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"evalf(eigenvals(A5, 'radical'));" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 15 "eigenvects(A 5);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 1 " " }{TEXT 213 15 "DIRECTION FIELD" }{TEXT 210 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 84 "DEplot ( \{sys5\}, \{x(t),y(t)\}, t=-5..5, x=-10..10, arrows=thin, y=-3..3, s tepsize=.1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 210 "" 0 "" {TEXT 214 54 "Now, we can plot a family o f orbits on the phase plane" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 34 "init1 \+ := seq([0,0,0.5*i],i=0..20);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 212 88 "DEplot( \{sys5\}, \{x(t),y(t) \}, t=-5..5, x=-10..10, [init1,[0,0,1]], y=-3..3, stepsize=.1);" } {MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }} {PARA 211 "" 0 "" {TEXT 215 25 "Graphic of each component" }{TEXT 215 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 96 "DEplot(\{sys5\}, \{x(t),y(t)\}, t=0..5, \{[0,0,1]\} , scene=[t,x], x=-2.1..2.1, y=-3..3, stepsize= .1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 97 "DEplot(\{sys5\}, \{x(t),y(t)\}, t=0..5, \{[0,0,1]\}, scene=[t,y], x=-2.1..2.1, y=-1... 1, stepsize= .1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 83 "DEplot( \{sys5\}, \{x(t),y(t)\}, t=0..5, x=-2.1..2. 1, [[0,0,1]], y=-1..1, stepsize=.1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }} {PARA 212 "" 0 "" {TEXT 211 16 "EXACT SOLUTION" }{TEXT 211 0 "" }}} {EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 28 "dsolve(\{sys5\}, \{x(t),y(t)\});" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 213 "" 0 "" {TEXT 216 16 "Another example:" }{TEXT 216 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{PARA 207 "" 0 "" {TEXT 210 0 "" }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 71 "sys4 := diff(x(t),t) = 2*x(t)+3*y(t), dif f(y(t),t) = -4*x(t) - 3*y(t); " }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 32 "A4:= matrix([[ 2,3], [-4, -3]]);" } {MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }} {PARA 209 "> " 0 "" {MPLTEXT 1 212 32 "evalf(eigenvals(A4, 'radical')) ;" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 1 " " } {TEXT 213 15 "DIRECTION FIELD" }{TEXT 210 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 82 "DEplot ( \{sys4\}, \{x(t),y(t)\}, t=-5..5, x=-5..5, arrows=thin, y=-5..5, ste psize=.1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 210 "" 0 "" {TEXT 214 54 "Now, we can plot a family o f orbits on the phase plane" }{TEXT 214 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 34 "init1 \+ := seq([0,0.5*i,2],i=-4..4);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 212 78 "DEplot( \{sys4\}, \{x(t),y(t) \}, t=-5..5, x=-5..5, [init1], y=-5..5, stepsize=.1);" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 207 "" 0 " " {TEXT 210 0 "" }{TEXT 210 1 "\n" }{TEXT 210 133 "\nAll orbits approa ch the origin. This can be made clear by observing that in the exact s olution there is a exponential decaying term." }{TEXT 210 0 "" }} {PARA 207 "" 0 "" {TEXT 210 0 "" }}}{EXCHG {PARA 214 "" 0 "" {TEXT 217 14 "EXACT SOLUTION" }{TEXT 217 0 "" }}{PARA 207 "" 0 "" {TEXT 210 0 "" }}{PARA 209 "> " 0 "" {MPLTEXT 1 212 28 "dsolve(\{sys4\}, \{x(t), y(t)\});" }{MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 212 0 "" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 210 0 "" }}}{PARA 215 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }