1、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 7a20a22a21a22a23a22a24a22a25a22a26a22a27a22a28a22a29a22a30a22a31a32a25a22a33a34a36a35a36a37a36a38a36a39a36a40a36a41a32a42a22a43a22a44a22a45a22a46a48a47a22a49a22a50a22a51a22a52a22a53a22a54a22a55a57a56a59a58a22a39a32
2、a60a22a61a48a62a36a63a32a64a22a65a22a66a22a67a22a55a22a60a44a22a68a22a46a69a58a22a49a22a34a22a35a22a70a22a71a22a40a32a41a22a42a32a43a22a44a32a45a32a72a22a73a32a74a32a39a22a617.1 a75a77a76a77a78a77a79a77a80a81a83a82a85a84 a39a36a37a36a67a36a86a32a68a22a72a22a46a48a87a22a88 a82a90a89a22a91a22a92a22a93
3、a22a94a22a95a22a96 a39a22a97a22a98a32a39 a89a22a99 a46a48a100a22a44a22a45a22a101a22a102a22a103a104a36a105 a55a36a60 a92a36a106 a35a36a55a36a60a22a39a22a107a36a108a22a61a110a109a36a111a22a112a22a66a22a113a22a114a36a115a22a116a22a117a22a118a36a119a22a120a22a121a22a122a36a54a22a123 a89a22a99 a105a22a12
4、4a36a125a22a126a127 a39a129a128a90a39a22a61a69a47a22a49a22a130a22a131a22a87a32a88 a126 a71a22a114a32a115a32a132a22a133 a125 a112a22a37a32a134a22a135a32a136a32a74a22a61a137a22a138a22a139a22a140a22a109a22a111a22a39a22a44a22a45dnxdtn + a1(t)dn 1xdtn 1 + + an(t)x = f(t): (7.1)a119a22a141a22a44a22a45a22a
5、39a22a142 a96 n = 1,a143a22a144a87a22a60 a104a22a105a22a145a32a146a148a147 a51a32a54a32a55a32a149a32a61a150a37a32a38a32a151a32a116a148a46a69a44a148a45a32a39a32a142a32a152a153a22a46a69a55a22a60a22a154a22a152 a147 a51a22a61 a82a155a84a32a156 a86 a125a22a157 a87a32a88a32a97a22a98a32a39 a89a22a99 a46a14
6、8a158a159a119a32a87a32a88a22a107a32a119x = (t)y (7.2)a39 a89a22a99 a46a69a100a22a44a22a45a22a39a22a142a22a160a32a161a22a46 a126a32a162 (t)a72a22a163a22a164a22a39 a95a22a96 a61a69a100 (7.2)a165a22a166 (7.1),a149 a125(t)dnydtn + b1(t)dn 1ydtn 1 + + bn 1(t)dydt + bn(t)y = f(t); (7.3)a144a57a167bn(t) =
7、dn(t)dtn + a1(t)dn 1(t)dtn 1 + + an(t)(t):a137a22a119 (t) a72 (7.1) a34a22a168a22a39a22a169a22a170a22a44a22a45a32a39a148a60a32a46a143 bn(t) 0. a171a22a172a22a173a22a174 (t) 6= 0 a39a22a175a120a57a176a22a177a59a178a22a46 (7.3)a154a22a101a22a179a22a140a22a180a22a35 z = dydta39 n 1a142a22a39a22a181a22a
8、182a22a107a22a44a22a45dn 1zdtn 1 + c1(t)dn 2zdtn 2 + + cn 1(t)z = g(t); (7.4)a144a57a167c1(t) = b1(t)(t) ; ; cn 1(t) = bn 1(t)(t) ; g(t) = f(t)(t):a183 a178a185a184a185a186a185a46a187a137a36a119a189a188a191a190a36a55a22a149 (7.1)a34a185a168a185a39a185a169a185a170a185a44a36a45a36a39a22a37a36a113a36a1
9、92a36a193a36a194a36a60a36a46a143a87a36a88 a89a36a99 (7.2)a104 a100a22a60 (7.1)a39a57a195a59a196a22a197a22a198a22a103a22a60a22a161a22a37a22a142a32a39a32a44a22a45 (7.4). a126 a67a22a44a22a68a22a154a22a199a22a103a201a200a22a202a33 .a105a22a203 a142a22a40a22a41a22a44a22a45d2xdt2 + a(t)dxdt + b(t)x = f(t
10、) (7.5)1a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a103a22a114a22a46a69a137a22a119a57a188 a94 a34a32a168a148a39a32a169a32a170a32a44a148a45a32a39a32a37a32a113a32a192a148a193a32a194a32a60 (t),a143a190a22a88 a89a22a99 (7.2)a204a46a69a149a125(t)d2ydt2 + (20(t) + a(
11、t)(t)dydt = f(t); (7.6)a205 (a173a22a174 (t) 6= 0 a39a22a175a22a120a57a176a22a177a59a178 )dzdt +(20(t) + a(t)(t)z(t) =f(t)(t):a126 a72a22a37a22a113a22a37a22a142a22a40a22a41a22a44a22a45a22a46a150a206 a106 a55a207a56a159a58a22a39a32a87a32a60a22a103z = eR a(t)dt2(t)c1 +Zf(t)(t)eR a(t)dtdt:a208a22a209 a
12、43a22a37a22a170a22a154a22a149 a125a22a210 a44a22a45a32a39a32a87a22a60a32a61a211 7.1 a60a22a44a22a45t(t 1)d2xdt2 2tdxdt + 2x = 0:a31a37a22a212a22a154a22a213a57a56a59a46a69a58a22a73a22a214a22a60 x = t. a35a22a72a22a123 a89a22a99 x = tya172a100a22a44a22a45a22a101a22a179d2ydt2 2t 1 2tdydt = 0:a60a22a136
13、a22a149dydt = c11 1t2; y = c1t 2 lnjtj 1t+ c2: a105 a178a36a184a36a186a36a160a36a142a36a68a22a72a22a215a22a216 (t)a103 a210 a44a36a45a36a39a36a169a36a170a22a44a22a45a22a39a22a60a22a46a174a149a22a190 a89a22a99 (7.2)a204a39a36a44a36a45a167a191a217a39 a93a22a94a32a95a22a96a22a218 y(t)a39a36a219 a96 bn(
14、t)a103a36a220a36a46 a126 a127 a154 a124a32a125 a140a22a160a32a142a22a39a129a128a155a39a22a61a77a47a49a22a221 a104a22a105 a215a22a216 (t),a174a149a22a190 a89a22a99 (7.2)a204a39a22a44a22a45a167a59a144a58a32a70 a218 a39a32a219 a96 a103a32a220a32a46 a105a32a222 a100a44a36a45a36a101a36a102a36a103a36a223a
15、36a35a22a55a22a60a32a39a22a107a22a108a22a61a48a98 a210 a44a22a45a22a39a22a169a22a170a32a44a22a45a22a39a22a192a22a193a22a194a22a214a22a60a22a224 a106 a55a22a149a22a225a22a46 a126a67a32a44a32a68 a104a32a105 a37a32a226a32a61 a105a148a203 a142a32a44a148a45 (7.5) a103a32a114a32a46a69a137a32a119 (t) a224a
16、32a72a32a58a32a34a32a168a32a39a32a169a32a170a148a44a148a45a148a39a60a22a46a143a190 a89a22a99 (7.2)a204a39a22a44a22a45(t)d2ydt2 + (20(t) + a(t)(t) dydt + (00(t) + a(t)0(t) + b(t)(t) y = f(t)a167 y a39a22a219 a96 a154a22a224a22a72a22a220a22a61 a126 a225a32a47a22a49 a104 a215a22a216 (t), a174a149 dydta
17、39a22a219 a96 a103a22a220 a15720(t) + a(t)(t) = 0:a114a22a119a22a216 (t) = e 12R a(t)dt, a126 a127 a184a22a149a22a44a22a45a22a154 a89 a179d2ydt2 +b(t) a2(t) + 2a0(t)4y = f(t)e12R a(t)dt: (7.7)2a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a211 7.2 a60a22a44a22a45d
18、2xdt2 + sin tdxdt +14(sin2 t + 2 cost + 4)x = 0:a31a224a22a206 a106a22a227 a125 a58a22a39a22a192a32a193a32a194a22a60a32a46a150a228 a146 a68a32a229a32a160a32a142a32a68a22a61a150a226a32a215a32a216 (t) = e 12R a(t)dt= e 12R sin tdt= e12 cos t, a143a44a22a45a22a102a22a101a22a179 (7.7)a39a22a107a22a230a2
19、2a46a172a22a144a57a167a39 b(t) a2(t)+2a0(t)4 a231a232a22a72a22a233 a96 1,a172a233a22a219 a96 a40a22a41a22a44a22a45d2ydt2 + y = 0a39a22a87a22a60a22a50a22a206 a106 a55a57a56 (a234a22a235a135a22a236 ).a109a22a111 a208a22a237a22a238 a37a22a67a22a40a22a41a22a44a32a45tn dnxdtn + a1tn 1 dn 1xdtn 1 + + an 1
20、tdxdt + anx = 0;a144a189a167 a1; ; ana239a72a36a233 a96 a61 a126 a67a36a44a22a45a22a39a22a214a22a240a22a72a235 n ka218 an ktk dkxdtk (k = 0; 1, ,n)a39a36a219 a96 a73a189a241a191a115 tk. a126 a127 a39a36a44a36a45a36a199a36a103a243a242a36a244 a25a36a26 (Euler, 1707-1783). a245a36a246 D = ddt,a143a44a2
21、2a45 a104a22a247 a179tnDnx + a1tn 1Dn 1x + + an 1tDx + anx = 0: (7.8)a126 a127 a39a22a44a22a45a22a248a22a249a22a55a22a60a22a72a32a50 a147 a51a22a39a32a46 a104a32a250 a39a32a72a22a46a150a190a32a88 a82a90a89a32a91 a39 a89a32a99t =8 0;es; t 0:3a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a1
22、5a17a16a14a18a14a6a14a7a31 a17a35a22a47a22a49a5a18a22a35a22a55 t 0 a225a22a44a22a45a22a39a22a60a22a46a148a241a20a19a22a46a69a123 a82a90a89a32a91 a39 a89a32a99t = es:a35a22a72a5a21a22a229a5a22a22a108 (7.10)a149 a125(D0(D0 1) 4D0 + 6)x = t + t2 ln t;a205(D20 5D0 + 6)x = t + t2 ln t:a58a22a39a22a87a22a
23、60a22a50a22a206 a106 a55a22a149 (a234a22a235a135a22a236 ).a23 a241. a255a25a0 a157 a245a44a22a45y00 + Q(t)y = 0;a144a57a167 Q(t) b(t) a2(t)+2a0(t)4 a39a22a184a22a73a22a60a22a35 t a39a22a181a5a26a5a27a22a73a3a28a32a46a30a29a3a31 t ! +1a225a22a46Rt0 a(s)ds ! +1,a143a44a22a45x00 + a(t)x0 + b(t)x = 0a39
24、a22a184a22a73a22a60a22a98 t ! +1a225a239a5a32a35a22a220a22a612. a55a22a109a5a33a22a44a22a45a22a39a22a87a22a60 a157(1) t2x00 + t(t 2)x0 (t 2)x = t3;(2) t(t + 1)x00 + (t + 2)x0 x = 0;(3) x00 8tx0 + 16t2x = 0;(4) tx00 + 2x0 tx = 0.7.2 a34a36a35a36a37a36a38a36a39a40a5a41 a96 a60a22a68a22a97a22a229a22a35
25、a5a42a5a43a32a39a22a37a3a44a22a44a32a45a32a61a69a109a32a111 a105a22a203 a142a32a40a22a41a32a44a32a45d2xdt2 + a(t)dxdt + b(t)x = 0 (7.11)a103a22a114a22a116a22a64a22a65 a126 a67a32a44a32a68a32a61a150a137a32a138 a(t)a45 b(t)a239a173 t = t0a46a5a47a72a31a5a48a22a27 ,a205a173 t = t0a46a5a47a104a5a49 a179
26、 t t0a39a5a40a5a41 a96 a157a(t) =1Xk=0ak(t t0)k; b(t) =1Xk=0bk(t t0)k: (7.12)4a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a104a22a105 a255a25a0 a157 a34a22a175a5a50a22a39 x0; x00,a44a22a45 (7.11)a239a173 t = t0a46a5a47a73a5a51a5a52a5a53a5a54a5a55a5a56x(t0) = x0;
27、 dx(t0)dt = x00 (7.13)a39a36a60a9a57a36a60a36a46 a205a9a58a173a95a22a96 x = x(t),a58a9a51a9a52a9a53a9a54a5a55a5a56 (7.13), a29a9a31a173 t = t0a46a9a47a104a9a49a179a5a40a5a41 a96x(t) =1Xk=0ck(t t0)k: (7.14)a255a1a0 a39a32a44a32a68a32a72 a157 a138a148a86a3a53a59a54 a195a69a196 (7.11), (7.13) a39a32a60
28、 x(t)a173 t = t0a46a3a47a104a3a49a179 (7.14). a100 a(t); b(t)a45 x(t) a39 a49 a108 a165a22a166 (7.11) a60a5a61a46a69a107a22a108a22a54a3a62 a218 a216a32a42a59a63a32a46a150a190a64a9a65 a46a155a223a36a149 a125 a37a22a113a5a40a5a41 a96 a61 a84a204a5a66a58a3a67a22a37 a218 a39a22a219 a96a239a3a68a35a22a22
29、0a22a46 a223a22a149 a125 a37a22a219a5a33a3a69a22a219a70 ck; k = 0; 1; a45 ak; bk; k = 0; 1; a39a165a96 a180a22a219a22a108a22a61a71a21a32a229 a126 a71a32a180a32a219a22a108a45a53a3a54a55a3a56 (7.13), a47a32a49 a104a3a72 a170a32a100 ck; k = 0; 1; a87a32a88 x0; x00 a45 ak; bk; k = 0; 1; a73a105a3a74 a16
30、4a32a61 a74 a164a32a140a32a219 a96 ck; k = 0; 1; ,a221a32a154 a74 a164a32a140a3a40a3a41 a96 (7.14). a104a32a105 a255a1a0 a157a69a126a127 a74 a164a57a56a59a39a5a40a3a41 a96 (7.14)a173 t = t0a46a5a47a72a22a133a5a75a32a39a32a61 a126 a37a148a240a32a37a32a190 a255a1a0 a46a150a47a32a49a148a154 a104a105a5a
31、76 a164 a157a17a126 a37a5a40a3a41 a96 (7.14)a77a56a59a39a22a60a5a57 a95a22a96 x = x(t)a173 t = t0a46a5a47a154a22a37a22a164a22a72a53a5a54a57a195a59a196 (7.11), (7.13) a39a22a60a22a61 a126 a72a57a241a69a103a32a46a30a40a59a41 a96a173a32a144a133a3a75a3a78a80a79 a104a148a105 a62 a218 a216a32a42a59a63a175
32、a5a50a5a81a22a170a22a46a71a82a3a83a32a219 a96 ck; k = 0; 1; a39 a74 a164a22a44a22a108a22a154 a94a3a84 a46a150a100 x = x(t) a39a77a124 a108(a85a87a86 a(t); b(t)a39 a77a124 a108 (7.12)a165a36a166 (7.11) a60a9a61a36a204a9a88a9a89a68a35a22a220a22a61a91a90a32a35a5a51a5a52a5a53a3a54a5a55a56 (7.13), a82a9a
33、83a36a219 a96 c0; c1a39a36a216a36a68a36a221a36a72 a0a20a92 a39a22a61a173a126a22a162 a46 a47a22a49a22a224a5a93a5a94a22a154a22a37a32a38a22a39a22a44a22a45(7.11) a116 a255a25a0a20a95 a178a32a186a148a44a32a108 a74 a164a32a39a59a40a3a41 a96 (7.14) a173 t = t0a46a5a47a39a22a133a5a75a32a41a32a46 a241a69a103
34、a173a62a5a96a235a25a97a236 x4a167a46a69a47a22a49a22a100a22a34a5a98a22a37a22a38a32a39a22a44a32a45a22a139a207a56 a126 a37a3a7 a76 a39 a255a25a0 a61a211 7.4a173 t = 0a46a5a47a55a22a60a100a99a5a101a5a102 (Legendre, 1752-1833)a25a22a26(1 t2)d2xdt2 2tdxdt + ( + 1)x = 0; (7.15)a144a57a167 a72a22a233 a96 a6
35、1a31a92a22a84a173 t = 0a46a5a47a46a69a44a22a45a22a39a22a219 a96a239a72a32a60a3a57a22a39a32a61a69a100x = c0 + c1t + c2t2 + + cktk + =1Xk=0cktka165a22a166 (7.15),a149(1 t2)(2 1c2 + 3 2c3t + 4 3c4t2 + + k(k 1)cktk 2 + )2t(1 c1 + 2 c2t + + kcktk 1 + )+ ( + 1)(c0 + c1t + + cktk + ) 0:5a0a2a1a4a3a4a5a4a6a
36、4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a103a60a5a61a39a86a40 a218a5a104a173a37a5a105a32a46a30a29a5a106a3a107 a218 a39a32a219 a96 a103a32a220a22a46a150a149 a125 a109a5a33a32a180a32a219a22a1082 1c2 + ( + 1)c0 = 0;3 2c3 2 1c1 + ( + 1)c1 = 0;(k + 2)(k + 1)ck+2 k(k 1)ck 2kck + ( +
37、 1)ck = 0; k 2:a17a19a22a149c2 = ( + 1)2 1 c0;c3 = (1 )( + 2)3 2 c1;ck+2 = (k )( + k + 1)(k + 2)(k + 1) ck; k 2:a35a22a72a22a98 m = 1; 2; a225c2 = ( + 1)2! c0;c4 = (2 )( )( + 1)( + 3)4! c0;c2m = (2m 2 ) ( )( + 1) ( + 2m 1)(2m)! c0;c3 = (1 )( + 2)3! c1;c5 = (3 )(1 )( + 2)( + 4)5! c1;c2m+1 = (2m 1 ) (
38、1 )( + 2) ( + 2m)(2m + 1)! c1:a216 c0 = 1; c1 = 0 a45 c0 = 0; c1 = 1,a43a5a108a22a149 a125x1(t) = 1 +1Pm=1(2m 2 )(2m 4 ) ( )( +1)( +3) ( +2m 1)(2m)! t2m;x2(t) = t +1Pm=1(2m 1 )(2m 3 ) (1 )( +2)( +4) ( +2m)(2m+1)! t2m+1:(7.16)a21a22a229 a124a5a109a25a110a20a111 (D Alembert, 1717-1783)a112a108a22a68a2
39、2a206 a106a22a255a25a0 a46 a126a5a113 a113a3a40a5a41 a96 a39a32a133a5a75a26a5a114a22a103 1. a241a20a19a22a46a17a58a32a49a32a164a3a115a148a39 a113 a113 a95a32a96a173 jtj 1 a178a239a72a22a44a22a45 (7.15) a39a22a60a22a61a171c0; c1 a39a22a216a22a68 a94 a46 x1(0) = 1; x01(0) = 0; x2(0) = 0; x02(0) = 1, a
40、184 a105 x1(t) a116 x2(t)a40a22a41 a146 a180a22a46a148a241a172a44a22a45 (7.15)a39a22a87a22a60a22a103x = c1x1(t) + c2x2(t):6a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a171a3a77a124 a108 (7.16) a104a32a105 a213a207a56 a157 a98 a72 a82a155a84a32a96 a225a32a46 x1(t
41、) a45 x2(t) a167a73a32a37a32a113a32a72a3a81 a218a108a22a46a69a58a22a154a22a72a22a73a5a117a22a39a118a99a5a101a5a102a5a119a5a120a5a121 .a137a22a119 a(t); b(t)a173 t = t0 a39a5a122a5a78a25a79a59a224a22a72a22a60a5a57a22a39a22a46 a158a159a119a(t) = p(t)t t0; b(t) = q(t)(t t0)2;a144a57a167 p(t); q(t)a239a
42、72 t = t0a39a22a70a5a122a5a78a25a79a59a39a22a60a5a57 a95a32a96 a46a143a47a32a49 a104 a226a32a55a32a44a22a45 (7.11) a39a119a22a109a22a107a22a230a22a39a5a41 a96 a60x = (t t0) 1Xk=0ck(t t0)k;a144a57a167 a45 ck; k = 0; 1; a239a72a22a163a22a164a22a233 a96 a46 c0 6= 0.a109a22a111a22a213a22a37a22a113a5a123
43、a5a12a22a39a22a214a22a114a32a61a211 7.5a173 t = 0a46a5a47a60a125a124a5a126a5a127 (Bessel, 1784-1846)a25a22a26t2x00 + tx0 + (t2 n2)x = 0; (7.17)a144a57a167 na103a22a192a5a128a22a252 a96 a61a31a106x = c0t + c1t +1 + + ckt +k + ; c0 6= 0: (7.18)a103a22a58a165a22a166a44a22a45 (7.17)a149t2( ( 1)c0t 2 + (
44、 + 1) c1t 1+ + ( + k)( + k 1)ckt +k 2 + )+t( c0t 1 + ( + 1)c1t + + ( + k)ckt +k 1 + )+(t2 n2)(c0t + c1t +1 + + ckt +k + ) 0:a104 a29a86a40 a218 a46a71a29a5a106a5a107 a218 a219 a96 a103a22a220a32a46a150a154a32a139a57a56( 1)c0 + c0 n2c0 = 0;( + 1) c1 + ( + 1)c1 n2c1 = 0;( + k)( + k 1)ck + ( + k)ck n2c
45、k + ck 2 = 0; k 2:a241a191a103 c0 6= 0, a17a191a235a37a36a113a36a44a36a45 (a199a36a103a9a129a9a130a36a44a36a45 ) a104 a55a189a56 a39 a113 a113a9a54 a157 = n; = n.a15 a237a22a238 = na39a5a131a22a107a22a61 a126 a225a22a73c1 = 0; ck = 1k(2n + k)ck 2; k 2:7a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a1
46、2a14a13a14a15a17a16a14a18a14a6a14a7a35a22a72a22a98 m = 1; 2; a225a22a46 c2m 1 = 0,a31c2 = 14(n + 1)c0;c4 = 142(n + 1)(n + 2) 2!c0;c2m = ( 1)m4m(n + 1)(n + 2) (n + m) m!c0:a165a22a166 (7.18)a149 a125x1(t) = c0tn +1Xm=1( 1)mt2m+n4m(n + 1)(n + 2) (n + m) m!:a21a22a229 a124a5a109a25a110a20a111a112a108a2
47、2a68a22a206 a106a3a132a32a255 a46a150a58a173a64 a113 t a27a22a178a22a73a22a164a5a115a22a61a71a21a22a229 a95a22a96 a39a22a41a5a133(n + m + 1) = (n + m) (n + 2)(n + 1) (n + 1);(m + 1) = m!;a29a22a216a22a233 a96 c0 = 12n (n+1), a134a37a5a135 a104 a149 a125x1(t) =1Xm=0( 1)m(n + m + 1) (m + 1)t22m+n;a58a
48、22a199a22a103 n a202a5a124a5a126a5a127a5a136a5a137 .a34a22a35 = n a39a5a131a22a107a22a46a69a119a22a141 2n a224a68a35a22a175a22a120 a64 a96 a46 a92 2na68a35a22a70a22a113a5a138 a96 a46a150a47a49 a104a22a105 a44a5a139a22a54a22a55a22a149a116 x1(t)a40a22a41 a146 a180a22a39a5a140a22a37a22a60 a157x2(t) =1X
49、m=0( 1)m( n + m + 1) (m + 1)t22m n;a199a22a103 n a202a5a124a5a126a5a127a5a136a5a137 . a98 2na68a35a22a70a22a113a22a192a22a220a5a141 a96 a225a32a46a150a224 a104 a132a171a178a32a186a3a142a5a143a3a22a32a108a32a149a56a116 = na34a36a168a36a39a9a41 a96 a60a22a61 a241a144a19 a173a126 a67a5a131a32a107a22a46 a105a3a145a173 n = 0a39a9a131a36a107a36a46a36a146a9a12a3a140a22a86a147a22a68a22a61a69a114a22a119a22a168a22a229a5a148a22a186a22a160a32a142a22a68 a104 a149a207a56a159a44a22a45 (7.17)a39a22a87a22a60 a157x =
