{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 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 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "MyWork" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "MyWork" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0 " -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 259 1 {CSTYLE " " -1 -1 "Courier" 1 14 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 "R3 Font 3" -1 260 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Normal2" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Subtitles" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 26 " \+ " }}{PARA 257 "" 0 "" {TEXT -1 67 " Visualization of solutions of \+ first order linear systems of ODE" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 " " 0 "" {TEXT -1 54 "Consider the first order linear system of ODE (n \+ = 2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sys1 := diff(x(t),t) = -y(t), diff(y(t),t) = x(t); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ICS1 := x(0)=1,y(0)=0;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "soln1:= dsolve(\{sys1\} un ion \{ICS1\},\{x(t),y(t)\});" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT -1 35 "Graphic representation of solutions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 118 "No w, we will show three different graphic representation for the solutio n of first order linear systems of ODE (n = 2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT 256 36 "1.- Curve solu tion on the 3-D Space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "DEplot3d([sys1], [x(t),y(t)], t=-2 0..20, \{[0,1,0]\}, scene=[t,x(t),y(t)], x=-2..2, y=-2..2, stepsize= . 1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 34 "2.- Orbit on the x-y phase plane. " }}{PARA 262 "" 0 "" {TEXT -1 105 " We can consider the graphic of the solution as the \+ trajectory of a moving particle in the x-y plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot(\{sy s1\}, \{x(t),y(t)\}, t=0..5, \{[0,1,0]\}, x=-2..2, y=-2..2, arrows=non e,stepsize= .1);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 59 "The range o f \"t\" needs to be increased to complete a cycle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot(\{sy s1\}, \{x(t),y(t)\}, t=0..7, \{[0,1,0]\}, x=-2..2, y=-2..2, arrows=non e,stepsize= .1);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 58 "2.- Solutio n components. Plots on the t-x and t-y planes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "DEplot(\{sy s1\}, \{x(t),y(t)\}, t=-20..20, \{[0,1,0]\}, scene=[t,x], x=-2..2, y=- 2..2, stepsize= .1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "DEp lot([sys1], [x(t),y(t)], t=-20..20, \{[0,1,0]\}, scene=[t,y], x=-2..2, y=-2..2, stepsize= .1);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 38 "Sol ution procedure for linear systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 261 "" 0 "" {TEXT -1 46 "First we need to load a linear algebra library" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with (linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 54 "Consider the first order linear s ystem of ODE (n = 2)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "sys2 := diff(x(t),t) = -2*x(t)+ y(t), dif f(y(t),t) = x(t) -2*y(t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 72 "Let's find the eigenvalues and eigenvec tors of the associated matrix A2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A2:= matrix([[-2,1],[1,-2]]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A2);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 17 "General solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "soln2:=dsol ve(\{sys2\},\{x(t),y(t)\});" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 37 " \nDirection Field for the given system" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "DEplot(\{sys2\}, \{x(t),y(t)\}, t=-20..20, x=-2..2, arrows=thin, dirgrid=[30,30],y=-2..2,stepsize= .1 );" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 14 "Phase Portrait" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 70 "First, we nee d to define a set of initial values for different orbits." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "init 1 := seq([0,0.5*i,2],i=-4..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 261 "" 0 "" {TEXT -1 54 "Now, we can plot a family of orbits on the phase plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "DEplot(\{sys2\}, \{x(t),y(t)\}, t=-20..2 0, x=-2..2, [init1], arrows=thin,y=-2..2, stepsize=.1,linecolour=COLOR (HUE,.7));" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 71 "More orbits are required for a better description of the phase portarit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "init2 := seq([0,0.5*i,-2],i=-4..4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "init3 := seq([0,2,0.5*i],i=- 4..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "init4 := seq([0,- 2,0.5*i],i=-4..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "DEplot(\{sys2\}, \{x(t),y(t)\}, t= -20..20, x=-2..2, [init1,init2,init3,init4], arrows=thin,y=-2..2, step size=.1,linecolour=COLOR(HUE,.7), title=`NODAL SINK`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 257 12 "SADDLE POINT" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "sys3 := diff(x(t),t) = x(t)- 3*y(t), diff(y(t),t) = - 3*x(t) + y(t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 " " 0 "" {TEXT -1 72 "Let's find the eigenvalues and eigenvectors of the associated matrix A2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A3:= matrix([[1,-3],[-3,1]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A3);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 17 "General solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "soln3:=dsolve(\{s ys3\},\{x(t),y(t)\});" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 37 "\nDire ction Field for the given system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "DEplot(\{sys3\}, \{x(t),y(t)\}, t=- 20..20, x=-2..2, arrows=thin, dirgrid=[30,30],y=-2..2,stepsize= .1);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 14 "Phase Portrait" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 70 "First, we need to \+ define a set of initial values for different orbits." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "init1 := \+ seq([0,0.5*i,2],i=-4..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 261 "" 0 "" {TEXT -1 54 "Now, we can plot a family of orbits on \+ the phase plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 116 "DEplot(\{sys3\}, \{x(t),y(t)\}, t=-20..20, x= -2..2, [init1], arrows=thin,y=-2..2, stepsize=.1,linecolour=COLOR(HUE, .7));" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 71 "More orbits are required for a better description of the \+ phase portarit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "init2 := seq([0,0.5*i,-2],i=-4..4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "init3 := seq([0,2,0.5*i],i=- 4..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "init4 := seq([0,- 2,0.5*i],i=-4..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "DEplot(\{sys3\}, \{x(t),y(t)\}, t= -20..20, x=-3..3, [init1,init2,init3,init4], arrows=thin,y=-3..3, step size=.1,linecolour=COLOR(HUE,.7), title=`NODAL SINK`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "59" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }