{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 } {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 "Ti mes" 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 "MyWork" -1 258 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" -1 259 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" -1 260 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" -1 261 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 "Norma l2" -1 262 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" -1 263 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 258 "" 0 "" {TEXT -1 26 " \+ " }}{PARA 258 "" 0 "" {TEXT -1 49 " Problems on first order linea r systems of ODE" }}}{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 262 "" 0 "" {TEXT 256 33 "Problem 23 of Section 7.6 (Boyc e)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 54 "C onsider the first order linear system of ODE (n = 3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "sys1 := diff(x(t),t) = -(1/4)*x(t) + y(t), diff(y(t),t) = -x(t) -(1/4)*y(t), \+ diff(z(t),t) = -(1/4)*z(t); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Eingenvalues:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "A1:=matrix([[-1/4,1,0],[-1, -1/4,0],[0,0,-1/4]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ei genvals(A1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects( A1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "General Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "soln1:= dsolve(\{sys1\} ,\{x(t),y(t ),z(t)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Particular Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ICS1 := x(0)=1,y(0)=0,z(0)=1 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "soln1:= dsolve(\{sys1 \} union \{ICS1\},\{x(t),y(t),z(t)\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Graphic Representations: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Firs t we will represent the trajectory that pass through the point (1,0,1) in the 3-d space x1-x2-x3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "DEplot3d([sys1], [x(t),y(t),z(t)], t=-200..50, \{[0,1,0,1]\}, scene=[x(t),y(t),z(t)], x=-2..2, y=-2..2, \+ z=-2..2, stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Just \+ by moving this 3-d graph with the mouse you can find the projections o n the respective planes. Try it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "Now, to find the projections unto the pla nes x1-x2, x1-x3, and x2-x3," }}{PARA 0 "" 0 "" {TEXT -1 55 "we will u se the command DEplot (2-d) with scene option." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "x1-x3 plane:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "DEplot([sys1], [x(t),y(t),z(t)], t =-200..50, \{[0,1,0,1]\}, scene=[x(t),z(t)],x=-5..5, y=-5..5, z=-5..5 , stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 12 "x2-x3 plane:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "DEplot([sys1], [x(t),y(t),z(t)], t=-200..50, \{[0,1, 0,1]\}, scene=[y(t),z(t)],x=-5..5, y=-5..5, z=-5..5, stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "x1-x2 plane:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "DEplot([s ys1], [x(t),y(t),z(t)], t=-200..50, \{[0,1,0,1]\}, scene=[x(t),y(t)], x=-5..5, y=-5..5, z=-5..5, stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The plane projections ca n also be obtained using the exact solution and making parametric grap hs as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "x1-x3 plane:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([exp(-(1/4)*t)*cos(t),exp(-(1/4)*t), t=-20..20]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "x2-x3 plane:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot ([exp(-(1/4)*t)*sin(t),exp(-(1/4)*t), t=-20..20]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "x1-x2 plane:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([exp(-(1/4)*t)*cos(t),e xp(-(1/4)*t)*sin(t), t=-20..20]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 22 "Problem 20 Section 7. 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "syst2 := \{diff(x(t),t) = 4* x(t) + alpha*y(t),diff(y (t),t) = 8*x(t) -6*y(t)\}; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 12 "Eigenvalues:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "At2:=matrix([[4,a lpha],[8,-6]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenval s(At2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "I case \+ alpha > -(25/8). (Saddle point)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "syst3:=subs(\{alpha=-2\},sys t2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "At2:= matrix([[4,-2 ],[8,-6]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvals(At 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(At2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "DEplot(syst3, [x(t),y(t)], t=-10..10, x=-4..4, y=-5 ..5, stepsize= .1, dirgrid=[30,30]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "II case alpha = 25/8" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "syst4:=subs(\{alpha=- 25/8\},syst2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "DEplot(syst4, [x(t),y(t)], t=-10..10, x=-5..5, y=-5 ..5, stepsize= .1, dirgrid=[30,30]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "III case alpha < -25/8 (Spiral) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "syst5:=subs(\{alpha= -5\},syst2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "DEplot(syst5, [x(t),y(t)], t=-10..10, x=-5. .5, y=-5..5, stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Problem \+ 7 Section 7.5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "syst6 := \{diff(x(t),t) = 4* x(t) - 3*y(t),dif f(y(t),t) = 8*x(t) -6*y(t)\}; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A6:=matrix([[8,-6],[4,-3]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eigenvects(A6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "DEplot(syst6, [x(t),y(t)], t=-50..100, x=-5..5, y=-5..5, \{[0,1,1], [ 0,2,4], [0,-2,2], [0,4,-4]\},stepsize= .1);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "58" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }