{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 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 {EXCHG {PARA 256 "" 0 "" {TEXT -1 19 "Getting Started wit" } {TEXT 256 0 "" }{TEXT -1 33 "h Differential Equations in Maple" }} {PARA 19 "" 0 "" {TEXT -1 14 "September 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "In this Maple session, we see some of the basic tools for working with differential equation s in Maple. We will only deal with first order equations here." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "First, we need to load the \"DEtools\" library:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Let's define a differential equation. As an example, we consider " }}{PARA 0 "" 0 "" {TEXT -1 15 " y' = y(4-y)." }}{PARA 0 "" 0 "" {TEXT -1 101 "This is a first order ordinary differential equation. ( It is Problem 11 of Section 1.1 in the text.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The following command def ines a variable called \"eq\" that holds the differential equation:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eq := diff(y(t),t) = y(t)*(4-y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/-%%diffG6$-%\"yG6#%\"tGF,*&F)\"\"\",&\"\"%F.F)! \"\"F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "A few points:" }}{PARA 0 "" 0 "" {TEXT -1 63 " 1. The derivative of y is specified with the \"diff\" command." }}{PARA 0 "" 0 "" {TEXT -1 9 " 2. We " }{TEXT 257 7 "can not" }{TEXT -1 120 " drop the \"(t)\" from the dependent va riable y. Maple treats \"y\" and \"y(t)\" differently, and our equati on is for \"y(t)\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 46 "Two useful commands are \"DEplot\" and \"dsolve\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 259 28 "Direction Fields with DEplot" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 380 "Let's create a direction field for this \+ equation. One Maple command that does this is called \"DEplot\". In \+ the first example given here, the first argument is the differential e quation, the second is the unknown function in the differential equati on, the third is the range of the independent variable to show in the \+ plot, and the fourth is the range of the dependent variable. 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ELSG6$Q\"t6\"Q%y(t)Fbj\\l-%&TITLEG6#%0Direction~FieldG" 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 -1 294 "Notice that a quick glance at the diection fie ld let's us predict how the solutions will behave. Before proceeding, print a copy of the above direction field, and sketch the solution fo r which y(0)=1. Be sure to consider negative t as well as positive. \+ Also sketch the equilibrium solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 33 "How to print the direction field:" }}{PARA 259 "" 0 "" {TEXT -1 61 "(1) In the menu bar, select Options - > Plot Display -> Window" }}{PARA 260 "" 0 "" {TEXT -1 109 "When you s elect this option, subsequent plots will appear in a separate window, \+ rather than in Maple session." }}{PARA 261 "" 0 "" {TEXT -1 153 "(2) M ove the cursor back to the previous DEplot command and hit enter. Thi s will execute the command again, and now it will appear in a separate window." }}{PARA 262 "" 0 "" {TEXT -1 64 "(3) Click on the printer ic on, or use the menu bar File -> Print" }}{PARA 263 "" 0 "" {TEXT -1 210 "Note that whether you use the icon or the pulldown menu, the \"ac tive\" window is printed. If you have just created a plot, it will be the active window. Otherwise, click on the window first before print ing it." }}{PARA 257 "" 0 "" {TEXT -1 127 "(4) You may want to restore the default plot output to Inline by using the menu bar to select Opt ions -> Plot Display -> Inline" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 357 "Let's add a solution curve to thi s plot. To do this, we must specify an initial condition. Let's try \+ y(0)=1. To specify initial conditions in DEplot, you put them in a li st, contained in square brackets, just after the t range. The added o ption \"linecolor=blue\" tells the command to draw the solution curve \+ in black. 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This is because you can enter several initial conditi ons; each must be in square brackets. 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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 " " {TEXT -1 133 "Observe how the arrows drawn in the direction field ar e tangent to the solution curves--that is the whole point of a directi on field!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Important Note: The solution curves plotted by the DEplo t command have been computed " }{TEXT 258 11 "numerically" }{TEXT -1 167 ", using an algorithm similar to Euler's Method. They are genera lly good approximations, but they are not exact, and they may occasion ally be very bad approximations." }}}{EXCHG {PARA 18 "" 0 "" {TEXT 260 18 "The Dsolve Command" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The command \"dsolve\" can solve some differential equati ons " }{TEXT 261 12 "analytically" }{TEXT -1 41 ". Let's start out wi th a simple example:" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ y' = -y." }}{PARA 0 "" 0 "" {TEXT -1 104 "(If you haven't done so befo re, solve this by hand before continuing. It is both linear and separ able.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "First, we'll define the equation, and save it in a Maple variable cal led \"eq2\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eq2 := diff (y(t),t) = -y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/-%%diffG6 $-%\"yG6#%\"tGF,,$F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Now we'll use \"dsolve\" to find the general solution. The first argumen t of \"dsolve\" is the equation, and the second is the function to so lve for:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(eq2,y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&%$_C1G\"\"\"-%$e xpG6#,$F'!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 212 "This should \+ look familiar! Notice that Maple has called the arbitrary constant \" _C1\". This is a common Maple convention; when it creates a constant, it typically puts an underscore in the beginning of the name." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "We can a lso use \"dsolve\" to solve an initial value problem. For example, su ppose we have the initial condition y(0)=2. We specify this in the co mmand by combining the differential equation with the initial conditio n in \"curly brackets\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dsolve(\{eq2,y(0)=2\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" yG6#%\"tG,$-%$expG6#,$F'!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "Look at that example carefully. The first argument of dsolve \+ is \{eq2,y(0)=2\}. In Maple, curly brackets are commonly used to crea te lists of things. In this case, we make a list of the equation and the initial condition, and pass this to \"dsolve\" as its first argum ent. The second argument, y(t), stays the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 480 "How do we plot this solu tion? Unfortunately, the solution given above is not in a format that is easily plotted by the \"plot\" command. The plot command plots ex pressions, but the above solution is actually an equation (it has an e quals sign sitting the middle). The expression that we want, namely 2 *exp(-t), is the right-hand-side of the equation. Fortunately, there \+ is a command in Maple that lets us take the right-hand-side from an eq uation. The function is called \"rhs\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "So here is one way to plot the so lution above. First, save the solution in a variable, say \"sol2\" (b ecause it solves eq2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s ol2 := dsolve(\{eq2,y(0)=2\},y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%sol2G/-%\"yG6#%\"tG,$-%$expG6#,$F)!\"\"\"\"#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 79 "Now take out the right-hand-side, and save it in a new variable called rhsSol2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rhsSol2 := rhs(sol2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(rhsS ol2G,$-%$expG6#,$%\"tG!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Now plot it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(rh sSol2,t=-1..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 255 202 202 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$\"3=!4=pljlV&!#<7$$!3[*****\\P&3Y$*!#=$ \"3QMInGIV#4&F-7$$!3C++Dcx6x()F1$\"3^nTmn'y2\"[F-7$$!3b++]iTDP\")F1$\" 3'*4Z6'*ff7XF-7$$!3A****\\P\"\\J\\(F1$\"3@CFRr.5JUF-7$$!3g***\\7V0@&oF 1$\"3U>51)4z$oRF-7$$!3w++DcexdiF1$\"3+VL<([)RRPF-7$$!3j***\\i+#QUcF1$ \"31jo\"Q*e@;NF-7$$!3$****\\i!3%f+&F1$\"3gYv7q?S*H$F-7$$!3;++D\"oS:P%F 1$\"33G^0.#*e'4$F-7$$!3h*****\\<#)*=PF1$\"3\"*f5^z1(4!HF-7$$!3#*****\\ (G3U9$F1$\"3tT#o=&=$*QFF-7$$!3Y*****\\-\\r\\#F1$\"3UL%fGz=tc#F-7$$!3?+ ++vGVZ=F1$\"3'=l9r>>eS#F-7$$!3_*****\\(4J@7F1$\"3m#ya)QW!)fAF-7$$!3;,+ ]iIKFl!#>$\"36&y)\\*G,\\8#F-7$$\"3(R,++]siL#!#?$\"3Q&3@*3HL&*>F-7$$\"3 K,+++!R5'fFbp$\"3QoH!>0jU)=F-7$$\"3!)***\\P/QBE\"F1$\"3/8'R4\\jYY\"F-7$$\"3#4++vVy!ePF1$\"3kkuHh &oMP\"F-7$$\"3'4+](=WU[VF1$\"3kA'RQJLZH\"F-7$$\"3s****\\7B>&)\\F1$\"3/ i3I8*e[@\"F-7$$\"3w***\\P>:mk&F1$\"33dVV;^5P6F-7$$\"3d***\\iv&QAiF1$\" 3C%3E.6$[t5F-7$$\"3j++]PPBWoF1$\"32j#3,(>w35F-7$$\"3%*)*****\\Nm'[(F1$ \"3o)*\\$f%)Q*f%*F17$$\"36****\\(yb^6)F1$\"3GCg&*)=(o$)))F17$$\"3')*** \\PMaKs)F1$\"3s*zKhto&f$)F17$$\"3a****\\7TW)R*F1$\"3zN.nZBx8yF17$$\"3z *****\\@80+\"F-$\"3C89FnP\"QN(F17$$\"31++]7,Hl5F-$\"3OG@E)=dD*oF17$$\" 3()**\\P4w)R7\"F-$\"3sgH0yhi*\\'F17$$\"3;++]x%f\")=\"F-$\"3o3\\&[6Mc4' F17$$\"3!)**\\P/-a[7F-$\"3W@Be9oYQdF17$$\"3/+](=Yb;J\"F-$\"3')\\-&=)\\ Z(Q&F17$$\"3')****\\i@Ot8F-$\"3Yb3o043l]F17$$\"3')**\\PfL'zV\"F-$\"3eK k#=u:#[ZF17$$\"3>+++!*>=+:F-$\"3W<(3x7\"zhWF17$$\"3-++DE&4Qc\"F-$\"3/$ p5pi\"F-$\"3X/7-$*yrIRF17$$\"39+++bJ*[o\"F-$\"3%Q )o\\O\\G4PF17$$\"33++Dr\"[8v\"F-$\"3SXWN&['zqMF17$$\"3++++Ijy5=F-$\"3e C*QX-50F$F17$$\"31+]P/)fT(=F-$\"3+LGA$Qx'pIF17$$\"31+]i0j\"[$>F-$\"3AK PsBg,*)GF17$$\"\"#F*$\"30aAtkcq1FF1-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l- %+AXESLABELSG6$Q\"t6\"Q!Ff[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "You can combine several of those steps:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol2 := rhs(dsolve(\{eq2,y(0 )=2\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G,$-%$expG6#,$ %\"tG!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(sol 2,t=-1..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 299 183 183 {PLOTDATA 2 "6%- %'CURVESG6$7S7$$!\"\"\"\"!$\"3=!4=pljlV&!#<7$$!3[*****\\P&3Y$*!#=$\"3Q MInGIV#4&F-7$$!3C++Dcx6x()F1$\"3^nTmn'y2\"[F-7$$!3b++]iTDP\")F1$\"3'*4 Z6'*ff7XF-7$$!3A****\\P\"\\J\\(F1$\"3@CFRr.5JUF-7$$!3g***\\7V0@&oF1$\" 3U>51)4z$oRF-7$$!3w++DcexdiF1$\"3+VL<([)RRPF-7$$!3j***\\i+#QUcF1$\"31j o\"Q*e@;NF-7$$!3$****\\i!3%f+&F1$\"3gYv7q?S*H$F-7$$!3;++D\"oS:P%F1$\"3 3G^0.#*e'4$F-7$$!3h*****\\<#)*=PF1$\"3\"*f5^z1(4!HF-7$$!3#*****\\(G3U9 $F1$\"3tT#o=&=$*QFF-7$$!3Y*****\\-\\r\\#F1$\"3UL%fGz=tc#F-7$$!3?+++vGV Z=F1$\"3'=l9r>>eS#F-7$$!3_*****\\(4J@7F1$\"3m#ya)QW!)fAF-7$$!3;,+]iIKF l!#>$\"36&y)\\*G,\\8#F-7$$\"3(R,++]siL#!#?$\"3Q&3@*3HL&*>F-7$$\"3K,+++ !R5'fFbp$\"3QoH!>0jU)=F-7$$\"3!)***\\P/QBE\"F1$\"3/8'R4\\jYY\"F-7$$\"3#4++vVy!ePF1$\"3kkuHh&oMP \"F-7$$\"3'4+](=WU[VF1$\"3kA'RQJLZH\"F-7$$\"3s****\\7B>&)\\F1$\"3/i3I8 *e[@\"F-7$$\"3w***\\P>:mk&F1$\"33dVV;^5P6F-7$$\"3d***\\iv&QAiF1$\"3C%3 E.6$[t5F-7$$\"3j++]PPBWoF1$\"32j#3,(>w35F-7$$\"3%*)*****\\Nm'[(F1$\"3o )*\\$f%)Q*f%*F17$$\"36****\\(yb^6)F1$\"3GCg&*)=(o$)))F17$$\"3')***\\PM aKs)F1$\"3s*zKhto&f$)F17$$\"3a****\\7TW)R*F1$\"3zN.nZBx8yF17$$\"3z**** *\\@80+\"F-$\"3C89FnP\"QN(F17$$\"31++]7,Hl5F-$\"3OG@E)=dD*oF17$$\"3()* *\\P4w)R7\"F-$\"3sgH0yhi*\\'F17$$\"3;++]x%f\")=\"F-$\"3o3\\&[6Mc4'F17$ $\"3!)**\\P/-a[7F-$\"3W@Be9oYQdF17$$\"3/+](=Yb;J\"F-$\"3')\\-&=)\\Z(Q& F17$$\"3')****\\i@Ot8F-$\"3Yb3o043l]F17$$\"3')**\\PfL'zV\"F-$\"3eKk#=u :#[ZF17$$\"3>+++!*>=+:F-$\"3W<(3x7\"zhWF17$$\"3-++DE&4Qc\"F-$\"3/$p5pi\"F-$\"3X/7-$*yrIRF17$$\"39+++bJ*[o\"F-$\"3%Q)o\\ O\\G4PF17$$\"33++Dr\"[8v\"F-$\"3SXWN&['zqMF17$$\"3++++Ijy5=F-$\"3eC*QX -50F$F17$$\"31+]P/)fT(=F-$\"3+LGA$Qx'pIF17$$\"31+]i0j\"[$>F-$\"3AKPsBg ,*)GF17$$\"\"#F*$\"30aAtkcq1FF1-%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%+AX ESLABELSG6$Q\"t6\"Q!Ff[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 62 "Let's try another example. Recall that \"eq\" is \+ still defined:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "eq;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"tGF**&F'\"\"\",& \"\"%F,F'!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "This equatio n is separable, and can be solved by hand. The y integral can be done using partial fractions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 28 "Let's give this to \"dsolve\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol1 := rhs(dsolve(eq,y(t)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%sol1G,$*&\"\"\"F',&F'F'*(\"\"%F'-%$expG6#,$% \"tG!\"%F'%$_C1GF'F'!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "T his is probably simiar to what you would get if you solved it by hand. Note, however, that Maple has given no indication that the equilibri um solution y(t)=0 is also a possibility." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Let's try the initial cond ition y(0)=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol1a := r hs(dsolve(\{eq,y(0)=1\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& sol1aG,$*&\"\"\"F',&F'F'*&\"\"$F'-%$expG6#,$%\"tG!\"%F'F'!\"\"\"\"%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sol1a,t=-1..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 294 168 168 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $!\"\"\"\"!$\"3e`n/3iEFC!#>7$$!3ommm;p0k&*!#=$\"3wW(yQ*[L')GF-7$$!3wKL $3s%HaF1$\"3G'>=]?FSY\"F17$$!3Q +++]$*4)*\\F1$\"3E>CFYx$ys\"F17$$!39+++]_&\\c%F1$\"3&>\\r5%G969Fgs7$ $\"3I*******pvxl\"F1$\"3!e\"Ge$3`7d\"Fgs7$$\"3#z****\\_qn2#F1$\"3<5_P^ ooLFu?Fgs7$$\"3ElmmmZvOLF1$\"3(4f:I4**\\B#Fgs7$$\"3i******\\2goPF1 $\"3p]Ru)HFKS#Fgs7$$\"3UKL$eR<*fTF1$\"3]x>&)*Ri2b#Fgs7$$\"3m******\\)H xe%F1$\"3*\\j8]H-\\q#Fgs7$$\"3ckm;H!o-*\\F1$\"3ruoYYxtTGFgs7$$\"3y)*** \\7k.6aF1$\"3KR!z@A\">vHFgs7$$\"3#emmmT9C#eF1$\"3-cNzS<_&4$Fgs7$$\"33* ***\\i!*3`iF1$\"3;-9vp$*R5KFgs7$$\"3%QLLL$*zym'F1$\"3gf4W;8I5LFgs7$$\" 3wKLL3N1#4(F1$\"37y%*GA)R=S$Fgs7$$\"3Nmm;HYt7vF1$\"3]!QlST$\\#[$Fgs7$$ \"3Y*******p(G**yF1$\"3C;f)Hye#[NFgs7$$\"3]mmmT6KU$)F1$\"3O_#3Mu^Xh$Fg s7$$\"3fKLLLbdQ()F1$\"35*=`\"HtLmOFgs7$$\"3[++]i`1h\"*F1$\"3io%ph)\\^9 PFgs7$$\"3W++]P?Wl&*F1$\"39F%\\+\"[`aPFgs7$$\"\"\"F*$\"3;i+Jv*f;z$Fgs- %'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%+AXESLABELSG6$Q\"t6\"Q!Ff[l-%%VIEWG 6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "That l ooks like what we expected from the direction field." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Let's try the initial \+ condition y(0)=0, to make sure that Maple will give us the correct sol ution if we specifically ask for it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol1b := rhs(dsolve(\{eq,y(0)=0\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol1bG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "OK, that worked." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Let's try one more initial condition: y(0 )=-1/2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sol1c := rhs(dsol ve(\{eq,y(0)=-1/2\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol1 cG,$*&\"\"\"F',&F'F'*&\"\"*F'-%$expG6#,$%\"tG!\"%F'!\"\"F1\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(sol1c,t=-1..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 333 223 223 {PLOTDATA 2 "6%-%'CURVESG6$7`p7 $$!\"\"\"\"!$!3g0_4)y$)o:)!#?7$$!3ommm;p0k&*!#=$!3)4O13V\"e9(*F-7$$!3w KL$37$$!3mmmmT%p\"e()F1$!3UGLv$=;AM\"F97$$!3 :mmm\"4m(G$)F1$!3!GOJ4zUZf\"F97$$!3\"QLL3i.9!zF1$!3;!>\\`K,>EF97$$!3+n m;/siqmF1$!3R?Ji')e@2JF97$$!3[++](y$pZiF1$!3e9AfA:B&o$F97$$!33LLL$yaE \"eF1$!3g,PGwAR$R%F97$$!3hmmm\">s%HaF1$!3Xu**Qv=YI^F97$$!3Q+++]$*4)*\\ F1$!3^!='3``W6hF97$$!39+++]_&\\c%F1$!3I;Mtw'G')G(F97$$!31+++]1aZTF1$!3 6,k<,uoT')F97$$!3umm;/#)[oPF1$!3&Rp06G>#45F17$$!3hLLL$=exJ$F1$!30_fyfC m97F17$$!3*RLLLtIf$HF1$!3U+_#HGAAU\"F17$$!3]++]PYx\"\\#F1$!3!HYMQRe0r \"F17$$!3EMLLL7i)4#F1$!3Ytl]c]b;?F17$$!3c****\\P'psm\"F1$!3c/&4!e9G>CF 17$$!3')****\\74_c7F1$!3C8$3>E;C)GF17$$!3)3LLL3x%z#)F9$!3+.:kYK;oMF17$ $!3KMLL3s$QM%F9$!3h*4)[)\\s.7%F17$$!3]^omm;zr)*!#@$!3]^*=%zJ%y(\\F17$$ \"3%pJL$ezw5VF9$!3;Cbk?u*4 \"!#<7$$\"3#z****\\_qn2#F1$!3AHcNZA3p8Fgt7$$\"3U)***\\i&p@[#F1$!3!*f\" o&odM8R$3=(Q$3(Fgt7$$\"3m******\\)Hxe%F1 $!3[tt+rW7m\"*Fgt7$$\"37KLeR*)**)y%F1$!3CK!)odNqH7!#;7$$\"3ckm;H!o-*\\ F1$!3/c\"p*=ye&z\"Few7$$\"3c(****\\7ga4&F1$!3,!Rz7K\"Q?BFew7$$\"3nJL$3 A_1?&F1$!3pr&zhniPA$Few7$$\"3G***\\(o#[KD&F1$!390%>*3(4J(RFew7$$\"3ylm m;V%eI&F1$!31='y-m()Q9&Few7$$\"3.**\\iSB9K`F1$!3g-c')\\![k,'Few7$$\"3G KLek.We`F1$!3<5h!Q*3L!*Few7$$\"39K3_+u)yR&F1$!3sM6$)[N$3.\"!#:7$$\"3y )***\\7k.6aF1$!3OB\\I7%\\#*>\"Fhz7$$\"3F2-8v>*QU&F1$!3:IuK*z;eU\"Fhz7$ $\"3v:/wPvuOaF1$!3%p\"oWAa$ev\"Fhz7$$\"3CC1R+Jg\\aF1$!3)p@H]376G#Fhz7$ $\"3sK3-j'eCY&F1$!3o]%*GexrZKFhz7$$\"3?T5lDUJvaF1$!3V\"3Sir4Zh&Fhz7$$ \"3o\\7G)yp\")[&F1$!3[P!R%=&)HU?!#97$$\"31d9\"4ND5]&F1$\"3wn5&=1mwD\"F \\]l7$$\"3al;a84)Q^&F1$\"3g*en<&RAB[Fhz7$$\"3-u=\"Fhz7$$\"3M:HK*G95f&F1$\"3QuE=$)*H5/\"Fhz7$$\"3#Q7`>&)pQg&F1$ \"3t&=G$zK1E#*Few7$$\"3IKLe9as;cF1$\"3_Iz$eSy!)G)Few7$$\"3E\\P%)RlVUcF 1$\"37G\"=R*ya'*oFew7$$\"3AmT5lw9ocF1$\"3!z1c$))f!Q\"fFew7$$\"3?$ek.ze Qp&F1$\"3AE$f#*[AG=&Few7$$\"31**\\i:*p&>dF1$\"3n+#[!*flyh%Few7$$\"3)=$ e9m@*4x&F1$\"3u+a3.Rs,QFew7$$\"3#emmmT9C#eF1$\"3Bz5B\"GY1C$Few7$$\"3WK LeR&Fgt7$$\"3W++]P?Wl&*F1$\"3A (3rRqff(\\Fgt7$$\"\"\"F*$\"3K/?-9d]*y%Fgt-%'COLOURG6&%$RGBG$\"#5F)$F*F *Fdfl-%+AXESLABELSG6$Q\"t6\"Q!Fifl-%%VIEWG6$;F(Fjel%(DEFAULTG" 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 14 "What happened?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Let's force the plot comm and to use a smaller vertical range." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(sol1c,t=-4..4,y=-3..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 343 197 197 {PLOTDATA 2 "6%-%'CURVESG6$7jp7$$!\"%\"\"!$!3;^] R$Qj:+&!#D7$$!3ommmmFiDQ!#<$!3(>x5_L&o/5!#C7$$!35LLLo!)*Qn$F1$!3YC%ez0 HL%=F47$$!3nmmmwxE.NF1$!3Eb\"H\"*>\"pZOF47$$!3YmmmOk]JLF1$!3w1m%HZ.5D( F47$$!3_LLL[9cgJF1$!3z'=Kk,!oO9!#B7$$!3smmmhN2-IF1$!3s&3n`%p?3FFI7$$!3 !******\\`oz$GF1$!3rQg)R'o4@_FI7$$!3!omm;)3DoEF1$!3\\`QFA(=%H5!#A7$$!3 ?+++:v2*\\#F1$!3-**H3JTCD?FY7$$!3BLLL8>1DBF1$!3k3xznULiSFY7$$!3kmmmw)) yr@F1$!3[&)GWOgm*\\(FY7$$!3;+++S(R#**>F1$!3klUV^=a&\\\"!#@7$$!30++++@) f#=F1$!3A,8o)pY3*HFho7$$!3-+++gi,f;F1$!3;;P\"4jjF$eFho7$$!3qmmm\"G&R2: F1$!3%*\\e%RTU)p5!#?7$$!3XLLLtK5F8F1$!34g3:)p56?#Fhp7$$!3eLLL$HsV<\"F1 $!3O=\"e$GwhcSFhp7$$!3+-++]&)4n**!#=$!3sn4[*)e;l#)Fhp7$$!37PLLL\\[%R)F fq$!39#3tN[1Kb\"!#>7$$!3G)*****\\&y!pmFfq$!3#z\">&*\\i:4JF^r7$$!3Y**** **\\O3E]Ffq$!3B]D,m$*QUgF^r7$$!3NKLLL3z6LFfq$!3RlC,9Ol<7Ffq7$$!3sLLL$) [`P.9$H=C?\"\\Ffq7$$\"3'oHLL3+TU)F^r$!3]XJ-ONrstFfq7$$\"3yELL$=2V s\"Ffq$!3/)ojS,Ry8\"F17$$\"3Khmmm7+#\\#Ffq$!3%3S#R9_+B[PG%>F_w7$$\"3CH$ek3k&y]Ffq$!3_\") \\G?Z3=AF_w7$$\"3iim;/**4K^Ffq$!3c:F'oS*=vDF_w7$$\"3+'*\\(=sNc=&Ffq$!3 /2'*pNi\"p0$F_w7$$\"3PHLeR:7N9F_w7$ $\"3ojm;H&fpT'Ffq$\"3-#4vg:hYH\"F_w7$$\"3VILek6.ClFfq$\"3M_Yd[!oO=\"F_ w7$$\"3>(******z-6j'Ffq$\"37l6mCW#Q4\"F_w7$$\"3#e****\\i(4]qFfq$\"3TGO HwKtG')F17$$\"3W%*****\\C4puFfq$\"3Om7?>pT@tF17$$\"31$****\\F(3))yFfq$ \"31^XELn()*['F17$$\"3q\"******4#32$)Ffq$\"36b=yqM8F1$\"3g6`'\\jk1=% F17$$\"3%)*******HSu]\"F1$\"3]r4Crc`)3%F17$$\"3'HLL$ep'Rm\"F1$\"3SvmMJ Z&o/%F17$$\"3')******R>4N=F1$\"3#4VQ%=W\\BSF17$$\"3#emm;@2h*>F1$\"3lR \")=/RI7SF17$$\"3]*****\\c9W;#F1$\"3e'37,Fmi+%F17$$\"3Lmmmmd'*GBF1$\"3 u&)yD+@C.SF17$$\"3j*****\\iN7]#F1$\"3[&z04+F;+%F17$$\"3aLLLt>:nEF1$\"3 :TEnvw$3+%F17$$\"35LLL.a#o$GF1$\"3=z'4;*[U+SF17$$\"3ammm^Q40IF1$\"3M[I 7Un@+SF17$$\"3y******z]rfJF1$\"3H$)o3pn6+SF17$$\"3gmmmc%GpL$F1$\"3>T'[ QZd++%F17$$\"3/LLL8-V&\\$F1$\"3&*)z%[([I++%F17$$\"3=+++XhUkOF1$\"3Yxxd 2b,+SF17$$\"3=+++:o_*>\"3++%F17$$\"\"%F*$\"3%HRq70/++%F1- %'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fjil-%+AXESLABELSG6$Q\"t6\"Q\"yF_jl-%% VIEWG6$;F(F_il;$!\"$F*$\"\"&F*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 341 "It appears that the solution has a vertical asymptote. If you lo ok at the solution found above, you'll see that this is the case. In \+ fact, we can find that value where it occurs by finding where the deno minator is zero. In the following, we use the \"denom\" command to ge t the denominator, and then find the root using the \"solve\" command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "den := denom(sol1c);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$denG,&!\"\"\"\"\"*&\"\"*F'-%$expG6 #,$%\"tG!\"%F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r := so lve(den=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,$-%#lnG6#\"\"* #\"\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(r);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+U91$\\&!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Can Maple solve any first order differential equation? Unfortunately, no. Here is a sim ple example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eq3 := diff (y(t),t) = sin(y(t))+cos(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3 G/-%%diffG6$-%\"yG6#%\"tGF,,&-%$sinG6#F)\"\"\"-%$cosGF+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dsolve(eq3,y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The function returns nothing; Maple can not sol ve this equation." }}}}{MARK "40 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }