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mathematica_note

# mathematica_note - 1 MATHEMATICA>í l jX  Ð ß ...

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Unformatted text preview: ] 1 ] MATHEMATICA >í l jX  Ð ß  V1â  1.1 5 mathematica Nñ e × 5 Ñ 9 D © F Ä e x3 − 2x + 1 = 0_ ` ½K.   ¨Ð H¦ Solve[x^3 - 2 x + 1 == 0] ªË Õa 1:  ½ l > H ¨  1.2 2π 0  \P & e x3 sin xdx_ &ì°` ½K. mathematica_ "§ë Aü °  hú ¨Ð r¯¦  îîÉ < ú Hr .  Integrate[x^3*Sin[x], {x, 0, 2*Pi}] ªË Õa 2: &ì l > r h 1.3 System of equations e ·    H  H  \$ s e(consistent) system` Û#. ¦¦  QÐ 2x − 3y + 4z = 2 3x − 2y + z = 0 x+y−z = 1 6 _  (7/10, 9/5, 3/2)T s. Mathematica SK.  É Hr  Ð XÐ  Integrate[x^3*Sin[x], {x, 0, 2*Pi}] Õa 3: Nonsingular system Ûl ªË > ¦   ß9 ë{ systems inconsistent  Mathematica #*ô ° Ø§ t ·>    H  Q ú¸ 4 ú Ç ¯ ¦ § a  . singular d` {§K.   94Ð  ¦  2x − 2y + 2z = −2 −x + y + 3z = 0 −3x + 3y − 2z = 1 sys={2x - 2y - 2z == -2, -x +y +3z==0,-3x +3y -2z ==1}; sols=Solve[sys, {x, y, z}] ªË Õa 4: Singular system Ûl > ¦  1.4 Second-Order Linear Equations 4 d2 y − y = t − 2 − 5 cos t − e−t/2 . dt2 DSolve\ 6  ` ½½ Ã e. Mathematica 6õ °s {§ ¦ x  ¦ É    H ¨+ º   Ð £ ú 94 §  Ð K. 1 ] MATHEMATICA >í l jX  Ð ß  7 sol1 = DSolve[4 y’’[t]- y[t]== t -2 -5Cos[t] -Exp[-t/2], y[t], t] Õo d` çéy K. ª¦  ßß Ð ¦  Simplify[sol1] ªË Õa 5: Second Order Equation > 1.5 «« « Ü8è Ühe y = sin x for −π ≤ x ≤ 2π _ ÕAá\ Õ.  ªÔ ª9Ð ¦ sys = {2 x - 3 y + 4 z == 2, 3 x - 2 y + z == 0, x + y - z == 1}; sols = Solve[sys, {x, y, z}] Õa 6: Sine Ã Õol ªË > Ê <º ª s] # >_ Ã\ ô ÕAá\  ?. j Q h <º  ªÔ / Ê ¦Ç p1 = Plot[Sin[x], x, -Pi, 2 Pi] 8 p2 = Plot[ArcSin[x], x,-1, 1, PlotStyle → GrayLevel[0.3], DisplayFunction → Identity]; p3 = Plot[x, x, -Pi, 2 Pi, PlotStyle → Dashing[0.01], DisplayFunction → Identity]; ªQ £ úÉ  : Õ 6õ ° õ .  § r  r ªË Õa 7: Ã # > Õol > Ê <º Q h ª 3 x Ü8è Ühe j «« « S   ¶f Q ª9 Ð A_ ds 3 "\" #b> Õt .  é G H f (x, y ) = x2 y . x4 + 4y 2  ¦ Ê Í :  <º ' P ×\ Ã f (x, y )\ &_ô. ¦ Ç  ñ f[x_, y_] = x^2 y/(x^4 + 4 y^2) ª¦ ¶ ªÔ ª A £ îîQ 94 Õo 3 " ÕAá\ Õol 0K 6_ "§#\ {§ô. é ¦ § ¦ Ç Plot3D[f[x, y], {x, -1/2, 1/2}, {y, -1/2, 1/2}, Axes -> Automatic, Boxed -> False, PlotPoints -> {50, 50}] 1 ] MATHEMATICA >í l jX  Ð ß  9 ªË Õa 8: 3 " ÕAá > ¶ ªÔ é Õ Õas Ø§. s Õa\ ContourPlot` æ 1 +I  ªQ ªË 4  ªË  > ¦a > ¦ x A  ¼ p¦ þÐ  è Ã e. 0ü °s &_ Ã\ ­ º¸  A< ú ñ <º q  aÊ ContourPlot[f[x, y], {x, -1/2, 1/2}, {y, -1/2, 1/2}, PlotPoints -> {50, 50}] ContourPlot` 6K" Õ. ¦x   f ª9: r ªË Õa 9: Contour ÕAá > ªÔ 10 Parametric equations Ü8è Ühe «« « T\ ô Parametric equations e.   aÇ    x = t + sin 2t y = t + sin 3t where − 2π ≤ t ≤ 2π Mathematica parameter\ &_ # 6  . Ð H ¦  ñ    x a x[t_] = t + Sin[2 t]; y[t_] = t + Sin[3 t]; ParametricPlot[{x[t], y[t]}, {t, -2 Pi, 2 Pi}, AspectRatio -> Automatic] ªË Õa 10: Contour ÕAá > ªÔ 1.6 Nonlinear Diﬀerential EquationU" Nñ c+ 5 Ñ £ f  1 Ð 6 d\" ` ¹K . § H¦ aÏ dx dt dy dt = x(1 − x − y ), = y (0.5 − 0.25y − 0.75x) where x and y are nonnegative. (1) 1 ] MATHEMATICA >í l jX  Ð ß  11 Mathematica 0_ Å# d` F (x, y )ü G(x, y ) &_ô. Ð A ÒQ   ¦ < Ð ñ Ç F[x_,y_] := x(1 - x - y) G[x_,y_] := y(0.5 - 0.25 x - 0.75 x) s] s d(algebraic equation)` Û# critical points\ ½ô. j    ¦¦  Q ¦Ç  ¨ x(1 − x − y ) = 0 y (0.5 − 0.25x − 0.75y ) = 0 Õ critical points (0, 0), (1, 0), (0, 2) Õo (0.5, 0.5)s . ªQ  H  ª¦  : r Mathematica (the solutions of the system)` SolofSys ¿ Aü Ð  HH ¦  Ð º¦ < °s {§ Ê z'ô.  ú 94 ê  ´Ç SolofSys = Solve[{F[x,y] ==0, G[x,y]==0},{x,y}] ·_ dÜ ó õü ° critical points\ Ø§ô. critical points\ ½ ¡ ú ¼Ð  < úÉ r r ¦ ¦Ç  4 ¦ ¨ ªË Õa 11: Mathematica critical points\ ½ô õ° > Ð ¦ Ç ¯  ¨ ú ¡ H þ¼  h f  ÙÜ y & %\" (linear approximated)_ _` ¹ . H  ¸þ 1 v¦ aÏ 7, Jacobian matrix J  ¤ £ H  −x Fx (x, y ) Fy (x, y ) 1 − 2x − y J = = Gx (x, y ) Gy (x, y ) −0.75y 0.5 − 0.5y − 0.75x (2) 12 x = 0, y = 0s 6 linear system` %> .  £ § ¦ a  3  d x 1 0 x = dt y 0 0.5 y (3) A  0 d(3)_ eigenvaluesü eigenvectors` ½   < ¦  ¨  1 0 r1 = 1, ξ 1 = ; r2 = 0.5, ξ 2 = . 0 1 f " general solution r É 1 x 0 0.5t t = c1 e + c2 e 1 0 y x = 1, y = 0 Ä, 6 linear system` %> .   âº £ § ¦ a  3  −1 x d x −1 = dt y 0 −0.25 y A  0 d(6)_ eigenvaluesü eigenvectors` ½   < ¦  ¨  1 4 r1 = −1, ξ 1 = ; r2 = −0.25, ξ 2 = . 0 3 f " general solution r É x 1 −t 4 −0.25t . = c1 e + c2 e y 0 −3 (4) 1 ] MATHEMATICA >í l jX  Ð ß  13 x = 0, y = 2 Ä\, Ä > 6 linear system` %> .  âº¸ » £  § ¦ a  3  x −1 0 x d = dt y −1.5 −0.5 y (5) 0 d(6)_ eigenvaluesü eigenvectors` ½  A   < ¦  ¨  1 0 r1 = −1, ξ 1 = ; r2 = −0.5, ξ 2 = . 3 1 " general solution f r É 1 x 0 −0.5t −t . = c1 e + c2 e 1 3 y x = 0.5, y = 0.5 Ä, linear system 6õ °.   âº r§ É £ ú −0.5 x d x −0.5 = dt y −0.375 −0.125 y 0 d(6)_ eigenvaluesü eigenvectors` ½  A   < ¦  ¨  √ 1 1 −5 + 57 r1 = 0.1594, ξ 1 = , √ 16 (−3 − 57)/8 −1.3187 √ 1 −5 − 57 1 r2 = − −0.7844, ξ 2 = . √ 16 0.5687 (−3 + 57)/8 " general solution f r É 1 x −0.1594t 1 −0.7844t + c2 . = c1 e e y −1.3187 0.5687 (6) 14 mathematica Õ. Ð¸ ª9Ð º Ä critical points\ &_K . (x0, y 0)\ ' P &(0, 0)Ü ° d  ¦  ñ Ð ¦    Í : h ¼Ð úÉ  r  ¼Ð Ü (x1, y 1) &(0, 2), (x2, y 2) &(0.5, 0.5)Ü t} & (x3, y 3) r Éh r Éh ¼Ð  hÉ  r r É  h &(0, 2)Ü &_ô. ¼Ð ñ Ç x0=SolofSys[[1,1,2]]; y0=SolofSys[[1,2,2]]; x1=SolofSys[[2,1,2]]; y1=SolofSys[[2,2,2]]; x2=SolofSys[[3,1,2]]; y2=SolofSys[[3,2,2]]; x3=SolofSys[[4,1,2]]; y3=SolofSys[[4,2,2]]; ª¦ Õo critical points_ |½` SSETCRs &_ô.  9+ Ë¦ ¦ ñ Ç SETCR={{x0,y0},{x1,y1},{x2,y2},{x3,y3}}; ªË Õa 12: critical pointsü Õ |½` &_ô. > < ª 9+ ñ Ë¦ Ç £¼Ð 9¼ 6Ü 7'×(vectorﬁeld)\ Õ. 7'×_ #0 0 ≤ §  ¦  ª9Ð 9¼ 3A  H x ≤ 1.5ü 0 ≤ y ≤ 2 ¿. Õa` Õo Mathematica "§ë A < Ð º ªË ª >¦ H Hr îîÉ  < ú ü °. Fig = VectorFieldPlot[{F[x, y], G[x, y]}, {x, 0, 1.5}, {y, 0, 2}, 1 ] MATHEMATICA >í l jX  Ð ß  15 Axes -> True, AxesLabel -> {x, y}, AxesOrigin -> {0, 0}, Background -> LightBlue, Epilog -> {PointSize[Large], Point[SSETCR]}, ImageMargins -> 5, PlotRange -> {{-.1, 1.6}, {-0.1, 2.1}}, Ticks -> {{0, 0.25, 0.5, 0.75, 1, 1.25, 1.5}, {0, 0.5, 1, 1.5, 2}}] Õa 13: 7'×\ Õ. ªË > 9¼ ª9: ¦ r 16 mathematica Critical Points Å_ ` ¶(. d(2) Aü Ð Ò H úRÐ  É <  ¦  r ú ß+ º  °s >í½ Ã e. J  LMÜ &_ %. Õo MatrixForm`  É  H  ¼Ð ñi ª¦  ¦  x    6 # d` ó. ¦ r Õa 14: Jacobian` ½ô. ªË > ¦Ç  ¨ Af ¨ 0\" ½ô LM` s6 # critical point(x0, y 0)\ @{ # s '§ Ç ¦x    ¦  /9  > = ¦  ` LM0 ô. Õo Eigensystem "§# eigenvalueü eigenvector\   ª¦ Ç îîQÐ < ¦  ¨ Í : ú ½ . ' P õ°s eigenvalues Qt ¿ s eigenvectors°   ¯  ©¦   º ½ Ó ¯ ú    hþ¸ úÉ ½O¼Ð  ¨+ º  s. Qt &[ ° ~ZÜ  ½½ Ã e. t r Ó É  Õa 15: eigenvalueü eigenvector\ ½ô. ªË > < ¦Ç  ¨ 1.7 mathematica ÃÂ#  · ZZaæ ´É H + : z'½ M shift+enter\ ô. ¦Ç   → ³r − >\ æ 1Ü ë[# . ð H ¦x  ¼ l¼Ð ßþQ  t  ...
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