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(2) 1ÊÙ 5©§| .1 <E¥($Ä Áïá/¥<E¥(7/¥$Ä©§"(ÑÙ§UNé< E¥(K§O9/¥Úå|é<E¥(^) ¥($ħ f mM x d2 x m 2 =− 3 dt (x2 + y 2 ) 2. (1). d2 y f mM y m 2 =− 3 dt (x2 + y 2 m) 2. ùÒ´¹kü¼ê©§|" (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 2 / 49.
(3) 1ÊÙ 5©§| .1 <E¥($Ä Áïá/¥<E¥(7/¥$Ä©§"(ÑÙ§UNé< E¥(K§O9/¥Úå|é<E¥(^) ¥($ħ f mM x d2 x m 2 =− 3 dt (x2 + y 2 ) 2. (1). d2 y f mM y m 2 =− 3 dt (x2 + y 2 m) 2. ùÒ´¹kü¼ê©§|" (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 2 / 49.
(4) 1ÊÙ 5©§| .2 Ó ¨Ó . kÓ «+ÚÓ (½¡ )«+)¹3Ó¸¥§d u)!)kÚp^§ü«+NêþòmCz"Á ïáü«+NêþmCzêÆ ." Ó Ó ü«+p^êÆ . dx dt = x(r1 − ax − by). Ù¥d > 0, r. 2. (. (2). dy = y(−r2 + cx − dy) dt. "ù´¹kü¼ê©§|". > 0, c > 0. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 3 / 49.
(5) 1ÊÙ 5©§| .2 Ó ¨Ó . kÓ «+ÚÓ (½¡ )«+)¹3Ó¸¥§d u)!)kÚp^§ü«+NêþòmCz"Á ïáü«+NêþmCzêÆ ." Ó Ó ü«+p^êÆ . dx dt = x(r1 − ax − by). Ù¥d > 0, r. 2. (. (2). dy = y(−r2 + cx − dy) dt. "ù´¹kü¼ê©§|". > 0, c > 0. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 3 / 49.
(6) 5.1. 5©§|nØ. ©§| ¡ dx1 = f1 (t, x1 , x2 , · · · , xn ) dt dx2 = f2 (t, x1 , x2 , · · · , xn ) dt ··· ··· dxn = fn (t, x1 , x2 , · · · , xn ) dt. (3). ¹kn¼êx , x , · · · , x ©§|" 1. (. þ°ã²ÆA^êÆX). 2. n. ~©§. 1ÊÙ. March 24, 2010. 4 / 49.
(7) 5.1. 5©§|nØ. ©§|) XJ3|¼êx (t), x (t), · · · , x (t)§¦3[a, b]þke ¡nªf 1. 2. n. dxi (t) = fi (t, x1 , x2 , · · · , xn ), dt. i = 1, 2, · · · , n. ð¤á§K¡ù|¼ê´ ©§|(3)3[a, b]þ). (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 5 / 49.
(8) 5.1. 5©§|nØ. ©§|Ï) kn?¿~êc , c , · · · , c ) 1. 2. n. x1 = φ1 (t, c1 , c2 , · · · , cn ) x2 = φ2 (t, c1 , c2 , · · · , cn ) ··· ··· xn = φn (t, c1 , c2 , · · · , cn ). ¡ ©§|(3)Ï)" (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 6 / 49.
(9) 5.1. 5©§|nØ. 5©§| kXJ ©§|(3)¥zf (t, x , x , · · · , x )é¤k ¼êÑ´g§= i. 1. 2. n. dx1 = a11 (t)x1 + a12 (t)x2 + · · · + a1n (t)xn + f1 (t) dt dx2 = a21 (t)x1 + a22 (t)x2 + · · · + a2n (t)xn + f2 (t) dt ··· ··· dxn = an1 (t)x1 + an2 (t)x2 + · · · + ann (t)xn + fn (t) dt. ¡(4) 5©§|"Ù¥a m[a, b]þëY" (. þ°ã²ÆA^êÆX). ~©§. ij , fi (t)(i, j. 1ÊÙ. (4). 3«. = 1, 2, · · · , n). March 24, 2010. 7 / 49.
(10) 5.1. 5©§|nØ. 5©§|þ/ª. X 0 = A(t)X + F (t). . · · · a1n (t) · · · a2n (t) , F (t) = ··· ··· · · · ann (t) x1 (t) x2 (t) X(t) = ··· xn (t). a11 (t) a12 (t) a21 (t) a22 (t) A(t) = ··· ··· an1 (t) an2 (t). (. þ°ã²ÆA^êÆX). (5). ~©§. 1ÊÙ. f1 (t) f2 (t) ··· fn (t). March 24, 2010. 8 / 49.
(11) 5.1. 5©§|nØ. 5©§|þ/ª. X 0 = A(t)X + F (t). . · · · a1n (t) · · · a2n (t) , F (t) = ··· ··· · · · ann (t) x1 (t) x2 (t) X(t) = ··· xn (t). a11 (t) a12 (t) a21 (t) a22 (t) A(t) = ··· ··· an1 (t) an2 (t). (. þ°ã²ÆA^êÆX). (5). ~©§. 1ÊÙ. f1 (t) f2 (t) ··· fn (t). March 24, 2010. 8 / 49.
(12) 5©§|p 5©§ d dn x dn−1 x dx + a (t) + · · · + a (t) + an (t)x = f (t) 1 n−1 n n−1 dt dt dt x(t ) = η , x0 (t ) = η , · · · , x(n−1) (t ) = η 0 1 0 2 0 n . 0 1 0 0 0 1 X0 = ··· ··· ··· −an (t) −an−1 (t) −an−2 (t) X(t0 ) = . η1 η2 .. .. ··· ··· ··· ···. . 0 0 X + ··· −a1 (t). 0 0 .. . 0 f (t). =η . ηn (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 9 / 49.
(13) 5©§|p 5©§ d dn x dn−1 x dx + a (t) + · · · + a (t) + an (t)x = f (t) 1 n−1 n n−1 dt dt dt x(t ) = η , x0 (t ) = η , · · · , x(n−1) (t ) = η 0 1 0 2 0 n . 0 1 0 0 0 1 X0 = ··· ··· ··· −an (t) −an−1 (t) −an−2 (t) X(t0 ) = . η1 η2 .. .. ··· ··· ··· ···. . 0 0 X + ··· −a1 (t). 0 0 .. . 0 f (t). =η . ηn (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 9 / 49.
(14) p 5©§|=z ~ 3 dx dx d2 x dy = a (t)x + a (t) + a (t) + b (t)y + b (t) + f1 (t) 1 2 3 1 2 dt3 dt dt2 dt. (6). 2 2 d y = c1 (t)x + c2 (t) dx + c3 (t) d x + d1 (t)y + d2 (t) dy + f2 (t) dt2 dt dt2 dt. C§-. x = x1 ,. (. þ°ã²ÆA^êÆX). dx1 dx2 dy1 = x2 , = x3 ; y = y1 , = y2 dt dt dt. ~©§. 1ÊÙ. March 24, 2010. 10 / 49.
(15) p 5©§|=z ~ 3 dx dx d2 x dy = a (t)x + a (t) + a (t) + b (t)y + b (t) + f1 (t) 1 2 3 1 2 dt3 dt dt2 dt. (6). 2 2 d y = c1 (t)x + c2 (t) dx + c3 (t) d x + d1 (t)y + d2 (t) dy + f2 (t) dt2 dt dt2 dt. C§-. x = x1 ,. (. þ°ã²ÆA^êÆX). dx1 dx2 dy1 = x2 , = x3 ; y = y1 , = y2 dt dt dt. ~©§. 1ÊÙ. March 24, 2010. 10 / 49.
(16) p 5©§|=z ¹k5¼êx , x , x , y , y 5©§| 1. . x01 x02 x03 y10 y20. (. . . = . 2. 3. 1. 2. 0 1 0 0 0 0 0 1 0 0 a1 (t) a2 (t) a3 (t) b1 (t) b2 (t) 0 0 0 0 1 c1 (t) c2 (t) c3 (t) d1 (t) d2 (t). þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. . x1 x2 x3 y1 y2. + . March 24, 2010. 0 0 f1 (t) 0 f2 (t). . . 11 / 49.
(17) 5.1. 5©§|nØ. 35½n A(t)´n × nÝ §F (t)´nþ§§Ñ3«m[a, b]þë Y"Kéu«m[a, b]þ?Ûêt 9?~ê þη = (η , η , · · · , η ) §§|X = A(t)X + F (t)3 )φ(t)§½Âu«m[a, b]þ§ ÷vЩ^φ(t ) = η" 1. 2. n. 0 0. T. 0. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 12 / 49.
(18) àg5©§|)m( àg5©§| e5©§|(5)¥F (t) ≡ 0§= X 0 = A(t)X. (7). ¡(7)àg5©§|"eF (t) 6≡ 0§¡(5)àg5 ©§|". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 13 / 49.
(19) àg5©§|)m( U\n ½n5.2 XJX (t), X (t), · · · , X (t)P´àg5©§ |(7)m)§K§5|Ü c X (t)´(7))"Ù ¥c , c , · · · , c ´?¿~ê" 1. 2. m. m. i. i. i=1. 1. (. 2. m. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 14 / 49.
(20) àg5©§|)m( þ¼ê|5'5 X (t), X (t), · · · , X (t)´«m[a, b]þþ¼ê§XJ3Ø "~êc , c , · · · , c §¦eªð¤á 1. 2. n. 1. 2. n. c1 X1 (t) + c2 X2 (t) + · · · + cn Xn (t) ≡ 0, t ∈ [a.b]. K¡d|þ¼ê3«m[a, b]þ5'§ÄK¡ù|¼ê5 Ã'". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 15 / 49.
(21) àg5©§|)m( þ¼ê|Wronski1ª ¡«m[a, b]þnþ¼ê X1 (t) = . x11 (t) x21 (t) .. .. . . , X2 (t) = . xn1 (t). x12 (t) x22 (t) .. .. . . x1n (t) x2n (t) , · · · , Xn (t) = .. . xn2 (t) xnn (t). ¤1ªù|þ¼êWronski1ª§= . x11 (t) x12 (t) · · · x21 (t) x22 (t) · · · W [X1 (t), X2 (t), · · · , Xn (t)] ≡ ··· ··· ··· xn1 (t) xn2 (t) · · · (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. x1n (t) x2n (t) ··· xnn (t). March 24, 2010. 16 / 49.
(22) àg5©§|)m( þ¼ê5'5O ½n5.3 XJþ¼êX (t), X (t), · · · , X (t)3«m[a, b]þ5 '§K§3[a, b]þWronski1ªð u"" þ¼ê5'5O ½n5.4 XJþ¼êX (t), X (t), · · · , X (t)´§|(7)n )§K§3«m[a, b]þ5Ã'¿©7^ÙWronski1 ªW (t) 6= 0, t ∈ [a, b]" þ¼ê5'5O ½n5.5 §|(7)½3n5Ã')" (. þ°ã²ÆA^êÆX). 1. 2. n. 1. 2. n. ~©§. 1ÊÙ. March 24, 2010. 17 / 49.
(23) àg5©§|)m( Ï)(½n ½n5.6 XJX (t), X (t), · · · , X (t)´§|(7)n5Ã' )§K§|(7)Ï)±L« 1. 2. n. X(t) = c1 X1 (t) + c2 X2 (t) + · · · + cn Xn (t). (8). Ù¥c , c , · · · , c ´?¿~ê" Ï)(8)¹ §|(7)¤k )" 1. (. 2. n. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 18 / 49.
(24) àg5©§|)m( úª ½n 1ª. Liouville 5.7 X1 (t), X2 (t), · · · , Xn (t) Wronski W (t). ´§|(7)?¿n)§K§ ÷v 5§. W 0 (t) = [a11 (t) + a22 (t) + · · · + ann (t)]W (t). Ï k Rt. W (t) = W (t0 ) · e. (. þ°ã²ÆA^êÆX). t0 [a11 (s)+a22 (s)+···+ann (s)]ds. ~©§. 1ÊÙ. ,. t, t0 ∈ [a, b]. (9). March 24, 2010. 19 / 49.
(25) àg5©§|)m( àg5©§|)Ý. XJn × nÝ zÑ´(7))§K¡ùÝ (7) )Ý " àg5©§|Ä)Ý. ¡(7)n5Ã')|¤)Ý (7)Ä)Ý "P Φ(t)§= Φ(t) = [φ (t), φ (t), · · · , φ (t)] Ù¥φ (t), i = 1, 2, · · · , n´(7)n5Ã')" 1. 2. n. i. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 20 / 49.
(26) àg5©§|)m( )Ý 5 ½n5.8 (7)½3Ä)Ý Φ(t)"XJψ(t)´(7)? )§K ψ(t) = Φ(t)c Ù¥c´(½~êþ" )Ý 5 ½n5.9 (7))Ý Φ(t)´Ä)Ý ¿©7^3« m[a, b]þ,:t ?§det Φ(t ) 6= 0" 0. (. þ°ã²ÆA^êÆX). 0. ~©§. 1ÊÙ. March 24, 2010. 21 / 49.
(27) àg5©§|)m( )Ý 5 ½n5.10 XJΦ(t)´(7)3«m[a, b]þÄ)Ý §C ´Û Én × n~êÝ §KΦ(t)C ´(7)3«m[a, b]þÄ)Ý " )Ý 5 ½n5.11 XJΦ(t), Ψ(t)´(7)3«m[a, b]þüÄ)Ý §K 3ÛÉn × n~êÝ C §¦3«m[a, b]þΨ(t) = Φ(t)C ". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 22 / 49.
(28) àg5©§|)m( ~1 y. . et −et. e3t e3t. . 2 1. . Φ(t) =. ´§| 0. X =. Ä)Ý §¿ÑÏ)" (. þ°ã²ÆA^êÆX). ~©§. 1 2. 1ÊÙ. . X. March 24, 2010. 23 / 49.
(29) àg5©§|)m( ~2 ÁѱÝ. e−t Φ(t) = 0 −e−t . 0 e−t −e−t. e2t e2t e2t. Ä)Ý àg5©§|". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 24 / 49.
(30) àg5©§|)8Ü5 ½n5.12 ¯ ´àg5©§|(5))§X(t)´§éAàg XJX(t) ¯ + X(t) E´àg5© 5©§|(7))§KX(t) §|(5))" ½n5.13 XJX (t), X (t)´àg5©§|(5)ü)§ KX (t) − X (t) ´éAàg5©§|(7))" 1. 1. (. 2. 2. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 25 / 49.
(31) àg5©§|)8Ü5 Ï)(½n ¯ ´àg Φ(t)´àg5©§|(7)Ä)Ý §X(t) 5©§|(5),)§Kàg5©§|(5)? )X(t)ÑL« ¯ X(t) = Φ(t)c + X(t). Ù¥c´(½~êþ". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 26 / 49.
(32) àg5©§|~êC´{. Φ(t)´§|(7)Ä)Ý §u´§|(7)Ï) X(t) = Φ(t)c. ·ßÿµ§|(5)kù«/ª)§cAt¼ê§=b X(t) = Φ(t)c(t). (10). Φ0 (t)c(t) + Φ(t)c0 (t) = A(t)Φ(t)c(t) + F (t). (11). ´§|(5))"Ù¥c(t)´½þ¼ê" ò(10)\§|(5)¥§ ÏΦ(t)´§|(7)Ä)Ý §¤± Ïd§(11)C (. þ°ã²ÆA^êÆX). Φ0 (t) = A(t)Φ(t) Φ(t)c0 (t) = F (t) ~©§ 1ÊÙ. March 24, 2010. 27 / 49.
(33) àg5©§|~êC´{ qÏΦ(t)´_§l ü>ÓÈ©§. c0 (t) = Φ−1 (t)F (t) Z. t. c(t) = c(t0 ) +. ò(12)£(10)¥§. Φ−1 (s)F (s)ds. (12). t0. Z. t. Φ−1 (s)F (s)ds. X(t) = Φ(t)c(t0 ) + Φ(t). u´§|(5)Ï). t0. Z. t. X(t) = Φ(t)˜ c + Φ(t)c(t0 ) + Φ(t) Φ−1 (s)F (s)ds t0 Z t = Φ(t)c + Φ(t) Φ−1 (s)F (s)ds (. þ°ã²ÆA^êÆX). t0. ~©§. 1ÊÙ. March 24, 2010. (13) 28 / 49.
(34) àg5©§|~êC´{ ~3 Á¦§|X = A(t)X + F (t)Ï)"Ù¥ 0. A(t) =. 2 1. 1 2. . ,. F (t) =. e2t 0. . 9éA§|X = A(t)X Ä)Ý 0. Φ(t) =. (. þ°ã²ÆA^êÆX). et −et. ~©§. e3t e3t. 1ÊÙ. . March 24, 2010. 29 / 49.
(35) 5.2. ~Xê5©§|. ~Xê5©§| A´n × n~êÝ §K~Xêàg5©§| X 0 = AX + F (t). (14). ÙéA~Xêàg5©§| X 0 = AX. (. þ°ã²ÆA^êÆX). ~©§. (15). 1ÊÙ. March 24, 2010. 30 / 49.
(36) 5.2. ~Xê5©§|. Ý êexp(A) A´n × n~êÝ §¡ exp A =. Ý ê". (. ∞ X An n=0. þ°ã²ÆA^êÆX). n!. =E+A+. ~©§. A2 An + ··· + + ··· 2! n!. 1ÊÙ. March 24, 2010. (16). 31 / 49.
(37) 5.2. ~Xê5©§|. Ý êexp A5 (1) XJÝ A, B §=AB = BA§K exp A · exp B = exp(A + B). (17). é?ÛÝ A§(exp A) 3§ −1. (2). (exp A)−1 = exp(−A). (18). XJT ´ÛÉÝ §K. (3). exp(T −1 AT ) = T −1 (exp A)T. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. (19). March 24, 2010. 32 / 49.
(38) 5.2. ~Xê5©§|. Ý ê¼êexp(At) ¡ exp(At) =. Ý ê¼ê". (. þ°ã²ÆA^êÆX). ∞ X An tn n=0. ~©§. (20). n!. 1ÊÙ. March 24, 2010. 33 / 49.
(39) 5.2. ~Xê5©§|. 5 ½n5.15 Ý. Φ(t) = exp(At) ´§|(15)Ä)Ý § Φ(0) = E" (1) §|(15)Ï)±L« exp(At). X(t) = exp(At)c. XJΦ(t)´§|(15), ØÓuexp(At)Ä)Ý. §K. (2). exp(At) = Φ(t)Φ−1 (0). (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. (21). March 24, 2010. 34 / 49.
(40) 5.2. ~Xê5©§|. 5 ½n5.15 Ý. Φ(t) = exp(At) ´§|(15)Ä)Ý § Φ(0) = E" (1) §|(15)Ï)±L« exp(At). X(t) = exp(At)c. XJΦ(t)´§|(15), ØÓuexp(At)Ä)Ý. §K. (2). exp(At) = Φ(t)Φ−1 (0). (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. (21). March 24, 2010. 34 / 49.
(41) 5.2. ~Xê5©§|. ~1 A´éÝ. . a1 0 A= ··· 0. 0 a2 ··· 0. ··· ··· ··· ···. 0 0 ··· an. Á¦X = AX Ä)Ý exp(At)" ~2 Á¦X = 20 12 X Ä)Ý exp(At)" 0. 0. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 35 / 49.
(42) ~Xêàg5©§|){ ©Ûµ§|(15)¬Ø¬k/X X(t) = eλt · c,. ,. (λ, c. )Qº ò(22)\§|(15)¥§. c 6= 0). (22). (λE − A)c = 0. ù¿X§e · c´§|(15))¿©7^µλ´Ý. AA c´éAuλAþ" λt. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 36 / 49.
(43) ~Xêàg5©§|){ Ý AA´ü/ ½n5.16 XJ§|(15)XêÝ AnA λ , λ , · · · , λ *dpɧ v , v , · · · , v ´§éAA þ§K§|(15)Ä)Ý 1. 2. n. 1. 2. n. Φ(t) = [eλ1 t v1 , eλ2 t v2 , · · · , eλn t vn ]. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. (23). March 24, 2010. 37 / 49.
(44) Ý AA´ü/ ~3 ¦§|X = AX Ä)Ý "Ù¥ 0. A=. 6 −3 2 1. . ~4 ¦§|X = AX Ä)Ý "Ù¥ 0. A=. (. þ°ã²ÆA^êÆX). 3 5 −5 3. ~©§. 1ÊÙ. . March 24, 2010. 38 / 49.
(45) ~Xêàg5©§|){ Ý AAk/ m©){ ½Xê{". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 39 / 49.
(46) Ý AAk/ m©){ ½n5.18 XJ§|(15)XêÝ AkkØÓA λ , λ , · · · , λ §§ê©On , n , · · · , n § n + n + · · · + n = n"-η©On ü þe , e , · · · , e § K÷vЯK(24))d(25)(½§©OP X (t), X (t), · · · , X (t)§u´§|(15)Ä)Ý 1. 1. 1. 2. k. 2. 1. 2. k. k. 2. 1. 2. n. n. exp(At) = [X1 (t), X2 (t), · · · , Xn (t)]. . (. þ°ã²ÆA^êÆX). X 0 = AX X(0) = η ~©§. 1ÊÙ. (24). March 24, 2010. 40 / 49.
(47) Ý AAk/ m©){ ½n5.18 XJ§|(15)XêÝ AkkØÓA λ , λ , · · · , λ §§ê©On , n , · · · , n § n + n + · · · + n = n"-η©On ü þe , e , · · · , e § K÷vЯK(24))d(25)(½§©OP X (t), X (t), · · · , X (t)§u´§|(15)Ä)Ý 1. 1. 1. 2. k. 2. 1. 2. k. k. 2. 1. 2. n. n. exp(At) = [X1 (t), X2 (t), · · · , Xn (t)]. . (. þ°ã²ÆA^êÆX). X 0 = AX X(0) = η ~©§. 1ÊÙ. (24). March 24, 2010. 40 / 49.
(48) m©){ X(t) = exp(At)η =. k P. exp(At)vi =. i=1. = = =. k P i=1 k P i=1 k P. exp(At) · E · vi. i=1. exp(At) · eλi t · exp(−λi Et) · vi eλi t · exp(A − λi E)t · vi eλi t {E + t(A − λi E) +. i=1. +. (. k P. (25) t2 (A − λi E)2 + · · · 2!. tni −1 (A − λi E)ni −1 }vi (ni − 1)!. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 41 / 49.
(49) m©){ 5 XJÝ AknA§KnîªmØ7©)"d (25)C. t2 tn−1 2 X(t) = e {E + t(A − λE) + (A − λE) + · · · + (A − λE)n 2! (n − 1)! λt. 2â½n5.18§§|(15)Ä)Ý. exp(At) = eλt {E + t(A − λE) +. (. þ°ã²ÆA^êÆX). ~©§. t2 tn−1 (A − λE)2 + · · · + (A − λ 2! (n − 1)!. 1ÊÙ. March 24, 2010. 42 / 49.
(50) m©){ ~5 ¦§|X = AX Ä)Ý "Ù¥ 0. . 1 A= 1 0. 1 1 3 −1 2 2. ~6 ¦§|X = AX Ä)Ý "Ù¥ 0. . 3 4 −10 A = 2 1 −2 2 2 −5 (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 43 / 49.
(51) Ý AAk/. ½Xê{ ½n5.19 XJ§|(15)XêÝ AkkØÓA λ , λ , · · · , λ §ê©On , n , · · · , n § n + n + · · · + n = n"Kén Aλ §§|(15)kn 5Ã')§/X 1. 1. 2. k. 2. 1. k. 2. k. i. i. i. X(t) = (R0 + R1 t + · · · + Rni −1 tni −1 )eλi t. (28). Ù¥þR , R , · · · , R dÝ §(½: 0. 1. ni −1. (A − λi E)R0 = R1 (A − λi E)R1 = 2R2 ··· ··· (A − λi E)Rni −2 = (ni − 1)Rni −1 (A − λi E)ni R0 = 0 (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. (29). March 24, 2010. 44 / 49.
(52) Ý AAk/ ½Xê{ ^½Xê{¦)c¡~5Ú~6". (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 45 / 49.
(53) ~Xêàg5©§|){ ~êC´úª Z. t. exp[(t − s)A]F (s)ds. X(t) =. (30). t0. Ú. Z. t. X(t) = exp[(t − t0 )A]η +. exp[(t − s)A]F (s)ds. (31). 1ÊÙ. 46 / 49. t0. (. þ°ã²ÆA^êÆX). ~©§. March 24, 2010.
(54) ~Xêàg5©§|){ ~7 ¦§|X = AX + F (t)÷vЩ^X(0) = η)"Ù¥ 0. A=. (. 3 −5. þ°ã²ÆA^êÆX). 5 3. . ,. F (t) =. ~©§. e−t 0. 1ÊÙ. . ,. η=. 0 1. . March 24, 2010. 47 / 49.
(55) 5.3. A^¢~. ÿU6Äó¿ . 3ïÄD/¾DÂ!«+)!¸À/!Ô3<N© Ù ¯K¥§²~r¤ïįÔw¤dkÜ©|¤X Ú§ zÜ©¡ó¿"§äk±eA:µ (1) zó¿k½Nþ§S¹zÑþ!©ÙXÔ (½Uþ)¶ (2) ó¿m±9ó¿ ܸmþ?1Ô(½Uþ) §¿ÑlÔ(½Uþ)Åð½Æ" ùXÚ¡ó¿XÚ"e¡â<NSÿUáÂ!Ü ¤!ü Ån5ïáÿU6Äó¿ ." (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 48 / 49.
(56) 5.3. A^¢~. <E¥($Ä ¦)¥($ħ f mM x d2 x =− m 3 2 dt (x2 + y 2 ) 2. (32). d2 y f mM y m 2 =− 3 2 dt (x + y 2 m) 2. (. þ°ã²ÆA^êÆX). ~©§. 1ÊÙ. March 24, 2010. 49 / 49.
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