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0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } 3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT 256 48 "Direction/Slope Fields For Firs t-Order Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 30 "Differential Equations Project" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" } {TEXT 260 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 101 "(a) To un derstand the construction of direction fields and numerically computed approximations to " }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ solution curves for first-order equations " }{XPPEDIT 18 0 "dy/dt = f(t,x);" "6#/*&%#dyG\"\"\"%#dtG!\"\"-%\"fG6$%\"tG%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "(b) To introduce the \+ use of the " }{TEXT 261 6 "dsolve" }{TEXT -1 9 " command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 209 " It is ve ry tedious to draw a direction field by hand, evaluating f(t, x) at \+ a large number of points and then drawing a line segment at each point having the corresponding slope, but it is easy with " }{TEXT 262 5 "M aple" }{TEXT -1 40 ", even for quite complicated equations. " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 17 "Solved Example 1:" }}{PARA 0 "" 0 "" {TEXT -1 49 " (a) Draw a direction field for the equation " }{XPPEDIT 18 0 "dx/dt = \+ -x/2+t/3;" "6#/*&%#dxG\"\"\"%#dtG!\"\",&*&%\"xGF&\"\"#F(F(*&%\"tGF&\" \"$F(F&" }{TEXT -1 22 " in the rectangle " }{XPPEDIT 18 0 "0 <= t, t <= 6;" "6$1\"\"!%\"tG1F%\"\"'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "-3 <= x,x <= 3;" "6$1,$\"\"$!\"\" %\"xG1F'F%" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 83 " (b) Com ment on the behavior of the solutions in the first and fourth quadran ts." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 256 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 26 "We wil l use t he command " }{TEXT 258 8 "DEplot " }{TEXT 259 0 "" }{TEXT -1 75 "to draw the direction field. This command is located in a pack age called " }{TEXT 260 7 "DEtools" }{TEXT 261 0 "" }{TEXT -1 35 ", s o we first call up this package:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(DEtools):\n" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "N ext we enter the given differential equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Eq:=diff(x(t ), t) = -x(t)/2 + t/3;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#EqG/-%% diffG6$-%\"xG6#%\"tGF,,&*&#\"\"\"\"\"#F0F)F0!\"\"*&\"\"$F2F,F0F0" }}} {PARA 0 "" 0 "" {TEXT -1 18 "The next command, " }{TEXT 262 0 "" } {TEXT 263 6 "DEplot" }{TEXT -1 293 ", draws the direction field. The \+ first argument is the name of the equation, the second is the dependen t variable, the third and fourth arguments define the rectangle on whi ch the plot is to be drawn, and the fifth argument is an optional 'equ ation' specifying the type of arrows to be used (" }{TEXT 264 4 "THIN " }{TEXT -1 2 ", " }{TEXT 265 4 "SLIM" }{TEXT -1 5 ", or " }{TEXT 266 5 "THICK" }{TEXT -1 2 "):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "DEplot(Eq, x, t=0..6, x=-3..3, arro ws=SLIM);\n" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6%-%'CURVESG6]dl7'7$$!3wqY#4:C%e()!#>$!3&*pQEijPJJ!#<7$$ \"3wqY#4:C%e()F*$!3/IhtPOioGF-7$$\"39k9h,VK]#)F*$!3C\"4-!e+.BB!#=$!3!ecSoU'FA$!3e)Ruir\"z-HF-FY7'7$$\"3H&**z6_!*>r)FA$!3#QnG$QvIQJF-7$$\" 3`js)**yPN-\"F-$!3=E8nhCphGF-7$$\"3m@cKqB8D5F-$!3+X(oZsB1%HF-7$$\"3YZG h6gyf&*FA$!3%)G9IS!RD!HF-Fho7'7$$\"3c^]ECBD!>\"F-$!3[(f6!Qa0SJF-7$$\"3 ]K'3_YjgL\"F-$!3_-%))>cW*fGF-7$$\"3P[&emG3&R8F-$!36]WBQn\")QHF-7$$\"3U 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[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_ [mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_ [mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_ [mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_ [mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[mFg_ [mFg_[mFg_[mFg_[mFg_[mFg_[mFg_[m-%%VIEWG6$;$!+++++I!#5$\"+++++j!\"*;$! +++++LFa`[m$\"+++++LFa`[m-%+AXESLABELSG6$Q\"t6\"Q%x(t)F[a[m" 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 -1 254 "(b) Remembering that solutions are everywhere tangent to the arrows in the direction \+ field, we conclude that solutions appear to approach a line or curve i n the first and fourth quadrants. A more detailed conclusion would re quire further investigation." }}{PARA 0 "" 0 "" {TEXT -1 73 "_________ ________________________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 267 17 "Solved Example 2: " }}{PARA 0 "" 0 "" {TEXT -1 65 " (a) Sketch the direction field for the autonomous equation " }{XPPEDIT 18 0 "dx/dt = (x-1)*(x-4);" "6 #/*&%#dxG\"\"\"%#dtG!\"\"*&,&%\"xGF&F&F(F&,&F+F&\"\"%F(F&" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 0 "" }{TEXT 273 0 " " }{TEXT 274 0 "" }{TEXT -1 72 " (b) Describe any constant solutions indicated by the direction field." }}{PARA 0 "" 0 "" {TEXT -1 50 " ( c) Find the general solution of this equation." }}{PARA 0 "" 0 "" {TEXT -1 63 " (d) Solve the equation with the initial condition x(0 ) = 3." }}{PARA 0 "" 0 "" {TEXT -1 32 " (e) Plot the solution to (d) ." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 0 "" }{TEXT 271 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 39 " (a) The commands we need are in the " }{TEXT 275 0 "" }{TEXT 276 7 "DEtools" }{TEXT 277 0 "" }{TEXT -1 46 " package, which w e have already brought up in " }}{PARA 0 "" 0 "" {TEXT -1 56 " \+ our first solved example. Now we have" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "ode:=diff(x (t), t) = (x(t) - 1)* (x(t) - 4);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$odeG/-%%diffG6$-%\"xG6#%\"tGF,*&,&F)\"\"\"F/!\"\"F/,&F)F/\"\"%F0F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "DEplot(ode, x, t=-4..4, x=-5..5, arrows=SLIM);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6%-%'CURVESG6]dl7'7$$!3$4KE#*)>([+%!#<$!3K78A=u3j_F*7$$!3 _zOx5!G^*RF*$!3o(oyQe$F*F+7$$!3=KZ>z`2uNF*F07$$!3_ [*o(GvcBNF*F57$$!3jtu0'\\-)GOF*F:F@7'7$$!3yD%ogsmF;$F*F+7$$!3#[y:wuAI: $F*F07$$!3=,+>(*[^-JF*F57$$!3GE&yW')\\x?$F*F:FM7'7$$!3Xy%*[%49t#F*F07$$!3#Q06cEi9o#F*F57$$!3%*y&**GB(p'y#F*F:FZ7'7$$!35J 0\"HYh1K#F*F+7$$!39!*yX%[<4J#F*F07$$!3[1@.M'4/E#F*F57$$!3eJ1K,YklBF*F: Fao7'7$$!3)ReJ8$)3'**=F*F+7$$!3cU*yG&[')*)=F*F07$$!3\"*eJX-qNR=F*F57$$ !3-%oT(p>fW>F*F:F^p7'7$$!3kOEv*>c&y9F*F+7$$!3A&***H@A\")o9F*F07$$!3d6U (3P/$=9F*F57$$!3oOF;Q$RN_\"F*F:F[q7'7$$!3I*ot\"oN]d5F*F+7$$!3)y/@(*efx 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