1、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 3 (LH)a20a22a21a22a23a22a20a22a24a22a25a26a28a27a28a29a28a30a28a31a28a32a28a33a22a34a22a35a22a36a22a37 (LH)a38a28a39a28a40a28a41a28a42a28a43a28a38a22a44a22a45a22a38a22a46a22a47a22a48a50a49a52a51a22a53a22a54a22a41a2
2、2a55a56a22a57a22a58a22a59 Ia60a22a61a62a39a22a63a22a64a22a65a22a66a22a67a22a68a70a69a22a71a70a72 I a60a22a40a22a73a22a74a22a38a22a41a22a48a31a28a32a28a33a28a34a28a35a28a36a28a37 (LH)a38a28a75a28a76a26a38a34a28a77a22a78a22a79a38a22a41a22a80a22a40a22a81a22a82a22a83a34a61a85a84a22a86a22a87a22a38a22a46a
3、88a43a22a89a22a90a91a22a92 3.1 (a93a28a94a28a95a92 )a96 x = 1(t) a97 x = 2(t) a98a28a99 (LH) a100a102a101a28a61a104a103a102a105a22a106a22a107a108a52a109 c1; c2, x = c11(t) + c22(t) a98a22a99 (LH) a100a52a101a22a48a110a82a111a83a111a112a111a113a111a114a115a61a117a116a119a118a104a114 (LH) a38 n a120a1
4、11a41 1(t); ; n(t), a121a111a122a115a53a111a123a111a124a111a125c1; ; cn a38a127a126a52a128a22a129a22a125x = c11(t) + + cnn(t)a130 a78 (LH)a38a28a41a28a48a132a131a79a28a133a22a134a22a78 (LH)a38a28a135a28a41a28a61a132a136a28a137a22a138 1(t) 0 a139a28a61a132a60a28a140a28a141a28a43 x =c22(t)+ +cnn(t), a
5、79 a66a28a122a28a40 n 1 a120a28a53a28a123a28a124a28a125a28a61a143a142a22a137a22a138 1(t) 2(t)+3(t)a139a144a61a145a60a144a140a144a141a144a43 x = (c1+c2)2(t)+(c1+c3)3(t)+ +cnn(t),a79a144a146a144a147 a60a130a66a144a122 n 1a120a28a53a28a123a28a124a28a125a28a48a148a49a102a149a22a51a22a150a22a151a79 a56a4
6、7a22a43 (LH) a38a28a135a28a41a28a61a134a28a152a28a153a41a37 1(t); ; n(t) a154a155a38a22a156a22a157a22a42a22a158a22a138a22a159a22a160a22a61a62a161a70a162a70a163a127a164a165a150a70a166a37a126a165a128a22a129a70a125a33a70a34a70a167a156a70a168a33a22a34a70a169a156a70a38a70a170a22a171a70a48a172a126a52a128a
7、22a129a22a125 1(t); ; m(t) a72a127a173a155 Ia60a78a175a174a22a176a22a177a22a178a38a22a61a62a137a22a179a22a40a22a180a182a181a70a51a70a183a38a22a124a22a125 1; ; m a151a22a18411(t) + + mm(t) 0; t 2 I; (3.1)a185a121a28a61a50a84a28a186a28a187 (3.1) a43a28a89a28a61a50a188a28a189 1 = = m = 0, a190a172a149
8、m a120a191a126a102a128a28a129a28a125a28a192a193a173a155I a60a174a22a176a22a194a22a178 .a91a22a92 3.2a195 t0 2 I, 1(t); ; m(t) a99a197a196a182a198 I a199 (LH) a100 m a200a182a101a182a48a182a1031(t); ; m(t) a201a197a196a182a198 I a199a182a202a182a203a182a204a197a205a182a100a182a206a117a207a209a208a117
9、a210a111a99a209a211a117a212a111a213 1(t0); ; m(t0)a202a22a203a22a204a127a205a52a48a214a70a215a116a193a126a165a128a37 1(t0); ; m(t0)a33a70a34a70a167a156a70a61a62a121a111a216a182a72a182a180a111a181a111a51a182a183a182a38a111a124a182a125 1, m a151a22a18411(t0) + + mm(t0) = 0:a217a28a218a163a28a113 3.1,
10、11(t) + + mm(t)a78 (LH)a38a28a41a28a48a79a28a219a28a220a72 t = t0a221a51a28a183a28a61a110a163a22a113 2.1 a190a22a114 (3.1) a134 a43a22a89a22a61a62a84 1(t); ; m(t) a192 I a60a33a22a34a22a167a156a22a48a117a163a70a113a182a38a70a222a166a22a223a22a224a78a22a225a22a220a38a22a48 a226a22a92 3.1 (LH)a227a191
11、a196a28a198 I a199a28a228 n a200a28a202a28a203a28a229a191a205a22a100a52a101a70a48a231a230a22a232 1(t); ; n(t) a99a196a22a198 I a199 (LH) a100 n a200a22a202a22a203a22a229a127a205a22a100a52a101a22a61a182a103a165a233a22a106a70a107a108a165a109 c1; ; cn a100a52a234a22a235a127a236x = c11(t) + + cnn(t) (3.
12、2)1a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a99 (LH) a100a165a237a70a101a70a61a62a238a70a239a182a240a182a241a182a61a117a99 (LH) a100a165a242a70a243a115a101a193a100a165a244a197a245a62a234a182a235a197a236a62a61a111a246a62a105a111a106a182a107a108a62a109c1; ; cn,
13、 a211a52a212a22a247a109 (3.2)a98a22a99 (LH) a100a52a101a22a248a62a249a22a250a22a61 (LH) a100a52a106a22a251a22a101a22a61a62a98a22a252a127a253a70a254a255 (3.2)a0a2a1a2a3a127a236a52a48a214a22a215a53a22a123a2a4a22a73a22a166a2a5 t0 2 I a168 n a120a33a22a34a22a169a156a22a38 n a6a22a124a127a126a52a128 1; ;
14、 n(a225a22a220a216a70a72 ). a153a8a7 a166 i, i = 1; ; n, a63 xi(t) a9a8a10 (LH) a11a8a12a8a13a8a14a8a15a8a16 x(t0) = i a38a41a28a48a110a73a28a113 2.1 a114a28a61 a149a28a41a28a216a28a72a8a17a22a192 I a60a28a40a28a73a28a74a28a48a110a192a193a126a52a128a37 x1(t0); ; xn(t0)a78a33a22a34a22a169a156a22a38a2
15、2a61a217a22a218a163a22a113 3.2 a162a22a114a22a61 x1(t); ; xn(t) a192 I a60a33a22a34a22a169a156a22a48a18 1(t); ; n(t)a78 (LH)a72 I a60a28a38 n a120a33a28a34a28a169a156a28a38a28a41a22a48a143a121a2a19a2a20a110a163a22a113 3.1a114a28a61a153a53a28a123a28a124a28a125 c1; ; cn, (3.2) a78 (LH) a38a28a41a28a48
16、a22a21a32a61a132a116 x(t) a78 (LH) a53a28a166a28a41a28a61a4a22a73 t0 2 I, a110 a163a22a113 3.2 a114a22a61a182a126a52a128a37 1(t0); ; n(t0)a33a22a34a22a169a156a22a61a24a23a2a25a110a52a33a70a34a8a26a125a22a114a22a61a62a216a22a72a22a124a22a125 c01; ; c0n, a151a22a184c011(t0) + + c0nn(t0) = x(t0):a217a2
17、2a218a163a22a113 3.1, a126a52a128a22a129a22a125 x = c011(t) + + c0nn(t) x(t) a78 (LH) a38a22a41a22a61a24a25a22a60a140a2a9a28a27a52a61a79a72 t0a221a51a22a183a22a61a24a29a110a163a22a113 2.1 a114a22a61a134a40c011(t) + + c0nn(t) x(t) 0; t 2 I;a161a2a9a28a27a52a41 x(t) a81a22a63a22a135a2a30a22a72 (3.2) a
18、31a52a158a22a138a2a32a2a4 c1; ; cn a25a22a184a59a48a62a73a22a113a2a33a2a34a22a48 a35 3.1a36a2a37 3.1 a234a28a38a52a61 (LH) a100a52a101a127a100a52a242a2a39a22a61a24a40a255a251a22a200 n a41a22a202a22a203a28a42a22a198a52a48(LH) a38a28a53a28a123 n a120a33a28a34a28a169a156a28a38a28a41a22a45a2a43a2a44a172
19、a51a79a38a22a166a22a120a46a45a48a47a28a23a48a49 . a137a28a179a28a63a79a65a42a22a51a2a50a127a126a52a128a2a51a182a43a70a166a70a120 n a52a2a53a28a54a52a61a62a121a70a149a8a53a55a54a172a51 (LH) a38a22a166a22a120a56a45a2a47a22a23a2a57a2a58 . a116a22a63(t) a9a2a10 (LH) a38a22a166a22a120a22a76a26a41a2a53a28
20、a54a52a61a62a121a70a81a8a59a2a9a8a60a70a140 (3.2) a61a2a62a22a51x = (t)c;a21a28a31 c = (c1; ; cn) a78 a53a22a123a22a38 n a6a22a124a127a126a52a128a22a48a62a42a70a51 (LH) a38a22a76a26a41a2a53a28a54a165a61 (t)a11a2a12d (t)dt A(t) (t); t 2 I; (3.3)det (t) 6= 0; t 2 I: (3.4)(3.3) a154a22a39a22a63a22a43a2
21、2a89a22a61 a78 a49a52a51 (t) a38 a7 a166a2a50a22a55 a78 (LH) a72 I a60a22a38a22a41a22a248 (3.4) a154a22a39a22a63a43a28a89a28a61a78a49a102a51 (t) a38a7a166a48a50a28a180a48a63a48a63a78 (LH)a38a28a41a28a61a64a25a48a17a28a161 n a120a28a41a78a28a33a28a34a28a169a156a22a38a22a48a65a190a2a66a127a164a52a61a1
22、53a192 n a120a22a166a2a67a22a38 n a6a127a126a52a128a22a129a22a125a22a61a33a22a34a22a169a156a8a68a70a180a2a69a70a122a167a2a70a38 n a52a71a50a22a140a22a180a2a72a22a192a22a183a22a61a62a136a22a137a8a73a2a6a193a126a52a128a70a129a70a1250101A ;0t01A2a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14
23、a15a17a16a14a18a14a6a14a7a79a192a22a53a22a123a127a173a155a60a33a22a34a22a169a156a70a61a117a81a78a70a167a8a70a38a71a50a22a140a8a74a2a75a8a72a70a192a22a183a70a48a18a40 n a120a22a73a22a74a22a72a127a173a155 Ia60a22a38a127a126a52a128a22a129a22a125a22a901(t) =0BB11(t).n1(t)1CCA ; ; n(t) =0BB1n(t).nn(t)1CC
24、A ;a110a52a79a65a2a51a2a50a2a25a22a43a22a38a71a50a22a140W(t) =11(t) 1n(t)n1(t) nn(t)a172a51a22a161 n a120a127a126a52a128a22a129a22a125a22a38a77a76a2a78a2a45 (Wronski, 1776-1853)a79a2a80a2a81 .a91a22a92 3.3 (a82a48a83a48a84a48a85a48a81 ) a96 1(t); ; n(t) a99 (LH) a100a102a101a28a61a165a103a48a86a88a8
25、7a127a100a2a89a91a90a2a92a93a2a94a236 W(t) a252a22a234a2a95a28a96W(t) = W(t0)expZ tt0trA(s)ds; t 2 I;a97a28a98 t0 2 I a252a22a106a22a107a2a99a2a36a22a61a24a100 trA(s) = a11(s) + + ann(s).a214a22a215 a110 a71a50a22a140a22a38a2a101a2a102a2a103a22a140a22a114dW(t)dt = W1(t) + + Wn(t);a21a104a31 Wi(t) a7
26、8 a161a48a105a28a38 n a52a71a50a28a140a28a61a79a38a2a106 i a71a48a107a48a108 a78 W(t) a31a88a106 i a71 a167a48a70 a107a48a108 a38a101a48a102a28a61a109a25a28a188a2a106 i a71a48a107a48a108a48a110 a38a8a21a79 a107a2a108a8a111 a78 W(t)a31a102a38a167a48a70 a107a2a108a48a110 a71a50a70a140a22a38a34a70a77a1
27、14Wi(t) = aii(t)W(t); i = 1; ; n;a81a2a112 x = W(t) a78 a86a2a113a2a13a2a14a28a114a116a115a22a38a22a41a22a90dxdt = (a11(t) + + ann(t)x; x(t0) = W(t0):a39a22a63a22a163a22a113a22a38a22a46a88a43a22a89a22a48 a161a22a120a22a163a22a113a22a83a2a117a22a150a2a118a22a87a22a38a70a46a88 (3.4).a35 3.2a195 1(t) a
28、97 2(t) a99 (LH) a201a127a196a22a198 I a199a127a100a2a119a52a200a2a92a8a120a182a101a8a121a123a122a165a48a111a103a91a124a182a201a125a116a126a2a127 a108a121a28a122 C, a128a2a1292(t) 1(t)C; t 2 I:3a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a35 3.3a195 2 I. a96 (LH
29、) a100a116a92a2a120a22a101a2a121a55a122 (t) a130a2a131a2a132a2a133a127a208a165a210 ( ) = E (na134a48a135a48a136a122 ), a103 (t) a137a104a96a139a138a48a140a48a141a48a142a48a57a48a58 . a143a88a144a28a61 a105a28a106a28a251a48a92a48a120a28a101a2a121a28a122 (t), (t; ) =(t) 1( ) a99a2a145a28a146a116a147a2
30、a148a2a121a28a122a52a48a24a149a55a150a116a151a55a38a52a901. a145a28a146a116a147a2a148a2a121a28a122 (t; ) a152a116a92a2a120a22a101a2a121a28a122 (t) a100a116a153a2a154a22a229a127a205a52a2482. (t; ) a205a52a227a28a155a52a212 t a99a2a121a28a122a116a156a2a157dYdt = A(t)Ya130a2a131a2a132a2a133a127a208a52a
31、210 Y ( ) = E a100a116a158a22a251a22a101a22a2483. a105a22a106a22a107 t; 2 I, (t; ) a99a125a116a126a2a127a100a52a61a160a159a116a161 1(t; ) = ( ; t);4. a105a22a106a22a107 t; ; 2 I,(t; ) = (t; ) ( ; );5. (LH) a100a52a101 x(t; ; ) a228a22a234a22a235a127a236x(t; ; ) = (t; ) :a162 a1631. a164a127a164a35a2
32、2a36a22a37tdx1dt = 2x1 x2; tdx2dt = 2x1 x2a38a28a166a28a120a28a76a26a41a37a2a165a70a79a65a22a38a2a166a2a167a70a76a71a50a22a140a70a48 a161a22a120a71a50a22a140a22a72 t = 0a221a51a28a183a48a25a48a74a28a180a8a75a2a72a192a22a183a22a61a78 a185a2a168a163a22a113 3.3 a169a2a170a172a1712. a118a52a114 n n a129
33、a22a125a2a53a28a54 A1(t) a168 A2(t) a173a22a192a127a173a155 Ia174a2a175a22a48a24a176a2a33a22a90a62a116a35a22a36a70a37dxdt = A1(t)xa168 dxdt = A2(t)xa40a22a166a167a28a177a38a22a76a26a41a37a61a62a121 A1(t) A2(t); t 2 I.3. a118a52a114 1(t); ; n(t) a78 a173a155 Ia60 n a120a22a80a22a40a2a174a2a175a2a178a
34、70a125a70a38 n a6a22a129a22a125a127a126a52a128a22a61a79a65a28a38a48a166a48a167a28a76a71a50a22a140a221a22a221a180a22a51a22a183a70a48a50a121a22a192 I a134 a40 n n a38a48a174a48a175a28a129a28a125a48a53a55a54 A(t), a151a28a1841(t); ; n(t) a47a22a43a35a22a36a22a37dxdt = A(t)x; t 2 Ia38a22a166a22a120a22a7
35、6a26a41a37a484. a33a28a27a116a179 3.2 a168a2a179 3.3.4a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a75. a72 (LH) a31a52a61a18 A(t + !) A(t).a116R !0 trA(s)ds 6= 0, a121 (LH) a38a22a39a22a40a22a41a180a56a55a78 !-a180a2a181a22a41a22a486. a18 (NH) a38a28a157a28a125a78 !-a180a48a181a28a38a28a48a182a33a104a27a165a90 (NH) a40a48a183a28a166 !- a180a48a181a28a41a28a38a48a184a22a186a8a15a16a78 (LH)a66a22a40a2a185a2a186a22a38 !- a180a2a181a22a41 x = 0.5
