{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "List \+ Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 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 "" 0 257 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 "" 0 258 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 "" 0 259 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 "" 0 260 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 "" 0 261 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 "" 0 262 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 "" 0 263 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 "" 0 264 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 "" 0 265 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 "" 0 266 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 "" 0 267 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 "" 0 268 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 "" 0 269 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 "" 0 270 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 "" 0 271 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 "" 0 272 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 "" 0 273 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 "List Subitem" 14 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 14 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 23 "Unit 12 -- Slope F ields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc.edu/~meade/" "" }} {PARA 257 "" 0 "" {URLLINK 17 "Industrial Mathematics Institute" 4 "ht tp://www.math.sc.edu/~IMI/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Depart ment of Mathematics" 4 "http://www.math.sc.edu/" "" }}{PARA 259 "" 0 " " {URLLINK 17 "University of South Carolina" 4 "http://www.sc.edu/" " " }}{PARA 260 "" 0 "" {TEXT -1 19 "Columbia, SC 29208\n" }}{PARA 262 " " 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://www.math.sc.edu/~mead e/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 263 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 38 "Copyright \251 2001 by Douglas B. Mea de" }}{PARA 265 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 67 "--------------- ----------------------------------------------------" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 12" }}{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "12.A" 1 "" "12.A" }{TEXT -1 13 " Slope \+ Fields" }}{PARA 274 "" 0 "" {HYPERLNK 17 "12.A-1" 1 "" "12.A-1" } {TEXT -1 10 " Example 1" }}{PARA 274 "" 0 "" {HYPERLNK 17 "12.A-2" 1 " " "12.A-2" }{TEXT -1 10 " Example 2" }}{PARA 274 "" 0 "" {HYPERLNK 17 "12.A-3" 1 "" "12.A-3" }{TEXT -1 10 " Example 3" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Init ialization" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 0 {PARA 3 "" 0 "12.A" {TEXT -1 17 "12.A Slope Fields" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "A slope field displays information about rates of change for ordinary differential equations. For a first-orde r ODE" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(y,t) = \+ f(t,y)" "6#/-%%DiffG6$%\"yG%\"tG-%\"fG6$F(F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 68 "the slope field shows the slope of the fu nction y for all values of " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 26 ". That is, at the p oint ( " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 52 " ) in the domain of f, the \"minitangent\" with slope " }{XPPEDIT 18 0 "f(t,y)" "6#-%\"fG6$%\"tG%\"yG" }{TEXT -1 150 " is displayed. A slope field should not be confused with a dir ection field. A direction field is a graphical tool for analyzing a sy stem of ODEs (see " }{HYPERLNK 17 "Unit 13" 1 "unit13.mws" "" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "12. A-1" {TEXT -1 16 "12.A-1 Example 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Consider the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ode1 := diff( y(t), t ) = 2*abs(y(t)) - t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Note that this equation is not li near. It is not solvable by any of the analytic methods discussed in t he ODE PowerTool. In fact, Maple is uable to find an analytic solution to this equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "dsolve( ode1, y(t) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "A portion of \+ the slope field for this ODE is created with the " }{HYPERLNK 17 "DEpl ot" 2 "DEtools,DEplot" "" }{TEXT -1 18 " command from the " } {HYPERLNK 17 "DEtools" 2 "DEtools" "" }{TEXT -1 9 " package:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "D Eplot( ode1, [y(t)], t=-4..4, y=-4..4, arrows=LINE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Solution curves can be added to a slope field with the inclusi on of a list of initial conditions. The initial conditions can appear \+ either as ordered pairs or as equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "IC1 := [ [ 0, i ] $ \+ i=-4..4, [ y(i)=0 ] $ i=-4..4 ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DEplot( ode1, [y(t)], t=-4..4, y=-4..4, IC1, arrows=LINE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 " Notice that the solution curves illustrate several curves with the sam e asymptotic behavior, but do not cross. Also note that the trajectori es are drawn for -4 < " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 48 " < \+ 4, that is both forward and backward in time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "12.A-2" {TEXT -1 16 "12.A-2 Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "When the differential equation has the form" }} {PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(y,t) = f(t)" "6# /-%%DiffG6$%\"yG%\"tG-%\"fG6#F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "the slope field has the special property that the slopes \+ are independent of " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For examp le," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ode2 := diff( y(t), t ) = t^2 - 2*t*sin(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "DEplot( ode2, [y(t)], t=-3..4, y=-3..3, arrows=LINE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 56 "When solution trajectories are added to the slope \+ field," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "IC2 := [ [ 0, i ] $ i=-3..3 ]:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "DEplot( ode2, [y(t)], t=-3..4, y=-3..3, IC2, a rrows=LINE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "it is readily appparent that all \+ solutions are vertical translations of the solution through the origin ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The \+ following animation emphasizes this point," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "IC2a := [ [0,0] ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "IC2b := [ [ 0, i/3 ] ] $ i=-9..9:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "p0 := DEplot( ode2, [y(t)], t=-3..4, y=-3..3, IC2a, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " arrows=LINE, line color=BLUE ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "p1 := seq( display ( p0," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " DEplot ( ode2, [y(t)], t=-3..4, y=-3..3, ic," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 " arrows=LINE, linecolor=GREEN ) )," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " ic=IC2b ):" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "display( p1, insequence=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "This geometric observation is a direct consequence of the fact the general solution of a first-order ODE of this type can be solved \+ by direct integration. In general, the solution is" }}{PARA 272 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y(t) = Int( f(t), t ) + C" "6#/-%\"yG 6#%\"tG,&-%$IntG6$-%\"fG6#F'F'\"\"\"%\"CGF/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 106 "where the presence of the integration constant explains the vertical translation of solution trajectories." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d solve( ode2, y(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 0 {PARA 4 "" 0 "12.A-3" {TEXT -1 16 "12.A-3 Example 3" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "A second special case is the auton omous first-order ODE" }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(y,t) = f(y)" "6#/-%%DiffG6$%\"yG%\"tG-%\"fG6#F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 133 "Autonomous ODEs are always sepa rable. However, finding an analytic expression for the solution can st ill be quite difficult. Consider" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ode3 := diff( y(t), t ) = y( t)^2 - 2*y(t)*sin(y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "DEplot( ode3, [y(t)], t =-3..3, y=-3..3, arrows=LINE );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Here, all slopes alon g a horizontal line are the same. A sample of the solution trajectorie s following this slope field is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "IC3 := [ [ 0 , i/2 ] $ i=-6. .6 ]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "DEplot( ode3, [y(t)], t=-3 ..3, y=-3..3, IC3, arrows=LINE, stepsize=0.2 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Withou t the " }{HYPERLNK 17 "stepsize=" 2 "DEtools,DEplot" "" }{TEXT -1 208 " optional argument, the approximate solutions computed by Maple cross . We know this cannot happen! Forcing Maple to use a smaller stepsize \+ in the computations produces approximate solutions that do not cross. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "A no teworthy geometric property of solutions to autonomous differential eq uations is that a horizontal translation of a solution trajectory is a nother solution trajectory. This is illustrated in the following anima tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "p0 := DEplot( ode3, [y(t)], t=-3..3, y=-3..3, IC3, ar rows=LINE, linecolor=BLUE, stepsize=0.2 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p_list := NULL:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for t0 in [ i/2 $ i=-6..6 ] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " ic := [ [ t0, i/2 ] $ i=-6..6 ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " p1 := DEplot( ode3, [y(t)], t=-3..3, y=-3..3, ic," }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 " arrows=LINE, stepsize=0.2, lin ecolor=GREEN ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " p_list := p_li st, display( [p0,p1] ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display( p_list, insequen ce=true );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Note that while this ODE is separable, M aple is unable to obtain an analytic expression for the solution." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[dsolve] := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ds olve( ode3, y(t) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[ds olve] := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Power tool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }