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with(plots): with(DEtools):" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 30 "a) Ein stabiler Fixpunkt (mit " }{TEXT 257 4 "vier" }{TEXT -1 34 " darauf zulaufenden L\366sungskurven)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "DEplot([diff(x(t), t) = -2*x(t), diff(y(t), t) = -3*y(t)], [x, \+ y], t=0..10,\{[0, 5, 2], [0, -3, -5], [0, -5, 5], [0, 5, -3]\}, arrows =large, linecolor=blue, color=red, thickness=10);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 46 "b) Ein in alle Richtungen absto\337ender Fixp unkt" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 177 "DEplot( [diff(x(t), t) = 2*x(t), diff(y(t), t) = 3*y(t)], [x, y], t=0..1,\{[0, 5, 2], [0, -3, -5], [0, -5, 5], [0, 5, -3]\}, arrows=large, linecolor =blue, color=red, thickness=10);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 18 "c) Ein Sattelpunkt" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "DEplot([diff(x(t), t) = -2*x(t), diff(y(t), t) = 3*y (t)], [x, y], t=0..1,\{[0, 5, 0.02], [0, -3, -0.02]\}, arrows=large, l inecolor=blue, color=red, thickness=10);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 69 "e) Ein anziehender Fixpunkt mit zwei konjugiert komplexen Eigenwerten" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "DEplot([diff(x(t), t) = -x(t)/5 - 5*y(t), diff(y(t), t) = 5*x(t)-y(t) /5], [x, y], t=0..12,\{[0, 5, 5]\}, stepsize=0.03, arrows=large, linec olor=blue, color=red, thickness=10);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 69 "f) Ein absto\337ender Fixpunkt mit zwei konjugiert komple xen Eigenwerten" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "DEplot([diff(x(t), t) = x(t)/5 - 5*y(t), diff(y(t), t) = 5*x(t)+y (t)/5], [x, y], t=0..12,\{[0, 5, 5]\}, stepsize=0.03, arrows=large, li necolor=blue, color=red, thickness=10);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 106 "g) Ein Fixpunkt im Dreidimensionalen mit einem negative n reellen und zwei konjugiert komplexen Eigenwerten" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "DEplot3d([diff(x(t), t) = -x(t)/5 - 5*y(t), diff( y(t), t) = 5*x(t)-y(t)/5,\ndiff(z(t), t) = -z(t)/10], [x, y, z], t=0.. 15,\{[0, 5, 5, 5]\}, stepsize=0.03, arrows=large, linecolor=blue, thic kness=5);" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 106 "h) Ein Fixpunkt i m Dreidimensionalen mit einem positiven reellen und zwei konjugiert ko mplexen Eigenwerten" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "DEplot3d([diff(x(t), t) = -x(t)/5 - 5*y(t), diff(y(t), t) = 5 *x(t)-y(t)/5,\ndiff(z(t), t) = z(t)/10], [x, y, z], t=0..20,\{[0, 5, 5 , 0.05]\}, stepsize=0.03, linecolor=blue, thickness=5);" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 48 "i) Ein strukturell instabiles Anfangswe rtproblem" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "AWP := y0 -> \{D(y)(t) = y(t) - 2*exp(-t), y(0) = y0\};" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "L\366sung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(dsolve(AWP(1), y(t)));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 18 "Numerische L\366sung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "kurve := dsolve(AWP(1), y(t), numeric):\nplots[odeplo t](kurve, [t, y(t)], 0..18, thickness=5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Grund:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "expand(dsolve (AWP(1+epsilon), y(t)));" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 46 "j) \+ Ein strukturell stabiles Anfangswertproblem" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "AWP := y0 -> \{D(y)(t) = -y(t) + 1, y(0) = y0\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "L\366sung:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(AWP(1), y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Numerische L\366sung:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 95 "kurve := dsolve(AWP(1), y(t), numeric):\nplots[odep lot](kurve, [t, y(t)], 0..1000, thickness=5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Grund:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(AWP (1+epsilon), y(t));" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }