range of the graph (here from -2..2). The fourth par ameter gives the " }{TEXT 314 0 "" }{TEXT -1 0 "" }{TEXT 315 2 "y-" } {TEXT -1 189 "axis range of the graph (from -2..2). The initial condit ions for the selected solutions are specified in brackets, which each \+ specified in its own separate bracket separated by commas. The " } {TEXT 316 7 "dirgrid" }{TEXT -1 108 " parameters specifies the spacing between and the number of direction field lines to be drawn. Finally, the " }{TEXT 317 20 "linecolor parameter " }{TEXT -1 53 "tells Maple \+ the color to draw the selected solutions." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "Analyzing Variations \+ of the Logistic Model Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 172 "Map le can be useful in analyzing models. To demonstrate, we consider the \+ logistic model that is subject to harvesting. The differential equatio n for this model is given by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG\"\"\"%#dtG!\"\"" } {TEXT -1 2 " =" }{XPPEDIT 18 0 "k*P*(1-P/K)-c;" "6#,&*(%\"kG\"\"\"%\"P GF&,&F&F&*&F'F&%\"KG!\"\"F+F&F&%\"cGF+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 242 "Here, t = the time a \+ population grows, P or P(t) = the population after time t, k = relativ e growth rate coefficient, K = the carrying capacity, and c represents to rate per time period the population is harvested. We start by read ing in the " }{TEXT 284 8 "DEtools " }{TEXT -1 48 "package and enterin g this differential equation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtoo ls):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "de := diff(P(t),t)= k*P(t)*(1 - P(t)/K) - c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%% diffG6$-%\"PG6#%\"tGF,,&*(%\"kG\"\"\"F)F0,&F0F0*&F)F0%\"KG!\"\"F4F0F0% \"cGF4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Suppose a species of f ish in a lake is modeled by a logistic population model with relative \+ growth rate of " }{TEXT 285 1 "k" }{TEXT -1 41 " = 0.3 per year and ca rrying capacity of " }{TEXT 286 1 "K" }{TEXT -1 106 " = 10000. In addi tion, suppose 400 fish are harvested from the lake each year. We assig n these parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k := 0.3; K := 10000; c := 400;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$ \"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"&++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"$+%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The following command shows the updated differential equa tion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "de;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"PG6#%\"tGF*,&*($\"\"$!\"\"\"\"\"F'F 0,&F0F0*&#F0\"&++\"F0F'F0F/F0F0\"$+%F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "The equilibrium solutions are important for understandin g the long term behavior of the solution. Recall that the equilibrium \+ solutions are solutions where " }{XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG\"\" \"%#dtG!\"\"" }{TEXT -1 5 " = 0." }}{PARA 0 "" 0 "" {TEXT -1 14 "The f ollowing " }{TEXT 287 6 "solve " }{TEXT -1 74 "command will find the e quilbrium solutions for this differential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(rhs(de) = 0, P(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+X(\\Ve\"!\"'$\"+b-l:%)F%" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 90 "The following command yields the general analytic \+ solution for this differential equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,P(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"PG6#%\"tG,&\"%+]\"\"\"*&#\"%+5\"\"$F**&\"$0\"#F*\"\"#-%%tanhG6#,&* (\"$+\"!\"\"F0F1F'F*F**(F8F9F0F1%$_C1GF*F*F*F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "In the previous command, the " }{TEXT 288 3 "rhs" }{TEXT -1 105 " accesses the right hand side of the differential equat ion, which is set equal to 0, and then solved for " }{TEXT 289 1 "P" } {TEXT -1 1 "(" }{TEXT 290 1 "t" }{TEXT -1 71 "). Thus, the equilibrium solutions occur when there are approximately " }{TEXT 318 1 "P" } {TEXT -1 17 " = 1584 fish and " }{TEXT 319 1 "P" }{TEXT -1 167 " = 841 6 fish. We can use this information when plotting the direction field \+ and selected solutions for this differential equation. The next comman d performs this plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "D Eplot(de, P(t), t = 0..50, P = 0..12000, [[P(0) = 1500],[P(0) = 2000], [P(0) = 5000],[P(0)= 11000]],dirgrid = [30,30], linecolor=blue, title \+ = \"Logistic Equation dP/dt = kP(1-P/K) - c\");" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6(-%'CURVESG6bcn7%7$$!3D%>WkB3`V%!# =$\"3axwd%HBTx\"!#:7$$\"3D%>WkB3`V%F*$!3axwd%HBTx\"F-7$$\"3k$4v<`3GO%F *$!3)3?g6zoyR(!#;7%7$$\"3Mj+R2rg!G\"!#!*yYvow;#F=F17$$\"3u \"*>@%y=/;#F=F67%7$$\"3*faC%Q]u/IF=F+7$$\"3g%Q8do1=*QF=F17$$\"3QukC:nb %)QF=F67%7$$\"33H!f%pH))GZF=F+7$$\"3rnyu;Y%fh&F=F17$$\"3\\d4GYYp3cF=F6 7%7$$\"3I6N\\+4-`kF=F+7$$\"3#*\\ByZD3StF=F17$$\"3pRaJxD$GL(F=F67%7$$\" 3]$*z_J)er<)F=F+7$$\"37Ko\")y/Ak!*F=F17$$\"3!>#*\\$30(p0*F=F67%7$$\"3r vCcinH,**F=F+7$$\"3WJ^)4%e$)y5F8F17$$\"3US%QR%36y5F8F67%7$$\"3!epf$pMa i6F8F+7$$\"3mz&)3M'\\7D\"F8F17$$\"3k))=/PY_]7F8F67%7$$\"3-WJYis&\\L\"F 8F+7$$\"3)y-#>FMmB9F8F17$$\"3'oLX,VQHU\"F8F67%7$$\"3B#fmb0rt]\"F8F+7$$ \"35waH?s2'f\"F8F17$$\"33&y[KA_`f\"F8F67%7$$\"3kS+n[[yz;F8F+7$$\"3]C*) R85\\o_=F8F+7$$\"3!HP-l![!4%>F 8F17$$\"30#ob%4)z,%>F8F67%7$$\"3VPp([V7Y-#F8F+7$$\"3H@eg*f=L6#F8F17$$ \"3XI\"fDg$f7@F8F67%7$$\"3#eQ!)zAEq>#F8F+7$$\"3qp#4FRKdG#F8F17$$\"3')y Dm&R2]G#F8F67%7$$\"3CMQ3@+WpBF8F+7$$\"35=F\"e=Y\"eCF8F17$$\"3CFgw)=@uX #F8F67%7$$\"3j#G(=9Q&=a#F8F+7$$\"3\\mh\"*y*f0j#F8F17$$\"3kv%p=)\\$)HEF 8F67%7$$\"3-J2H2wE9FF8F+7$$\"3)[h>?xtH!GF8F17$$\"3/CH(\\x[A!GF8F67%7$$ \"3UzTR+9o')GF8F+7$$\"3GjI7lvQvHF8F17$$\"3Wsj2oDmuHF8F67%7$$\"3#yi(\\$ >&4fIF8F+7$$\"3o6lAe8!y9$F8F17$$\"3%3#)z6Owq9$F8F67%7$$\"3'e2,m)*3:B$F 8F+7$$\"3Wg*H8::-K$F8F17$$\"3fpKGa,\\>LF8F67%7$$\"3ECXqzF#RS$F8F+7$$\" 3%)3MVW*GE\\$F8F17$$\"3)zr'QZR!>\\$F8F67%7$$\"3msz!GdOjd$F8F+7$$\"3Cdo `PF/lOF8F17$$\"3Sm,\\SxJkOF8F67%7$$\"31@9\"fO]([PF8F+7$$\"3j0.kIlXPQF8 F17$$\"3z9OfL:tOQF8F67%7$$\"3Yp[,fT;@RF8F+7$$\"3.aPuB.()4SF8F17$$\"3=j qpE`94SF8F67%7$$\"3'yJ=@&zd$4%F8F+7$$\"3W-s%o6%G#=%F8F17$$\"3e60!)>\"f :=%F8F67%7$$\"3Em3h%F8F+7$$\"3iZv:'\\D&*p%F8F17$$\"3yc36*\\+))p%F8F67%7$$\"3Y6@` CJB$y%F8F+7$$\"3-'*4E*GR>([F8F17$$\"3=0V@#H97([F8F67%7$$\"3')fbj&[F<6dF-7$$\"3!GVsAl[()f&F*$\"3!H(eW?$*okDF-7$$\"3f\"GeK#G@8[F*$\"33 xo!\\)y%=e$F-7%7$$\"3ERs!e1jU;\"F=Fj^l7$$\"3/EsTo*4S))GF=Fj^l7$$\"3p3iHF2:3SF=F__l7$$\"3o$z%RW rfHRF=Fd_l7%7$$\"3,0i(y#*QDh%F=Fj^l7$$\"3z\"pI$e')GKdF=F__l7$$\"3Mw#Ha 2NPl&F=Fd_l7%7$$\"3@(o5*eonOjF=Fj^l7$$\"3+u^O*eEkX(F=F__l7$$\"3aePY1I( yP(F=Fd_l7%7$$\"3Up^%**y931)F=Fj^l7$$\"3?c'*R?Xc!=*F=F__l7$$\"3vS#)\\P 4,-\"*F=Fd_l7%7$$\"3k^'z4s_\\y*F=Fj^l7$$\"3%QTV^Cq/4\"F8F__l7$$\"3IsK& o)[h#3\"F8Fd_l7%7$$\"3Q89?l!44:\"F8Fj^l7$$\"31ioCQS)GE\"F8F__l7$$\"3_? n&*z'G]D\"F8Fd_l7%7$$\"3hh[IeGKB8F8Fj^l7$$\"3H5.NJyHN9F8F__l7$$\"3uo,1 tCWF9F8Fd_l7%7$$\"3#)4$39lOd\\\"F8Fj^l7$$\"3]ePXC;r2;F8F__l7$$\"3'phjh Ec)*f\"F8Fd_l7%7$$\"3Ce<^W/:o;F8Fj^l7$$\"3!p?dvTD,y\"F8F__l7$$\"3alqEf +FsF8F__l7$$\"3%R^qB&QoW >F8Fd_l7%7$$\"3-b'=2.yH,#F8Fj^l7$$\"3q.Tw.I&\\7#F8F__l7$$\"3LiRZXw4<@F 8Fd_l7%7$$\"3U.@#Q#=R&=#F8Fj^l7$$\"35_v'ozmtH#F8F__l7$$\"3u5udQ9^*G#F8 Fd_l7%7$$\"3#=bDph0yN#F8Fj^l7$$\"3]+5(**e!ypCF8F__l7$$\"39f3oJ_#>Y#F8F d_l7%7$$\"3A+!H+T>-`#F8Fj^l7$$\"3!*[W2$Q%>UEF8F__l7$$\"3`2VyC!RVj#F8Fd _l7%7$$\"3h[C8.Kj-FF8Fj^l7$$\"3H(*y-()HF8F__l7$$\"3K/7*4hm\"zHF8 Fd_l7%7$$\"3UX$R$*ygu/$F8Fj^l7$$\"35%z%QidVfJF8F__l7$$\"3s_Y4//e^JF8Fd _l7%7$$\"3\"QzUCeu)>KF8Fj^l7$$\"3\\U#)[b&\\=L$F8F__l7$$\"37,\")>(>%*RK $F8Fd_l7%7$$\"3?Uiav$)G#R$F8Fj^l7$$\"3*3p\"f[LE/NF8F__l7$$\"3_\\:I!*zS '\\$F8Fd_l7%7$$\"3i!p\\'o@qkNF8Fj^l7$$\"3IR^pTrnwOF8F__l7$$\"3#z*\\S$y @)oOF8Fd_l7%7$$\"3,RJvhf6PPF8Fj^l7$$\"3o(e)zM44\\QF8F__l7$$\"3KY%3ldN7 %QF8Fd_l7%7$$\"3S(ec[vH&4RF8Fj^l7$$\"33O?!zs/:-%F8F__l7$$\"3r%*=hp$\\O ,%F8Fd_l7%7$$\"3!e.gzaV>3%F8Fj^l7$$\"3[%[05_=R>%F8F__l7$$\"37V`riJ1'=% F8Fd_l7%7$$\"3?%[j5MdVD%F8Fj^l7$$\"3)G$*3TJKjO%F8F__l7$$\"3_\"z=e&pZeV F8Fd_l7%7$$\"3gKp;M6xEWF8Fj^l7$$\"3G\"Q7s5Y(QXF8F__l7$$\"3\"*RA#*[2*3` %F8Fd_l7%7$$\"3+\"Qqs#\\=*f%F8Fj^l7$$\"3nHeJ+*f6r%F8F__l7$$\"3J)oD?a/L q%F8Fd_l7%7$$\"3SHQP?()frZF8Fj^l7$$\"33y#>MptN)[F8F__l7$$\"3sO\"H^L=d( [F8Fd_l7%7$$\"3!yFxM^7S%\\F8Fj^l7$$\"3[EF_'[()f0&F8F__l7$$\"35&eK#G@8[ ]F8Fd_l7%7$$!3mJ&)o`lH.qF*$\"3YP'*p5xK#[*F-7$$\"3mJ&)o`lH.qF*$\"3U(RJs U'RpqF-7$$\"3sLkHM_^F_F*$\"3-J,sn\"4?,)F-7%7$$\"3\\Hcmv#3Q-\"F=F\\bm7$ $\"3fNLS'enWU#F=Fabm7$$\"3!e7kWX*)oC#F=Ffbm7%7$$\"3#>6+n?Yzu#F=F\\bm7$ $\"3q=yV'F=F\\bm7$ $\"3)Ry1&z8)of(F=Fabm7$$\"3kuvcZKI>uF=Ffbm7%7$$\"3UfN!)***f.#zF=F\\bm7 $$\"3Jl7a5$>5K*F=Fabm7$$\"33b?gy6WV\"*F=Ffbm7%7$$\"3_U!Q3$z\\W'*F=F\\b m7$$\"3vuv:Cd^/6F8Fabm7$$\"3u`O'4\"zv'3\"F8Ffbm7%7$$\"3[_s='ejo8\"F8F \\bm7$$\"3)H-hs^HpF\"F8Fabm7$$\"3'>5nSqr\"f7F8Ffbm7%7$$\"3p+2HztF48F8F \\bm7$$\"3?rWO5LM\\9F8Fabm7$$\"3<]0<(\\&eJ9F8Ffbm7%7$$\"3#*[TRs6p\"[\" F8F\\bm7$$\"3f>zY.rv@;F8Fabm7$$\"3d)*RF!H**Rg\"F8Ffbm7%7$$\"39(f(\\l\\ 5a;F8F\\bm7$$\"3+o8d'*3<%z\"F8Fabm7$$\"3)pWxL38kx\"F8Ffbm7%7$$\"3aX5ge (=l#=F8F\\bm7$$\"3S;[n*o%em>F8Fabm7$$\"3Q&*3[wo#)[>F8Ffbm7%7$$\"3%R\\/ F8F\\bm7$$\"3zk#yF[)**Q@F8Fabm7$$\"3wVVep1C@@F8Ffbm7%7$$\"3MUz! 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$$\"+++++SFf[_m$\"3k'Gyi+[mQ)F`en7$$\"++++]UFf[_m$\"3'H1yD/Q#)R)F`en7$ $\"+++++XFf[_m$\"34RRA%[._S)F`en7$$\"++++]ZFf[_m$\"3(HFxg2&Q4%)F`en7$$ F[[_mFf[_m$\"3jNFK%z$*=T)F`en777$F`j^m$\"%+]Fgi^m7$Fdj^m$\"3avLJ&\\;j& eF`en7$Fjj^m$\"37;`.#p@8h'F`en7$F_[_m$\"3ca)3:yvl?(F`en7$Fd[_m$\"3h]J# eJfej(F`en7$Fj[_m$\"3>+3.'=kf#zF`en7$F_\\_m$\"3CTdLk:^8\")F`en7$F\\^_m $\"3*3aT]()38B)F`en7$Fa^_m$\"31-n#>zhRI)F`en7$Ff^_m$\"3]%Hvxcp#[$)F`en 7$F[__m$\"3O7$zei0^P)F`en7$F___m$\"3)R/+K&=H\"R)F`en7$Fd__m$\"3x'3(\\f +.,%)F`en7$Fi__m$\"3Z')**oN+)oS)F`en7$F^`_m$\"3\"R?;\"f5R5%)F`en7$Fc`_ m$\"3(=;$QWr\\7%)F`en7$Fh`_m$\"3'f\"f7k+w8%)F`en7$F]a_m$\"3#\\8e%Gs^9% )F`en7$Fba_m$\"3-#Q-87r\\T)F`en7$Fga_m$\"3LR=a%>V_T)F`en7$F\\b_m$\"3.Z tCuiS:%)F`en777$F`j^m$\"&+5\"Fgi^m7$Fdj^m$\"39'e&\\j\\Ri(*F`en7$Fjj^m$ \"3mtNZQk9k\"*F`en7$F_[_m$\"3qr]M^]^X))F`en7$Fd[_m$\"3I38rQ])F`en7$F\\^_m$\"3%*3N:8 aDo%)F`en7$Fa^_m$\"3#e'eF0J3Z%)F`en7$Ff^_m$\"3/#\\d@IbWV)F`en7$F[__m$ \"3%4Bq<')3pU)F`en7$F___m$\"3v1_RkPRA%)F`en7$Fd__m$\"3IBr!QW!p>%)F`en7 $Fi__m$\"3O.>@072=%)F`en7$F^`_m$\"3QGk(o1,rT)F`en7$Fc`_m$\"3L(yBxt>lT) F`en7$Fh`_m$\"3LFsTg8<;%)F`en7$F]a_m$\"3,pM^uD'fT)F`en7$Fba_m$\"3a*yCI WPeT)F`en7$Fga_m$\"3b_$3mWidT)F`en7$F\\b_m$\"3*eAlw\\ " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 284 "One can see the effects of harves ting. If there are intially less than 1584 fish, harvesting will event ually deplete the entire population. If the population starts above th e equilbruim solution of 1584 fish, the population will eventually app roach the equilibrium point of 8416 fish." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises " }}{PARA 0 "" 0 "" {TEXT -1 95 "1. Find the general solution and part icular solutions for the following differential equations." }}{PARA 0 "" 0 "" {TEXT -1 12 " a. y' = " }{XPPEDIT 18 0 "y/((x^2+1)^2);" "6# *&%\"yG\"\"\"*$,&*$%\"xG\"\"#F%F%F%F*!\"\"" }{TEXT -1 11 " , y(0) = 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " b. " }{XPPEDIT 18 0 "dy/dt;" "6#*&%#dyG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(y^2+y)*t*e^t;" "6#*(,&*$%\"yG\"\"#\"\"\"F&F(F(%\"t GF()%\"eGF)F(" }{TEXT -1 12 ", y(0) = 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "2. Plot the direction field for t he following differential equation. On your graph, use the interval " }{TEXT 292 4 "x = " }{TEXT -1 5 "-3..3" }{TEXT 295 6 ", y = " }{TEXT -1 5 "-3..3" }{TEXT 296 2 ", " }{TEXT 300 5 "color" }{TEXT -1 1 " " } {TEXT 291 7 "= green" }{TEXT -1 51 " for your line color with thicknes s of arrows of " }{TEXT 293 7 "LARGE, " }{TEXT -1 4 "and " }{TEXT 294 8 "dirgrid " }{TEXT -1 10 "= [30,30]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "y' = " }{XPPEDIT 18 0 "y/((x^2+1) ^2);" "6#*&%\"yG\"\"\"*$,&*$%\"xG\"\"#F%F%F%F*!\"\"" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "3. Plot \+ the direction field for the following differential equation with the f ollowing selected solutions. On your graph, use the interval " } {TEXT 297 4 "t = " }{TEXT -1 7 "-20..20" }{TEXT 298 6 ", y = " }{TEXT -1 7 "-10..10" }{TEXT 299 2 ", " }{TEXT -1 4 "the " }{TEXT 301 14 "lin ecolor blue" }{TEXT -1 126 " for your selected line color for your sol utions. Plot solutions for the initial conditions y(0) = -2, y(0) = 0, and y(0) = 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dy/dt;" "6#*&%#dyG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "y*sin(t^2);" "6#*&%\"yG\"\"\"-%$sinG6#*$%\"tG\"\"#F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "4. Suppose that when the population falls below a minimu m population m, the \011population becomes extinct. Then this populati on can be modeled by the differential\011equation\n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "dP/dt;" "6#*&%#dPG\"\"\"%#dtG!\"\"" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "k*P*(1-P/K)*(1-m/P);" "6#**%\"kG\"\"\"%\"PGF%,&F%F%* &F&F%%\"KG!\"\"F*F%,&F%F%*&%\"mGF%F&F*F*F%" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "where t = the tim e a population grows, P or P(t) = the population after time t, " } {TEXT 307 1 "k" }{TEXT -1 37 " = relative growth rate coefficient, " } {TEXT 308 1 "K" }{TEXT -1 30 " = the carrying capacity, and " }{TEXT 302 1 "m" }{TEXT -1 110 " is the minimum population such that the popu lation becomes extinct if the size of the population falls below " } {TEXT 303 1 "m" }{TEXT -1 37 ". Suppose we have a population where " } {TEXT 304 4 "k = " }{TEXT -1 4 "0.08" }{TEXT 306 6 ", K = " }{TEXT -1 11 "10000, and " }}{PARA 0 "" 0 "" {TEXT 305 2 "m " }{TEXT -1 43 "= 10 00. Use Maple to perform the following." }}{PARA 0 "" 0 "" {TEXT -1 33 "a. Find the equilbrium solutions." }}{PARA 0 "" 0 "" {TEXT -1 38 " b. Find the general analytic solution." }}{PARA 0 "" 0 "" {TEXT -1 217 "c. Plot the direction field of this differential equation along w ith selected solutions with the intital conditions of your choice demo nstrating what happens to the population for various starting initial \+ populations." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }