Wu Chong-shi a0a1a2a3a4 a5a6a7a8a9 ( a10) a111a12 a13a14a15a16a17 a18a19a20a21a22 ( a23) §32.1 a24a25a26a27a28a29a30 a31a32a33a34a35a36a37a38a39a40a41a42a43a44a45 ? a46a47a48 a37a38a39 f(x, y) a49a50a51 a41a42a40a52a53a54a55a56 df = ?f?xdx + ?f?ydy = 0. a57a58dx, dy a59a60a49a61a62a48 a37a38a39 f(x, y) a51 a41a42a40a52a53a54a55a63a64 a62a65a66 ?f ?x = 0, ?f ?y = 0. ? a67a47a68 a34a69 a48 a37a38a39a40a41a42a43a44 a49a48 a37a38a39a40a54a55a41a42a43a44 a49a70a71a72a73 a54a55 g(x, y) = C a35a74f(x, y)a40a41a42a45a75a76 a49a71a77a78a79a49 a64 a62a80a72a73 a54a55a81a82y = h(x) a49a83a84a85a86f(x, y) a87 a40y a45a75a88 a49a79a89 a54a55a41a42a43a44a90a91a92a58a34a37a38a39 f(x, h(x))a40a93a94a41a42a43a44 a49a50a51 a41a42a40 a52a53a54a55a90a56 ?f ?x + ?f ?yh prime(x) = 0. ? a95a96 a75a97a98a99 a67a47a68 a34a100a101a81a45a57a58 a79a102a103a104a105 a53a106a107a108a109 y = h(x)a40a110a111a112 a49 a113a114 a105 a53a108 a109 dy dx ≡ h prime(x). a75a88 a49a115a116a104 a52 ( a71a117 a36a39a118a119a35a120 a104 a64a121) a74a82y = h(x) a49 a90a64 a62a122a123a95a72a73 a54a55a124a125 ?g ?xdx + ?g ?ydy = 0, a126a113a74a82 dy dx = ? ?g/?x ?g/?y, a96 a56 a70 a64a127 a79a89a48 a37a38a39 a51 a41a42a40a52a53a54a55 a65a66 ?f ?x ? ?f ?y ?g/?x ?g/?y = 0. a128a129a130a131a132 a49a133a134a135a136a137a138a139a140a141a142a143a144 a145a146a130 a142a147a148a149 a130a150a151 a45 a152a153 a49 a154a155 a144 a145 a146 a149a156 a130a157 a142a49a158a159a160a161a162a163a162a164a165 a45 Wu Chong-shi §32.1 a166a167a168a169a170a171a172 a112a12 ? a71a173a174a87a49a175a176a174 Lagrange a177a178a179a180a181 a101a36a37a38a39a40a54a55a41a42a43a44a45 a182a183 a49a95a96a79a102 a40 a71a72a73 a54a55 g(x, y) = C a35a74a38a39f(x, y)a40a41a42a43a44 a49 a90a64 a62a184a185 Lagrangea186a187λa49 a113a188a189a34a97a190a40 a48 a37a38a39a191 h(x, y) = f(x, y)?λg(x, y). a192a127 x a193 y a194a66 a56a195a97a196a197a198a199 a49 a75a88 a49 a75a97 a48 a37a38a39 a51 a41a42a40a52a53a54a55a90a56 ( a200a201a194 a82 a49a85a86 λa49 a75a90a121a92a58 a79a102a202 a82a40a52a53a54a55) ?(f ?λg) ?x = 0, ?(f ?λg) ?y = 0. a80a203 a64 a62 a74a82 x = x(λ), y = y(λ), a204a32a205 a72a73 a54a55 a87a49 a188a82 Lagrange a186a187λa40a39a42a49a90a64a62a74a82a64a121a40a41a42a206 (x, y)a45 a207a208a153 a141a142a143a144 a145a146a130 a142a147a148a149a49a160a209a210a211a212a213a214a215 a45 a207a208a216a217 a142a143a218a219a220a221a49a160a161a222 a223a224a225 a142a143Lagrangea226a227a228a209 a45 a229 a71 a32a205a230a38a40a54a55a41a42a43a44a45 a183a99a53a74a230a38 J[y] = integraldisplay x1 x0 F(x, y, yprime)dx a71a231a232 a54a55 y(x0) = a, y(x1) = b a62a233a72a73 a54a55 J1[y] ≡ integraldisplay x1 x0 G(x, y, yprime)dx = C a35a40a41a42 a49a78 a64a188a189 J0[y] = J[y]?λJ1[y], a192a127δy a194a66 a56a196a197a40 a49a78 a230a38 J 0[y]a71a231a232 a54a55a35 a51 a41a42a40a52a53a54a55a90a56 parenleftBig ? ?y ? d dx ? ?yprime parenrightBig (F ?λG) = 0. a80a203 a124a125a234a235a236 a231a232 a54a55 a62a233a72a73 a54a55 a49 a52a53a76a237a238a239a240 a49 a90a64 a62 a74a82Lagrange a186a187 a40a42λ = λ 0 a236 a41a42a38a39y = y(x, λ 0)a49a62a233a241a242 a40a230a38 J 0[y] a40a54a55a41a42a45 a243 32.1 a74a230a38 I[y] = integraldisplay 1 0 xyprime2 dx a191 a244a245a246a247a248a249a250a251a252a253a248 Lagrange a254a255a0a1a2 a245a3a4a5a6a7 Wu Chong-shi a0a1a2a3a4 a5a6a7a8a9 ( a10) a113a12 a71a231a232 a54a55 y(0)a47a232, y(1) = 0 a193a72a73 a54a55 integraldisplay 1 0 xy2 dx = 1 a35a40a41a42a8a9a45 a10 a11 a174a79a102a12a89 a40 Lagrange a186a187a13a49 a64 a62a14 a205a52a53a54a55 parenleftBig ? ?y ? d dx ? ?yprime parenrightBigparenleftbig xyprime2 ?λxy2parenrightbig = 0, a70 d dx parenleftbigg xdydx parenrightbigg + λxy = 0. (#) a203 a234a235 a233a15a16 a40 a231a232 a54a55 a70a17a66 a34a97a18a19a42a43a44 a49a50 a40a18a19a42 λi = μ2i, μi a56a20a21a22a23a24a38a39 J0(x)a40a25ia97a107a20a206a49i = 1,2,3,··· a107a26a90a56Lagrange a186a187a49 a113a41a42a38a39a90a56 a241a242 a40a18a19a38a39 yi(x) = C J0 (μix). a176 a199C a64 a62a80a72a73 a54a55a188a82a45a57a58 C2 integraldisplay 1 0 xJ20(μix)dx = C 2 2 J 2 1(μi) = 1, a61a62 C = √2 J1(μi). a75a88 a49 a90a74a82a27a41a42a38a39 yi(x) = √2 J1(μi)J0(μix). a80a96Lagrangea186a187 a40 a184a185a49a71 Euler–Lagrangea234a235a82a229a27a28a188a29a199a49a193a15a16a231a232 a54a55a30a31 a71 a34a32 a49 a90 a17a66 a18a19a42a43a44a45a113a33a58a18a19a42a43a44 a49a50 a40a81 a49 a18a19a42 a193 a18a19a38a39 a49a47a34a35 a36a97a45a75a36 a47 a195a97a43a44 a105 a53a37a38a45 star a25a34a97a43a44a49 a75 a34a35 a36a97a18a19a38a39a39a56a41a42a38a39a45 a75a64 a62 a126a35 a102 a40a198a125a40a41 a194 a82a45 a80a231a232 a54a55 a62a233a80a203a42a14 a40 δy vextendsinglevextendsingle vextendsingle x=0 a47a232, δy vextendsinglevextendsingle vextendsingle x=1 = 0. a64 a62 a74a82I[y]a40a34a43a198a125 δI[y] = 2 integraldisplay 1 0 xyprime (δy)prime dx, a185 a113a64 a62 a74a82 I[y]a40 a48 a43a198a125 δ2I[y] = 2 integraldisplay 1 0 xparenleftbigδyprimeparenrightbig2 dx > 0. Wu Chong-shi §32.1 a166a167a168a169a170a171a172 a114a12 a57a58a230a38I[y]a40 a48 a43a198a125a44 a51 a107a42 a49a61a62 a75a45a41a42a38a39a46a47a230a38 a51 a41a48a45 star a25a48a97a43a44a56a49a75a34a35a97a18a19a42a107a26a120a90a56a230a38a40a41a42a45a75a56a57a58a49a127a234a235 (#)a186a62 a41 a42a38a39y(x) a49a49a50 a125 a49 a90 a47 λ integraldisplay 1 0 xy2 dx = ? integraldisplay 1 0 yparenleftbigxyprimeparenrightbigprime dx = ?y·xyprime vextendsinglevextendsingle vextendsingle 1 0 + integraldisplay 1 0 xyprime2 dx = integraldisplay 1 0 xyprime2 dx, a51a52 a72a73 a54a55 a49 a90a121 a14 a205 λ = integraldisplay 1 0 xyprime2 dx. a53a54 a49 a55a56a57 a140a49 a58a59a60a61 a148 a130 a220a221a62a63a64 a65a130a66a67 a49a209a210 a68a69 a140a70 a71a72a73a74a75a76a59a77a78 a129a79a80 a62a81a82 a130a66a83a84a85 a64 a65a45a86a87 a49 a61 a148 a130 a220a221a62a63a64 a65 a49 a88a89a90a91a75 a64 a65(Isoperimetric problem)a45 Wu Chong-shi a0a1a2a3a4 a5a6a7a8a9 ( a10) a115a12 §32.2 a92a93a94a95a96a97a98a99a100a101a102a30a98a99a26a103a93a104a105 a61 a148 a80 a62a63 a130a106a56 a220a221 a130a107a108a151 a159(Euler–Lagrangea109a110)a153a88a107a108a109a110a111a112 a107a108 a109a110a49 a113a114a145a146 a148a149 a130a77a115 a220a221a116 a72a117 a163a49a161a118a119 a88 a107a108 a109a110a111a112 a107a108 a109a110 a130a77a115 a64 a65a120 a121a122a61 a148 a130 a220a221a62a63a64 a65 a49a123 a106a56 a220a221a124a125a126a127 a77a128a146 (Lagrange a226a227) a49 a113a114a129a130a131 a132 a220a221a116 a72a117 a163a49a161a118a119 a107a108 a109a110a133a134a63a64 a65a45 a58a59a135a136a137a138a113a130a139 a64 a65a140a207a85a136a107a108 a109a110 a130a77a115 a64 a65 a111a133a134a63a64 a65a141a142a90a61 a148 a130 a62 a63a111a220a221a62a63a64 a65 a49a111a143a144a49 a207a85a136a107a108 a109a110 a130a77a115 a64 a65 a111a133a134a63a64 a65a145a145a108a146a147a148 a149a45 a243 32.2 a65 a82 a176 a124a125a234a235 a231 a42a43a44 d dx bracketleftbigg p(x)dydx bracketrightbigg + q(x)y(x) = f(x), x0 < x < x1, (#) y(x0) = y0, y(x1) = y1 (maltesecross) a40a230a38a119a112 a49a70a150 a82 a241a242 a40a230a38 a49a50a71a231a232 a54a55 (maltesecross)a35 a51 a41a42a40a52a53a54a55 a70 a58 (#)a45 a10 a151 a83 a230a38a41a42a52a53a54a55a40a124a125a119a112a90a56a234a235 (#) a49a152a153a49 a75a97a234a235a34a188 a180a154integraldisplay x1 x0 braceleftbigg d dx bracketleftbigg p(x)dydx bracketrightbigg + q(x)y(x)?f(x) bracerightbigg δy(x)dx = 0. a229 a71 a40a43a44a90a56a53a155 a79 a112a156a157a92 a66a158 a34 a50 a125a40a198a125 a49 a75 a95a96a159a50 a125a160 a50 a38a39a40a25 a48 a236a161a162a56a163 a200 a201a173 a229a40 a49 integraldisplay x1 x0 q(x)y(x)δy(x)dx =12δ integraldisplay x1 x0 q(x)y2(x)dx, integraldisplay x1 x0 f(x)δy(x)dx =δ integraldisplay x1 x0 f(x)y(x)dx. a164a165 a148a149q(x)a114f(x)a153a166y(x)a130a145a108a167a168a130a49 a86a87 a49 a169a145 a108a170a171 a124a49 a113a172a173a153a88a146a45 a95a96 a160 a50 a38a39 a87 a40a25a34a162 a49 a64 a62 a125a174 a50 a125 a49integraldisplay x1 x0 d dx bracketleftbigg p(x)dydx bracketrightbigg δy(x)dx = p(x)dydxδy(x) vextendsinglevextendsingle vextendsinglevextendsingle x1 x0 ? integraldisplay x1 x0 p(x)dydx d(δy)dx dx =? integraldisplay x1 x0 p(x)dydxδ parenleftbiggdy dx parenrightbigg dx =? 12δ integraldisplay x1 x0 p(x) parenleftbiggdy dx parenrightbigg2 dx, a175 a87a174 a205a27 δy(x)vextendsinglevextendsingle x0 = δy(x) vextendsinglevextendsingle x1 = 0 a45a155 a79a102 a40a98a99a176a31a32 a180a49 a90 a14 a205 integraldisplay x1 x0 braceleftbigg d dx bracketleftbigg p(x)dydx bracketrightbigg + q(x)y(x) ?f(x) bracerightbigg δy(x)dx = ?δ integraldisplay x1 x0 braceleftBigg 1 2 bracketleftBigg p(x) parenleftbiggdy dx parenrightbigg2 ?q(x)y2(x) bracketrightBigg + f(x)y(x) bracerightBigg dx Wu Chong-shi §32.2 a177 a6a178a179a180a181a182a183a184a185a186 a172 a182a183 a168 a5a6a187a188 a116a12 = 0. a75a90a189a190 a49 a234a235 (#)a34a188a90a56a230a38 J[y] = integraldisplay x1 x0 braceleftBigg 1 2 bracketleftBigg p(x) parenleftbiggdy dx parenrightbigg2 ?q(x)y2(x) bracketrightBigg + f(x)y(x) bracerightBigg dx a51 a41a42a40a52a53a54a55a45 a243 32.3 a65 a82a191a124a125a234a235a188a81a43a44 ?2u(r) + k2u(r) = ?ρ(r), r ∈ V, u(r)vextendsinglevextendsingleΣ = f(Σ) a40a198a125a119a112a45 a10 a64 a62a192a193a194a195 a182 32.2a40a196 a13a49a197a198a50 a125 integraldisplayintegraldisplayintegraldisplay V bracketleftbig?2u + k2u + ρ(r)bracketrightbigδudr, a95a96 a160 a50 a38a39 a87 a40 a84 a195a162 a49a47 integraldisplayintegraldisplayintegraldisplay V k2uδudr = 12δ integraldisplayintegraldisplayintegraldisplay V k2u2dr, integraldisplayintegraldisplayintegraldisplay V ρ(r)δudr =δ integraldisplayintegraldisplayintegraldisplay V ρ(r)udr. a95a96 a160 a50 a38a39 a87 a40a25a34a162 a49a78a105 a53 a242a174 Greena25a34a199a112a62a233a231a232 a54a55 δu(r)vextendsinglevextendsingle Σ = 0a49integraldisplayintegraldisplayintegraldisplay V ?2uδudr = integraldisplayintegraldisplay Σ δu?u·dΣ ? integraldisplayintegraldisplayintegraldisplay V ?u·?parenleftbigδuparenrightbigdr = ?12δ integraldisplayintegraldisplayintegraldisplay V parenleftbig?uparenrightbig2dr. a57 a203a49a77 a234a235a90a91a92a58 δ integraldisplayintegraldisplayintegraldisplay V braceleftbigg1 2 bracketleftbigparenleftbig?uparenrightbig2 ?k2u2bracketrightbig?ρubracerightbiggdr = 0. a75a189a190 a49a77a180 a40a188a81a43a44a90a200a201 a96a71a231a232 a54a55 u(r)vextendsinglevextendsingleΣ = f(Σ) a35a74a230a38 integraldisplayintegraldisplayintegraldisplay V braceleftbigg1 2 bracketleftbigparenleftbig?uparenrightbig2 ?k2u2bracketrightbig?ρubracerightbiggdr a40a41a42a43a44a45 a243 32.4 a65 a82a191a124a125a234a235a40a18a19a42a43a44 ?2u(r) + λu(r) = 0, r ∈ V Wu Chong-shi a0a1a2a3a4 a5a6a7a8a9 ( a10) a117a12 u(r)vextendsinglevextendsingleΣ = 0 a40a198a125a119a112a45 a10 a202a31 a49 a64 a62 a127a18a43a44 a194a66 a56a182 32.3a40a203a204a118a119a45a57 a203a49a203 a18a19a42a43a44a90a200a201 a96 a230a38 J[u] = integraldisplayintegraldisplayintegraldisplay V braceleftBigbracketleftbig ?u(r)bracketrightbig2 ?λbracketleftbigu(r)bracketrightbig2 bracerightBig dr a71a15a16a231a232 a54a55 u(r)vextendsinglevextendsingleΣ = 0 a35a40a41a42a43a44a45 a175a185 a34a205 a49 a155a18a19a42 λ a194a66 a56Lagrange a186a187a49a152a153a49 a75a97a230a38a41a42a43a44a63a200a201 a96 a230a38 J[u] = integraldisplayintegraldisplayintegraldisplay V bracketleftbig?u(r)bracketrightbig2 dr a71a79a89a15a16a231a232 a54a55 a193a72a73 a54a55 (a18a19a38a39a40a206a34a92a54a55) J1[u] ≡ integraldisplayintegraldisplayintegraldisplay V bracketleftbigu(r)bracketrightbig2 dr = 1 a35a40a54a55a41a42a43a44a45 a104a207 a101a81 a49 a75a45a18a19a38a39a107a26a90a56a230a38a40a41a42a38a39 a49 a113a18a19a42a107a26a56a230a38a40a41a42a45 a80a96 a230a38 J[u]a40a48 a43a198a125 δ2J[u] = 2 integraldisplayintegraldisplayintegraldisplay V bracketleftbig?parenleftbigδu(r)parenrightbigbracketrightbig2 dr a44a58a107 a49a61a62a49 a230a38a40a41a42a56a41a48a42a45a75a45a41a48a42 a87 a40a208a48a209 a49a210a83 a90a56a18a19a42a43a44a40a208a48a18a19 a42a45 Wu Chong-shi §32.3 Rayleigh–Ritza178a7 a118a12 §32.3 Rayleigh–Ritz a94a211 ? a198a125a13a71a212a101a213a87a40a242a174a49a64a62a125a58a195a97a214a53a40a234a102a45 ? a34a100a242a174 a56a33a58a215a18 a212 a101a216a217a40a110 a89a218a219 a45 – a64a62a174Hamiltona77 a101a220a175a221a69a222a40 a218a219a12a89a223 a213a224a225 ( a226 a206a236 a226 a206a30 ······)a40a227a228 a49 – a64a62a174Fermata77 a101 a12a89a229 a9 a71a230a226a87 a40a231a232 a49a233a234a71a232a102a79 a40a235a236 a193a237 a236 a49 – a120a64a62a174a198a125a40a218a219a12a89a238a239a240a241a116a124a242a243a187a40a227a228a49a200a200a45 a71a212 a101a213a40a75a45a125a244 a87a49 a244a245 a212a226 a227a228a40a246a100a203a188a119a112a40a215a18a216a217 a49a34 a34a182a247a248a39a64 a62 a110 a89 a58a246 a154 a40a230a38a41a42a43a44a45 a198a125 a13 a40a75a100 a242a174a49 a249 a47a250 a53a40a101a38 a60 a189a45 a50 a64 a62 a47a251a252 a175 a225a34a248a27a81 a212a226a253a232 a40a227a228 a49 a64 a62 a47a251a252 a175 a234a254a248a126a255a108a40 a212 a101a0a1a2a190a40a0a1a3a4a45 ? a198a125a13a40a25a48a100a242a174a78a56a5a229a82a50a40a173a174a201a42a140a50a58a74a81a249a5a40a212a101a43a44a6a7a27a34a100a190a40a8 a9a10a11a45 a12a13a251a252a14a76a192 a83 a56a15a16 a96 a47 a174 a124a125a234a235 a86a12a65 a75a45 a212 a101a43a44 a49 a17a18a19a114 a47a20 a39a40a43a44a21a121 a22a23a24a25a26a27a28a29a30a31a32a33a34a35a36a36a37a38a39a40a41a42a25a43a27a44a45a46a30a47a48a49a26a50a51a52a53a54a31a55 a30a41a42a25a46a43 a56a57a58a59a60a59a61a30a62a63a64a33a34 LX = λρX, α1X(a) + β1Xprime(a) = 0, α2X(b) + β2Xprime(b) = 0. a24a25a30a65a66a67a68a69a70a71 ( a72 a58a73a74a75a29a30) a76a77 a45a78a79a30a80a25a26a81a82a83a84a85a86a87a88a26a74a71a62a63a64a89 a62a63a90a29a43a58a91a92a93a38a94a59a92a93a38a68 a76a77 a45a78a79a54a95a96a24a25a97a98a59a92a93a38a68a99a100a101a55 a76a77 a45a78a79 a102a29a25a46a26a103a38a24a71 a76a77 a45a78a79a30a25a43a59a61a104a105a26a106a53a107a29a108a109a110a111a30a90a29a112a26a54a113a114a115a38a39a40 a62a63a64a30a116a23a117a118a119a43a120a121a122 Xprimeprime(x) + λX(x) = 0 a123a124a125a126a127a30a78a79a26a58a110a128a30a91a60a129a130a131a132a25 sin√λxa89cos√λxa26a27a59a61a30a133a134a135a85a86a87a88a136a26 a137a131a46a70a71a62a63a64a30a138a139a117a118a119a43a140a141a26a142a143a59a61a30a62a63a64a33a34a26a123a144a30a145a113a50a93a146a147a128a53a43a44 a45a46a50a148a149a150a151a152a53a24a25a62a63a64a30a41a42a78a46a43 ? a55Rayleigh–Ritza78a46a41a42a24a25a62a63a64a33a34a30a47a62a153a154a68a94 Wu Chong-shi a155a156a157a158a159 a160a161a162a163a164 ( a165) a1669a167 – a168 a69a169a62a63a64a33a34a170a171a148a172a90a30a87a88a173a64a33a34a26 – a81a82a27a59a74a30a90a29a174a175a35a24a25a26a176a147a169a33a34a177a170a171a90a29a30a87a88a173a64a33a34a43 ? a37a101a178a179a30a90a29a174a175(a142a143a180a62a63a64a33a34)a68a181a182a30a26a183a184a49a67a93a185a186a187a188a23a189a190a41a62a63a64a30 a188a23a64a43 ? a191 a31a55a30a192a193a194a26a50a68a101a178a179a59a60 a195a196a197a30a90a29a174a175 (a31a32a49a68a59a60a90a29a198a199) a26a59a78a200a201a143 a202a203a26a59a78a200a177a38a187a186a187a204a189a205a186a187a188a23a189a24a39a62a63a64a30a41a42a64a43 ? a123a50a101a24a90a29a198a199a206a58a62a63a90a29a207a101a24a30a208a101a47a62a209a63a26a101a24a149a150a210a69a191a211a212 a49a89a29a213a49a142 a143a62a63a90a29a30a130a214a215a71a116a23a30a216a217a43 a218 32.5 a24a62a63a64a33a34 1 x d dx parenleftbigg xdydx parenrightbigg + λy(x) = 0, y(0)a58a86, y(1) = 0 a30a125a219a62a63a64a43 a220 a123a60a62a63a64a33a34a27 32.1 a221 a30a22232.1 a35a108a109a223a224a225a43a226a227a223a224a30a68a172a90 I[y] = integraldisplay 1 0 xyprime2 dx a27a85a86a87a88 y(0)a58a86, y(1) = 0 a89a228a229a87a88 I1[y] ≡ integraldisplay 1 0 xy2 dx = 1 a136a30a87a88a173a64a33a34a26a230a30 Euler–Lagrangea78a79a50a68 1 x d dx parenleftbigg xdydx parenrightbigg + λy(x) = 0. a231a27a55Rayleigh–Ritza78a46a105a41a42a24a25a123a60a172a90a30a87a88a173a64a33a34a43 a210a69a26a149a150a142a143a62a63a90a29a30a53a25a68a26a230a106a53a232a233a234a186a85a86a87a88a235a112a26a99a236a237a206a58a238a239a130(a148a240 a141a241)a43a176a180a26a93a55a28a242a119a198a199 yn(x) =α1parenleftbig1?x2parenrightbig+ α2parenleftbig1?x2parenrightbig2 + α3parenleftbig1?x2parenrightbig3 + ···+ αnparenleftbig1?x2parenrightbign, n = 1,2,3,··· a243a190a41a62a63a90a29a43 a168 a69a244a41a42a30a62a63a90a29y 2(x) a26a120a27a49a119a35a244a245a91a242a26a83a84a172a90a246a228a229a87a88a26a39 I[y2] = integraldisplay 1 0 xyprime22 dx=α21 + 43α1α2 + 23α22, Wu Chong-shi §32.3 Rayleigh–Ritza247 a162 a16610a167 I1[y2] = integraldisplay 1 0 xy22 dx = 16α21 + 14α1α2 + 110α22 = 1. a123a93a185a194a248a68 α 1 a89α 2 a30a249a250a90a29a30a87a88a173a64a33a34a26a232a101a87a88a68 ?(I ?λI1) ?α1 = 2α1 + 4 3α2 ?λ parenleftBig1 3α1 + 1 4α2 parenrightBig =0, ?(I ?λI1) ?α2 = 4 3α2 + 4 3α1 ?λ parenleftBig1 5α2 + 1 4α1 parenrightBig =0. a123a177a68a132a143 α 1 a89α 2 a30a83a29a78a79a251a26a58a252a253a25a30a254a45a232a101a87a88a68 vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle 2? λ3 43 ? λ4 4 3 ? λ 4 4 3 ? λ 5 vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsinglevextendsingle = 0, a120 3λ2 ?128λ+ 640 = 0. a25a235a39 λ = 643 ± 83√34. a123a91a60a255a71a30a0a68 λa30a173a219a64a43 a1 32.1 a2 32.2 a3a4a5 a6a7a8a9 a26a10a11a12a13a11a14a15a16a17a18a10a11a12a19a20a14a43 a123a144a39a40a30a226a81a37a68a62a63a64a33a34a30a125a219a62a63a64a30a41a42a64 ˉλ1 = 5.7841···, a230a89a188a21a64 λ1 = (2.4048···)2 = 5.7831··· a30a22a142a23a24a25a40 2×10?4 a43a22a236a189a26a62a63a90a29a30a41a42a25a68 ˉy1(x) =α1parenleftbig1?x2parenrightbig+ α2parenleftbig1?x2parenrightbig2, α1 =2 radicalBig 12?33 radicalbig 2/17 = 1.6505676···, α2 = radicalBig 80?230 radicalbig 2/17 = 1.0538742···. a148a53a26a188a21a25 y1(x) = √2 J1(μ1)J0 (μ1x) a215a27a28a26a25a29a202a203 ? = integraldisplay 1 0 bracketleftbigy 1(x)? ˉy1(x) bracketrightbig2xdx = 2?2integraldisplay 1 0 y1(x) ˉy1(x)xdx =2 braceleftBigg 1? bracketleftBig4√2α1 μ31 + 8√2α2 μ31 parenleftBig 8 μ21 ?1 parenrightBigbracketrightBigbracerightBigg = 1.66×10?5. Wu Chong-shi a155a156a157a158a159 a160a161a162a163a164 ( a165) a16611a167 a30 32.1 a31a32a33a34a35a36a33a37a38 a39 a30 32.2 a31a32a33a34a35a36a33a37a40 a41a143a27a28a242a119a190a41a35a103a37a244a53a91a242a26a62a63a64a89a62a63a90a29a50a38a118a40a123a60a188a193a26a123a68a30a21a42a43a44a45a30a43 a41a180a93a185a194a71a26Rayleigh–Ritza78a46a30a21a25a46a148a59a92a196a30a41a42a78a46a43a93a185a146a122a26a47a48a244a30a242a29a49a28a26 a39a40a30a188a193a50a49a51a43 ? a191 a49a200a30a202a203a93a185a194a71a26a27a236a55 Rayleigh–Ritz a78a46a227a26a37a38a24a39a125a52a30a53a60a62a63a64a30a41a42 a64a26a62a63a64a30a60a29a89a121a55a30a190a41a90a29a35a30a75a29a29a54a22a55a43 ? a123a68a236a55Rayleigh–Ritza78a46a24a25a62a63a64a33a34a30a59a60a209a56a43a27a31a32a236a55a35a26a57a25a50a176a148Rayleigh– Ritza78a46a37a38a24a39a58a58a60a62a63a64a147a59a52a230a30a31a55a60a64a26a176a148a58a25a107a33a34a37a100a101a24a71a125a219a30a61 a62a60a62a63a64a43 Wu Chong-shi §32.3 Rayleigh–Ritza247 a162 a16612a167 a63a194a59a136a49a200a39a40a30a133a249a60a62a63a64 ˉλ2 = 36.883···, a230a89a188a21a64 λ2 = 30.471···, a235a175a30a23a24a64a65a225 20% a66 a148a53a101a24a39a186a187a188a21a30a133a249a60a62a63a64a26a226a81a232a233a67a68a190a41a90a29a35a30a75 a29a26a123a226a81a232a233a185a181a248a69a70a67a71a30a202a203a72a148a83a60a43a27a31a55a35a26a49a196a30a73a46a68a24a25a59a60a74a30a172a90a87 a88a173a64a33a34a26a230a89a183a105a30a172a90a87a88a173a64a33a34a30a24a75a37a27a143a76a106a77a133a59a60a62a63a64a43a123a37a101a27a183a105a30 a172a90a87a88a173a64a33a34a35a63a78a68a49a59a60a79a80a87a88 integraldisplay 1 0 y(x) ˉy1(x)xdx = 0 a120a93a43a123a124a26a27a123a60a74a30a172a90a87a88a173a64a33a34a35a26a125a219a30a62a63a64a226a81a50a68a183a105a30a133a249a60a62a63a64a53a43