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(2) 1oÙp ©§. ~XJ . 'é3,°S÷XÊ1§·ÔéTÐ u'é Ü1 úp?"·é'éu~X§'éÝ 0.42 úp/©§~XÝ'éÝ2"Á¯'éÊ1õ òÂ¥º 'éЩ:3Q (1, 0) ?§$IJ1y ¶§t Q :§~XЩ:3P (0, 0)?§÷y = y(x)J§' éÝv = 0.42§K3t §~X3:P (x, y)?§d'é 3:Q(1, v t)"du~X3JÂL§¥©ª'é§ ~X $Äд÷y = y(x) §@o§~X$Ä § 0. 0. 0. 0. dy v0 t − y = dx 1−x. (1). ~X1¦Ý2v §©Y²$ÄÚR$ħ (. þ°ã²ÆA^êÆX). 0. ~©§. 1oÙ. March 19, 2010. 2 / 52.
(3) 1oÙp ©§. ~XJ . 'é3,°S÷XÊ1§·ÔéTÐ u'é Ü1 úp?"·é'éu~X§'éÝ 0.42 úp/©§~XÝ'éÝ2"Á¯'éÊ1õ òÂ¥º 'éЩ:3Q (1, 0) ?§$IJ1y ¶§t Q :§~XЩ:3P (0, 0)?§÷y = y(x)J§' éÝv = 0.42§K3t §~X3:P (x, y)?§d'é 3:Q(1, v t)"du~X3JÂL§¥©ª'é§ ~X $Äд÷y = y(x) §@o§~X$Ä § 0. 0. 0. 0. dy v0 t − y = dx 1−x. (1). ~X1¦Ý2v §©Y²$ÄÚR$ħ (. þ°ã²ÆA^êÆX). 0. ~©§. 1oÙ. March 19, 2010. 2 / 52.
(4) ~XJ .. ÷v±e'Xª r (. ò(1)U. dx 2 dy ) + ( )2 = 2v0 dt dt. v0 t − y = (1 − x). ò(3)ü>Óéx¦ê§ d(2). v0. þ°ã²ÆA^êÆX). dy dx. (3). dt dy d2 y dy − = (1 − x) 2 − dx dx dx dx. ò(5)\(4)¥§ (. (2). dt 1 = dx 2v0. r 1+(. ~©§. (4). dy 2 ) dx. 1oÙ. (5). March 19, 2010. 3 / 52.
(5) ~XJ . . r. dy 2 ) dy dx = 2 2(1 − x) dx y(0) = 0, y 0 (0) = 0. (. þ°ã²ÆA^êÆX). 2. 1+(. ~©§. 1oÙ. (6). March 19, 2010. 4 / 52.
(6) 4.1. n. p ©§ü {. ©§/ª F (t, x, x0 , · · · , x(n) ) = 0. (7). Ù¥n ≥ 2§tgCþ§x¼ê". (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 5 / 52.
(7) 4.1. p ©§ü {. Øw¹¼êx§ XJ(7)¥Øw¹¼êx9Ùk − 1 (k ≥ 1) ê§ §(7) CþO§-x. F (t, x(k) , · · · , x(n) ) = 0 (k). =y. §K. x(k+1) =. u´(8)C. dy dn−k y , · · · , x(n) = n−k dt dt. F (t, y, · · · ,. § êü. k " þ°ã²ÆA^êÆX (. ). (8). dn−k y )=0 dtn−k. ~©§. 1oÙ. (9). March 19, 2010. 6 / 52.
(8) 4.1. p ©§ü {. Øw¹¼êx§ XJ(7)¥Øw¹¼êx9Ùk − 1 (k ≥ 1) ê§ §(7) CþO§-x. F (t, x(k) , · · · , x(n) ) = 0 (k). =y. §K. x(k+1) =. u´(8)C. dy dn−k y , · · · , x(n) = n−k dt dt. F (t, y, · · · ,. § êü. k " þ°ã²ÆA^êÆX (. ). (8). dn−k y )=0 dtn−k. ~©§. 1oÙ. (9). March 19, 2010. 6 / 52.
(9) Øw¹¼êx§ XJU¦Ñ(9)Ï) y = φ(t, c1 , · · · , cn−k ). ¿X x = φ(t, c , · · · , c ) éþªëYÈ©kg§§(8)Ï)" (k). (. þ°ã²ÆA^êÆX). 1. ~©§. n−k. 1oÙ. March 19, 2010. 7 / 52.
(10) 4.1. p ©§ü {. ~1 ¦§t ddtx − ddtx = 0Ï)" ~2 ¦§y = e − cos xÏ)" 5. 4. 5. 000. (. 4. 2x. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 8 / 52.
(11) p ©§ü {. 4.1. Øw¹gCþt§. F (x, x0 , · · · , x(n) ) = 0. ÏLCþO§rxw¤#gCþ§K§ü " -x = y§K. (10). 0. dx =y dt d2 x dy dy dx dy = = · =y· 2 dt dt dx dt dx dy dy ) d(y ) dx 2 dx dy 2 2d y dx dx = = · = y( ) + y dt3 dt dx dt dx dx2 (þ°ã²ÆA^êÆX) ~©§ 1oÙ March 19, 2010 3. d(y. 9 / 52.
(12) p ©§ü {. 4.1. Øw¹gCþt§. F (x, x0 , · · · , x(n) ) = 0. ÏLCþO§rxw¤#gCþ§K§ü " -x = y§K. (10). 0. dx =y dt d2 x dy dy dx dy = = · =y· 2 dt dt dx dt dx dy dy ) d(y ) dx 2 dx dy 2 2d y dx dx = = · = y( ) + y dt3 dt dx dt dx dx2 (þ°ã²ÆA^êÆX) ~©§ 1oÙ March 19, 2010 3. d(y. 9 / 52.
(13) Øw¹gCþt§ dy d ^êÆ8B{§x ^y, dx ,··· , dx §(10)C. k−1. (k). =k#§. y. k−1. 5L"u´. (k ≤ n). 2 dy dy 2 2d y F (x, y, y , y( ) + y ,···) = 0 dx dx dx2. H(x, y,. dy dn−1 y , · · · , n−1 ) = 0 dx dx. (11). ù´±xgCþ§y¼ên − 1 §" (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 10 / 52.
(14) 4.1. p ©§ü {. ~3 ¦)ЯK xx| − e= 0,= 0x | 00. t=0. (. þ°ã²ÆA^êÆX). 2x. 0. t=0. ~©§. =1. ". 1oÙ. March 19, 2010. 11 / 52.
(15) 4.2. p 5©§nØ. 5©§ ¡§ n. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = f (t) 1 n n−1 dt dt dt. (12). n 5©§"Ù¥a (t) (i = 1, 2, · · · , n)9f (t)3« m[a, b]þëY" XJf (t) ≡ 0§K§(12)C i. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = 0 1 n n−1 dt dt dt. (13). ¡n àg5©§§{¡àg5§" XJf (t) 6≡ 0§¡(12)n àg5©§§{¡àg 5§" þ°ã²ÆA^êÆX ~©§ 1oÙ (. ). March 19, 2010. 12 / 52.
(16) 4.2. p 5©§nØ. 5©§ ¡§ n. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = f (t) 1 n n−1 dt dt dt. (12). n 5©§"Ù¥a (t) (i = 1, 2, · · · , n)9f (t)3« m[a, b]þëY" XJf (t) ≡ 0§K§(12)C i. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = 0 1 n n−1 dt dt dt. (13). ¡n àg5©§§{¡àg5§" XJf (t) 6≡ 0§¡(12)n àg5©§§{¡àg 5§" þ°ã²ÆA^êÆX ~©§ 1oÙ (. ). March 19, 2010. 12 / 52.
(17) n. 5©§. (1) (1 − t2 ). ¶. d2 x dx − 2t + 2x = 0 dt2 dt. d2 y + y = x; dx2 d2 x dx (3) 2 + + x = sin t; dt dt 2 dy dy (4) x2 2 − (x + 2)(x − y) = x4 dx dx (2). (. þ°ã²ÆA^êÆX). ~©§. ". 1oÙ. March 19, 2010. 13 / 52.
(18) 4.2. p 5©§nØ. ЯK)35½n ½n4.1 XJ¼êa (t) (i = 1, 2, · · · , n)Úf (t)3«m[a, b]þëY§ Ké? t ∈ [a, b]9?¿x , x , · · · , x §Ð¯K i. 0. 0. (1) 0. (n−1) 0. n dn−1 x dx d x + a (t) + · · · + an−1 (t) + an (t)x = f (t) 1 n n−1 dt dt dt (1) (n−1) x(t0 ) = x0 , x0 (t0 ) = x0 , · · · , x(n−1) (t0 ) = x0. (14). 3)x = φ(t), t ∈ [a, b]" (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 14 / 52.
(19) p 5©§nØ. 4.2. àg5§)m( ½n4.2 XJx (t), x (t), · · · , x (t)´§(13)k)§K§ 5|Üc x (t) + c x (t) + · · · + c x (t) ´§(13))"Ù ¥c , c , · · · , c ´?¿~ê" ©Ûµn àg5§(13)) N|¤8 ÜV = {x(t)|x(t)§(13))}§ù)8Ü÷vµ (1) é?¿x(t) ∈ V §Kc x(t) ∈ V ¶ (2) é?¿x (t) ∈ V, x (t) ∈ V §Kc x (t) + c x (t) ∈ V § Ï V ¤5m§¡)m"@où)mê ´õQºÄ.´oQº 1. 1 1. 1. 2. 2. k. 2 2. k k. k. 1. 1. (. þ°ã²ÆA^êÆX). 2. 1 1. ~©§. 1oÙ. 2 2. March 19, 2010. 15 / 52.
(20) p 5©§nØ. 4.2. àg5§)m( ½n4.2 XJx (t), x (t), · · · , x (t)´§(13)k)§K§ 5|Üc x (t) + c x (t) + · · · + c x (t) ´§(13))"Ù ¥c , c , · · · , c ´?¿~ê" ©Ûµn àg5§(13)) N|¤8 ÜV = {x(t)|x(t)§(13))}§ù)8Ü÷vµ (1) é?¿x(t) ∈ V §Kc x(t) ∈ V ¶ (2) é?¿x (t) ∈ V, x (t) ∈ V §Kc x (t) + c x (t) ∈ V § Ï V ¤5m§¡)m"@où)mê ´õQºÄ.´oQº 1. 1 1. 1. 2. 2. k. 2 2. k k. k. 1. 1. (. þ°ã²ÆA^êÆX). 2. 1 1. ~©§. 1oÙ. 2 2. March 19, 2010. 15 / 52.
(21) àg5§)m( ¼ê5'5 ¼êx (t), x (t), · · · , x (t)´«m[a, b]þk¼ê§XJ3 Ø "~êc , c , · · · , c §¦eªð¤á 1. 2. k. 1. 2. k. c1 x1 (t) + c2 x2 (t) + · · · + ck xk (t) ≡ 0,. t ∈ [a, b]. K¡¼êx (t), x (t), · · · , x (t)3«m[a, b]þ5'§ÄK¡ù. ¼ê5Ã'" 1. (. 2. þ°ã²ÆA^êÆX). k. ~©§. 1oÙ. March 19, 2010. 16 / 52.
(22) àg5§)m( ¼êWronski1ª ¼êx (t), x (t), · · · , x (t)3«m[a, b]þ©O3k − 1 ê§ 1ª 1. 2. k.
(23)
(24) x1 (t) x2 (t) ···
(25) 0
(26) x01 (t) x2 (t) ··· W [x1 (t), x2 (t), · · · , xk (t)] ≡
(27)
(28) · · · · · · ···
(29) (k−1) (k−1)
(30) x (t) x2 (t) · · · 1. ¡ù ¼êWronski1ª" (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. xk (t) x0k (t) ··· (k−1) xk (t). March 19, 2010.
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(38). 17 / 52.
(39) àg5§)m( ¼ê5'5O ½n4.3 ¼êx (t), x (t), · · · , x (t)´«m[a, b]þk¼ê§X J3Ø "~êc , c , · · · , c §¦eªð¤á 1. 2. k. 1. 2. k. c1 x1 (t) + c2 x2 (t) + · · · + ck xk (t) ≡ 0,. t ∈ [a, b]. K¡¼êx (t), x (t), · · · , x (t)3«m[a, b]þ5'§ÄK¡ù. ¼ê5Ã'" 1. (. 2. þ°ã²ÆA^êÆX). k. ~©§. 1oÙ. March 19, 2010. 18 / 52.
(40) ¼ê5'5O 5¿ (1)½n4.3_½nؽ¤á"~X§~2¥Ñ¼ êx (t)Úx (t)Wronski1ªð u"§ x (t)Úx (t)3(−∞, +∞)þ%´5Ã'¶ (2)XJ¼êx (t), x (t), · · · , x (t)Wronski1ª3«m[a, b]þ ,:t ?Ø u"§=W (t ) 6= 0§Kù ¼ê3«m[a, b]þ7 5Ã'" 1. 2. 1. 2. 1. 0. (. þ°ã²ÆA^êÆX). 2. n. 0. ~©§. 1oÙ. March 19, 2010. 19 / 52.
(41) àg5§)m( ¼ê5'5O ½n4.4 ¼êx (t), x (t), · · · , x (t)´§(13)n)§K§ 3«m[a, b]þ5Ã'¿©7^ÙWronski 1 ªW (t) 6= 0, t ∈ [a, b]" ½n4.4`² XJ3t ∈ [a, b]§¦W (t ) = 0§Kùn)3«m[a, b]þ5 '¶ XJ3t ∈ [a, b]§¦W (t ) 6= 0§Kùn)3«m[a, b]þ5 Ã'" 1. (. 2. n. 0. 0. 0. 0. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 20 / 52.
(42) àg5§)m( ¼ê5'5O ½n4.5 n àg5§(13)½3n5Ã')" ¼ê5'5O ½n4.6 XJx (t), x (t), · · · , x (t)´§(13)n5Ã')§ K§(13)Ï)±L« 1. 2. n. x(t) = c1 x1 (t) + c2 x2 (t) + · · · + cn xn (t). (15). Ù¥c , c , · · · , c ´?¿~ê" Ï)(15)) §(13)¤k )" 1. (. 2. n. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 21 / 52.
(43) àg5§)m( úª ´§(13)?¿n)§W (t)´§ 1ª§K ÷v 5§. Liouville x1 (t), x2 (t), · · · , xn (t) Wronski W (t). . W 0 (t) = −a1 (t)W (t). Ï k −. W (t) = W (t0 ) · e. (. þ°ã²ÆA^êÆX). Rt t0. a1 (s)ds. ~©§. ,. 1oÙ. t, t0 ∈ [a, b]. March 19, 2010. (16). 22 / 52.
(44) àg5§)m( ~1 y¼êx (t) = cos t, x (t) = sin t´§ ddtx + x = 0 ü5 Ã')§¿ÑT§Ï)" ~2 àg5§3«m[a, b]þ?¿ü5Ã')|©O (x (t), x (t))Ú(x (t), x (t)) y²µ§Wronski1ª'´Ø"~ê" 2. 1. 2. (1) 1. (. þ°ã²ÆA^êÆX). 2. (1) 2. ~©§. (2) 1. 1oÙ. (2) 2. March 19, 2010. 23 / 52.
(45) p 5©§nØ. 4.2. àg5§)8Ü5 ½n4.8 XJx¯(t)´àg5§(12))§x(t)´àg5 §(13))§Kx¯(t) + x(t)E´àg5§(12))" àg5§)8Ü5 ½n4.9 XJx (t), x (t)´àg5§(12)ü)§ Kx (t) − x (t) ´éAàg5§(13))" 1. 1. (. 2. 2. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 24 / 52.
(46) 4.2. p 5©§nØ. àg5§)8Ü5 ½n4.10 x (t)x (t)©O´àg5§ 1. 2. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = f1 (t) 1 n n−1 dt dt dt. Ú. dn−1 x dx dn x + a (t) + · · · + a (t) + an (t)x = f2 (t) 1 n−1 dtn dtn−1 dt. )§Kx (t) + x (t)´§ 1. 2. dn−1 x dx dn x + a (t) + · · · + a (t) + an (t)x = f1 (t) + f2 (t) 1 n−1 dtn dtn−1 dt. )" (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 25 / 52.
(47) 4.2. p 5©§nØ. àg5§)8Ü5 ½n4.11 x (t), x (t), · · · , x (t)§(13)Ä )|§ x¯(t)´§(12),)§K§(12)Ï)L 1. 2. n. x(t) = c1 x1 (t) + c2 x2 (t) + · · · + cn xn (t) + x¯(t). (17). Ù¥c , c , · · · , c ?¿~ê§ dÏ)(17)) §(12)¤ k)" 1. (. 2. n. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 26 / 52.
(48) àg5§)8Ü5 ~3 àg§ d2 x dx + p(t) + q(t)x = f (t) 2 dt dt. kn)x (t) = t, x (t) = e , x (t) = e §¦d§÷vЩ^ x(0) = 1, x (0) = 3A)" 1. 2. t. 3. 2t. 0. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 27 / 52.
(49) àg5§~êC´{. x (t), x (t), · · · , x (t)´§(13)Ä )|§Ï 1. 2. n. x(t) = c1 x1 (t) + c2 x2 (t) + · · · + cn xn (t). ´§(13)Ï)"ßÿ§(12)´ù«/ª)§ c , c , · · · , c At¼ê§=b 1. 2. n. x(t) = c1 (t)x1 (t) + c2 (t)x2 (t) + · · · + cn (t)xn (t). (18). ´§(12))" ¦Ñ½¼êc (t), c (t), · · · , c (t)§ ò(18)\§(12)¥§=U¤÷v^"é(18)ü> ¦§ 1. 2. n. x0 (t) = c1 (t)x01 (t) + c2 (t)x02 (t) + · · · + cn (t)x0n (t) +c01 (t)x1 (t) + c02 (t)x2 (t) + · · · + c0n (t)xn (t) (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 28 / 52.
(50) àg5§~êC´{ -. . c01 (t)x1 (t) + c02 (t)x2 (t) + · · · + c0n (t)xn (t) = 0. (19). x0 (t) = c1 (t)x01 (t) + c2 (t)x02 (t) + · · · + cn (t)x0n (t). (20). éþªü>UY¦§¿þ¡{§-¹kc (t)Ü© "§ 0 i. c01 (t)x01 (t) + c02 (t)x02 (t) + · · · + c0n (t)x0n (t) = 0. (21). x00 (t) = c1 (t)x001 (t) + c2 (t)x002 (t) + · · · + cn (t)x00n (t). (22). ÚLª. UYþ¡{§¼1n − 1^ (n−2). c01 (t)x1. ÚLª þ°ã²ÆA^êÆX (. (n−2). (t) + c02 (t)x2 ). (t) + · · · + c0n (t)x(n−2) (t) = 0 n. ~©§. 1oÙ. March 19, 2010. (23) 29 / 52.
(51) àg5§~êC´{ (n−1). x(n−1) (t) = c1 (t)x1. (n−1). (t) + c2 (t)x2. §é(24)ü>2¦g§ (n). (t) + · · · + cn (t)x(n−1) (t) (24) n. (n). (n). x(n) (t) = c1 (t)x1 (t) + c2 (t)x2 (t) + · · · + cn (t)xn (t) (25) (n−1) (n−1) (n−1) +c01 (t)x1 (t) + c02 (t)x2 (t) + · · · + c0n (t)xn (t). ò(18)-(25) Ü\§(12)¥§¿5¿x (t), x (t), · · · , x (t)´ §(13))§ 1. (n−1). c01 (t)x1. (n−1). (t) + c02 (t)x2. 2. n. (t) = f (t) (t) + · · · + c0n (t)x(n−1) n. (26). ùn¼êc (t) (i = 1, 2, · · · , n)Ó÷v(19),(21),(23) Ú(26) n^§ùn^|¤5ê§|§ÙX ê1ªW [x (t), x (t), · · · , x (t)] 6= 0§Ï §|k )§Ø¦ 0 i. 1. (. þ°ã²ÆA^êÆX). 2. n. ~©§. 1oÙ. March 19, 2010. 30 / 52.
(52) àg5§~êC´{ c0i (t) = φi (t),. È©. i = 1, 2, · · · , n. Z ci (t) =. i = 1, 2, · · · , n. φi (t)dt + ri ,. ùpr ´?¿~ê"ò¤c (t) (i = 1, 2, · · · , n)Lª \(18)¥§§(12)Ï) i. i. x(t) =. n X. ri xi (t) +. i=1. (. þ°ã²ÆA^êÆX). n X. Z xi (t). φi (t)dt. i=1. ~©§. 1oÙ. March 19, 2010. 31 / 52.
(53) àg5§)8Ü5 ~4 àg5§(t − 1) ddtx − t dxdt + x = (t − 1) éAàg 5§Ï)x(t) = c t + c e §¦d§Ï)" 2. 2. 2. 1. (. þ°ã²ÆA^êÆX). 2. t. ~©§. 1oÙ. March 19, 2010. 32 / 52.
(54) 4.3. ~Xêàg5§½ê¼ê{. E¼ê φ(t)Úϕ(t)´«m[a, b]þ¢¼ê§¡z(t) = φ(t) + iϕ(t)T« mþE¼ê" Eê¼ê k = α + iβ´?Eê§α, β, t´¢ê§½ÂXeEê¼ê ekt = e(α+iβ)t = eαt (cos βt + i sin βt). E) XJ½Â3«m[a, b]þ¢CþE¼êz(t)÷v§(12)§=. dn z(t) dn−1 z(t) dz(t) +a (t) +· · ·+an−1 (t) +an (t)z(t) ≡ f (t), t ∈ [a, b] 1 n n−1 dt dt dt. ¡z(t) §(12)E)"~©§ þ°ã²ÆA^êÆX (. ). 1oÙ. March 19, 2010. 33 / 52.
(55) E)5 ½n4.12 XJ§(13)¥¤kXêa (t)Ñ´¢¼ê§ z(t) = φ(t) + iϕ(t)´T§E)§Kz(t)¢Üφ(t)ÚJ Üϕ(t)±9z(t)ÝEêÑ´§(13))" i. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 34 / 52.
(56) E)5 ½n4.13 XJ§. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = u(t) + iv(t) 1 n n−1 dt dt dt. kE)x=U(t)+i V(t)§Ù¥a (t) (i = 1, 2, · · · , n) 9u(t), v(t)Ñ ´¢¼ê§Kù)¢ÜU (t)ÚJÜV (t)©O´e¡ü§ ) i. Ú. dn x dn−1 x dx + a (t) + · · · + an−1 (t) + an (t)x = u(t) 1 n n−1 dt dt dt dn x dn−1 x dx + a (t) + · · · + a (t) + an (t)x = v(t) 1 n−1 dtn dtn−1 dt (þ°ã²ÆA^êÆX) ~©§ 1oÙ March 19, 2010. 35 / 52.
(57) 4.3. ~Xêàg5§½ê¼ê{. ~Xêàg5§ ¡§ n. L[x] ≡. dn x dn−1 x dx + a + · · · + an−1 + an x = 0 1 n n−1 dt dt dt. (27). n ~Xêàg5§§Ù¥a , a , · · · , a ~ê" 1. 2. n. 2. dy dy + 3 + 2y = 0; 2 dx dx d4 x d2 x (2) 4 + 2 2 + x = 0; dt dt d2 y dy d3 y (3) 3 − 6 2 + 11 − 6y = 0 dt dt dt (1). (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 36 / 52.
(58) 4.3. ~Xêàg5§½ê¼ê{. ~Xêàg5§ ¡§ n. L[x] ≡. dn x dn−1 x dx + a + · · · + an−1 + an x = 0 1 n n−1 dt dt dt. (27). n ~Xêàg5§§Ù¥a , a , · · · , a ~ê" 1. 2. n. 2. dy dy + 3 + 2y = 0; 2 dx dx d4 x d2 x (2) 4 + 2 2 + x = 0; dt dt d2 y dy d3 y (3) 3 − 6 2 + 11 − 6y = 0 dt dt dt (1). (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 36 / 52.
(59) 4.3. ~Xêàg5§½ê¼ê{. ©Ûµb§(27)3ê¼ê/ª) x = eλt. (28). ò(28)\§(27)¥§ L[eλt ] ≡ (λn + a1 λn−1 + · · · + an−1 λ + an )eλt = 0. ù¿X§e ´§(27))¿©7^µλ´ê§ λt. F (λ) ≡ λn + a1 λn−1 + · · · + an−1 λ + an = 0. (29). " (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 37 / 52.
(60) 4.3. ~Xêàg5§½ê¼ê{. A´ü/ ½n4.14 λ , λ , · · · , λ ´A§(29)n*dpÉA §K e , e , ··· , e §(27)Ä )|" 1. 2. n. λ1 t. (. þ°ã²ÆA^êÆX). λ2 t. ~©§. λn t. 1oÙ. March 19, 2010. 38 / 52.
(61) A´ü/. 5¿ (1)AλU´¢ê§U´Eê¶ (2)XJλ (i = 1, 2, · · · , n) ¢ê§Ke ¢)§§(27)Ï). λi t. i. . (i = 1, 2, · · · , n) n. x(t) = c1 eλ1 t + c2 eλ2 t + · · · + cn eλn t. XJλ (i = 1, 2, · · · , n)¥kEê§Øλ §Ä )|. (3). i. 1,2. = α ± iβ. §K. e(α+iβ)t , e(α−iβ)t , eλ3 t · · · , eλn t. Ù¥e Úe §(27)Ï) (α+iβ)t. (α−iβ)t. Ñ´E)§§¢ÜÚJÜ. x(t) = eαt (c1 cos βt + c2 sin βt) + c3 eλ3 t + · · · + cn eλn t (þ°ã²ÆA^êÆX) ~©§ 1oÙ March 19, 2010. 39 / 52.
(62) 4.3. ~Xêàg5§½ê¼ê{. ~1 ¦§ ddtx − 2 dxdt − 3x = 0Ï)" ~2 ¦§ ddtx − x = 0Ï)" 2. 2. 4. 4. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 40 / 52.
(63) 4.3. ~Xêàg5§½ê¼ê{. Ak/ ½n4.15 λ = 0´§(27)kA§K§(27)kk5 Ã') 1. 1, t, t2 , · · · , tk−1. Ak/ ½n4.16 λ 6= 0´§(27)kA§K§(27)kk5 Ã') 1. eλ1 t , teλ1 t , · · · , tk−1 eλ1 t. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 41 / 52.
(64) 4.3. ~Xêàg5§½ê¼ê{. ~3 ¦§ ddtx − 3 ddtx + 3 dxdt − x = 0Ï)" ~4 ¦§ ddtx + 2 ddtx + x = 0Ï)" 3. 2. 3. 4. 2. 2. 4. (. þ°ã²ÆA^êÆX). 2. ~©§. 1oÙ. March 19, 2010. 42 / 52.
(65) 4.3. ~Xêàg5§½ê¼ê{. § ¡§ Euler. xn. n−1 dy dn y y n−1 d + a x + · · · + an−1 x + an y = 0 1 n n−1 dx dx dx. Euler§§Ù¥a. (. þ°ã²ÆA^êÆX). i. (30). ~ê". (i = 1, 2, · · · , n). ~©§. 1oÙ. March 19, 2010. 43 / 52.
(66) Euler. §f){. PD = dtd §D. k. = x. dk dtk. §u´k. dy = Dy dx. d2 y = D2 y − Dy = D(D − 1)y dx2 ······ dk y xk k = D(D − 1) · · · (D − k + 1)y dx. x2. u´§(30)=z. Dn y + b1 Dn−1 y + · · · + bn y = 0 (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 44 / 52.
(67) Euler. §. ~5 dy dy − x + y = 0Ï)" ¦§x dx dx ~6 dy dy dy ¦§x dx +x − 4x = 0Ï)" dx dx 2. 2. 2. 2. 3. 2. 3. 3. (. þ°ã²ÆA^êÆX). 2. ~©§. 1oÙ. March 19, 2010. 45 / 52.
(68) 4.4. ~Xêàg5§½Xê{. ~Xêàg5§ ¡§ n. dn x dn−1 x dx L[x] ≡ n + a1 n−1 + · · · + an−1 + an x = f (t) dt dt dt. (31). n ~Xêàg5§"Ù¥a , a , · · · , a ~ê§f (t)´ ëY¼ê" 1. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. 2. n. March 19, 2010. 46 / 52.
(69) ~Xêàg5§½Xê{. 4.4. ´õªê¼ê¦È/ ½n4.17 f (t) = (b t + b t + · · · + b t + b )e §Ù ¥b (i = 0, 1, · · · , m), λ¢~ê"K§(31)kA) f (t). 0. m. 1. m−1. m−1. m. λt. i. x˜(t) = tk (B0 tm + B1 tm−1 + · · · + Bm−1 t + Bm )eλt. (32). Ù¥B , B , · · · , B ½~ê¶k dλ´ÄAû½" λØ´A§k = 0¶λ´A§kλê" 0. (. 1. þ°ã²ÆA^êÆX). m. ~©§. 1oÙ. March 19, 2010. 47 / 52.
(70) 4.4. ~Xêàg5§½Xê{. ~1 ¦§ ddtx − 2 dxdt − 3x = 3t + 1Ï)" ~2 ¦§ ddtx − 7 ddtx + 16 dxdt − 12x = −20t e Ï)" 2. 2. 3. 2. 3 2t. 3. (. þ°ã²ÆA^êÆX). 2. ~©§. 1oÙ. March 19, 2010. 48 / 52.
(71) ~Xêàg5§½Xê{. 4.4. ´õªê¼ê!{u¼ê£u¼ê¤È/ ½n4.18 f (t) = A(t) cos βt · e (½f (t) = B(t) sin βt · e )§Ù ¥A(t)(B(t))´¢Xêmgõª§K§(31)kA) f (t). αt. αt. x˜(t) = tk (P (t) cos βt + Q(t) sin βt)eαt. (33). Ù¥P (t), Q(t)þ½¢Xêõª§gêm§, gêØLm§kdα + iβ ´ÄAû½" ~3 ¦§ ddtx + x = 2 sin tÏ)" 2. 2. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 49 / 52.
(72) ~Xêàg5§½Xê{. 4.4. ´õªê¼ê9!{u¼êÈ/ ½n4.19 f (t) = (A(t) cos βt + B(t) sin βt)e §Ù¥A(t), B(t)´ ¢Xêõª§gêm §,gêØLm"K §(31)kA) f (t). αt. x˜(t) = tk (P (t) cos βt + Q(t) sin βt)eαt. Ù¥P (t), Q(t)þ½¢Xêõª§gêm§, gêØLm§kdα + iβ´ÄAû½" ~4 ¦§ ddtx − dxdt − 2x = (cos t − 7 sin t)e Ï)" 2. −t. 2. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 50 / 52.
(73) 4.5. A^¢~. Ï+nÚ ¥§ A õm 3Ï+n¥§½Ï/ ãm ´ 4@ 1¨ 3´þ½å´C±Ã{ÊeýÏL´" ¨C´f¨ §3wÚ&Ò Ñû½µ´ Ê´ÏL´"XJ¦±{½Ý(½$u{½Ý)1¨§ û½Ê§¦7Lkv Êål"û½ÏL´§ ¦7Lkv m¦¦U ÏL´§ù)ÑÊû ½m§±9ÏLʤIáålf¨m" u´§ GA±Ym)f¨ Am!¦Ï L´m±9ʤIm"XJ{½Ýv § ´°ÝI §;. ÝL§@o§ÏL´m (I + L)/v "ÁOáål" 0. 0. (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 51 / 52.
(74) 4.5. A^¢~. 5¢Ô?n¯K ãm§{IfU ¬(yØ+n ¬)´ù? nß 5¢Ô§§rù ¢ÔC3µ5UéÐ × ¥§, DY 300f t(1f t = 0.3038)°p"ù«{´Ä ¬E¤5À/§ég,/Úå )Æ[9¬.' 5"fU ¬2y§ ×~j§ûج»¦§ù« {´ýéS ", ó§%édL«~¦§¦@ ×3Ú°.EkUu)»" fU ¬k ; [E,j±gCw{"¯K' 3u ×.U«Éõ Ý-E§ ×Ú°.-EÝkõºÁïá ×e A÷v©§§¿¦)" (. þ°ã²ÆA^êÆX). ~©§. 1oÙ. March 19, 2010. 52 / 52.
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