1、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20a21 2a22a24a23a24a25a24a26a24a27a24a28a24a29a30a32a31a32a33a32a34a32a35a32a36a24a37a39a38a41a40a24a42a24a43a24a44a24a45a24a46a24a47a49a48a24a50a24a51a24a52a24a36a24a37a24a53a24a54a49a55a57a56a24a58a24a59a24a60a2
2、4a40a24a42a24a61a49a62 6a30a24a63a24a64a24a65a24a66a24a50a24a67a24a68a24a692.1 a70a72a71a72a73a72a74a72a75a72a76a72a77a78a24a79dydx = h(x)g(y) (2.1)a50a32a65a32a66a32a80a32a81a83a82a32a84a32a85a32a86a32a87a25a24a26 , a88a38a89a47a32a48 h(x)a90 g(y)a91a32a92a52a32a34a32a93a24a94a39a95a24a96a41a97a24a
3、98a49a99a24a69a100a24a30a24a65a24a66a24a50a24a67a24a68 a79 a61a24a1011. a79a24a102a24a103 y0a104a24a105g(y0) = 0,a106 y = y0a107(2.1)a50a24a67a24a1082. a79a24a102 g(y) 6= 0,a106a24a109a24a110a24a111 z = R 1g(y)dy. a100a24a112dzdx =dzdy dydx =1g(y) h(x)g(y) = h(x):a33a107a24a113a24a114a67 (2.1)a50a39
4、a115a41a116a24a117a24a118a24a119a114a24a120a47a24a48a24a50a121a115a122a116a24a69a123a67a49a124a24a55a105z =Zh(x)dx+C;a88a38 Ca107a24a125a24a126a24a127a48a24a69a129a128a39a130a41a131a49a132a105a49a133a24a120a65a49a66a24a50a49a134a49a54a24a135a49a67Z 1g(y)dy =Zh(x)dx +C:a136 2.1a137 g(y) 6= 0 a138a55a
5、129a139a24a140 (2.1)a141a41a142a24a143a24a144a140a24a145a121a146a129a147a49a148a49a149a151a150a121a152a49a153a122a101a123a154a151a155a121a156a157a41a1581g(y)dy = h(x)dx;a159 a152a41a160a24a161a24a155a24a69a136 2.2a162a129a163a161a49a155a49a164a121a165 R f(x)dx,a137a121a166a122a167a155a49a139a49a140a
6、49a168a141a129a169a151a170a172a171a129a137a151a173a172a174a155a151a175a151a168a141a169a24a170a24a176a24a177a39a162a24a178a69a137a39a166a41a167a155a24a139a49a140a49a168a24a55a180a179a49a181a41a182a49a183 f(x)a141a24a184a41a185a24a186a24a187a24a173a55a129a188a137a24a173a121a174a155a24a175a168a24a55a15
7、1a179a41a182a24a183a190a189 f(x)a141a24a191a41a192a24a186a24a187a24a173a69a193 2.1 a67a24a65a24a66dydx = 3x2y: (2.2)a28 a194a107a110a24a195a24a196a37a24a197a24a65a24a66a24a69a129a198 y 6= 0a112a24a55a129a37a24a197a110a24a195a1011ydy = 3x2dx:1a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a
8、15a17a16a14a18a14a19a14a5a14a20a199a24a200 a36a24a37a24a55a105a134a24a54a24a135a24a67lnjyj = x3 +C1;a88a38 C1 a107a24a125a24a126a24a201a48a24a69a194a24a196a24a202a24a203a24a204a81y = C2ex3;a88a38 C2 = eC1. a205 a126a24a133y = 0 a206a107a65a24a66 (2.2) a50a24a67a24a55a151a207a41a208a24a65a49a66a49a13
9、5a49a67a49a209a49a210a196a49a211a49a212a81y = Cex3;a88a38 Ca196a24a213a24a214a55a129a215a90a24a216a50a125a49a126a24a201a48a24a69a193 2.2 a67a24a65a24a66dydx = 2x(1 y2)12:a28 a194a107a110a24a195a24a196a37a24a197a24a65a24a66a24a69a129a198 y 6= 1 a112a24a55a129a37a24a197a110a24a195a101(1 y2) 12 dy = 2x
10、dx:a199a24a200 a36a24a37a24a55a105a134a24a54a24a135a24a67arcsiny = x2 +C;a88a38 Ca107a24a125a24a126a24a127a48a24a69a194a24a196a24a202a49a204a118a49a217 a78 a54y = sin(x2 +C):a208a24a218a24a55a194a24a219a103 a199a49a220a49a221 a67 y = 1(a100 a199a24a220a24a221 a67a24a51a151a222a49a34a49a97a49a54a121a
11、38a129a135a49a223a49a224a125a151a126a49a127a48 C a50a213a24a225a105a24a133). a226 a2271. a67a24a61a24a228a24a65a24a66a24a101(1) dydx = 1+y2xy+x3y;(2) dydx = xe2y+x2;(3) (x+ 1)dydx + 1 = 2e y;(4) dydx = xy+yxy+x.2a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a2
12、02.2 a229a72a230a72a76a72a77a78a24a79dydx = gyx(2.3)a50a24a65a24a66a24a80a24a81a83a231a24a232a25a24a26 . a61a24a228 a78 a54a24a50 a199 a30a24a65a24a66 a88a201a206a107a49a233a24a234a65a24a66a49a1011. a65a24a66dydx =M(x;y)N(x;y); (2.4)a88a38a122a47a49a48 M(x;y)a90 N(x;y) a235a107xa90 y a50a121a236a234
13、(a237a79 ma234)a233a49a234a47a49a48a49a55a129a238a49a224t 0a103M(tx;ty) tmM(x;y); N(tx;ty) tmN(x;y);2. a65a24a66dydx = f(x;y); (2.5)a88a38a41a239a24a240a24a47a24a48 f(x;y)a107xa90 ya50 a216a234a24a233a24a234a47a24a48a24a55a129a238a49a224 t 0a103f(tx;ty) f(x;y):a241a201a97a24a55a129a242a24a243a24a47a
14、24a48a24a50a233a49a234a24a244a55a123a245a49a246a213 t = 1x,a65a24a66 (2.4)a132 a196a24a203a24a204a118dydx =xmM(1; yx)xmN(1; yx) =M(1; yx)N(1; yx);a247 (2.5)a106a24a196a24a203a24a204a118dydx = f1;yx:a233a24a234a65a24a66 (2.3)a50a24a67a24a68 a79 a61a24a101a109a24a248a24a249a47a24a48a110a24a111 u = yx,
15、a238 y = xu. a100a24a112a24a65a24a66 (2.3)a117a24a81xdudx +u = g(u):a194a24a250a107a43 a220a110a24a195a24a196a37a24a197a24a65a49a66a49a101dudx =1x(g(u) u):a193 2.3 a67a24a65a24a66dydx =y + px2 +y2x :3a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20a28 a194a24
16、a196a24a202a24a203a24a204a81dydx =yx + r1 +yx2; (2.6)a88a38 a198 x 0a112a213 1,a198 x 0 a112a24a55a129a65a24a66dydx +ay = f(x)a50a1a122 a103 a67a24a198 x! +1a112a1a123a24a33 ba;a247 a198 a 0a112a24a55 y = 0a206a107a1a137a1a138a1a139a65a24a66a24a50a24a67a24a69a226 a2271. a67a24a61a24a228a24a65a24a66a
17、24a10110a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20(1) dydx = x4+y3xy2 ;(2) dydx = x3y3 xy;(3) dydx = y(y+2x)x2 1 .2.a114a62a1a63a24a61a24a62 a56a1a140 a54a6a33a30a220 xa141a98a24a99a24a50a24a47a24a48 y(x):(1) y(x) = x+ R x0 y(t)dt;(2) R x0 (x t)2y(t)dt
18、= x3;(3) y(x) +x2 = 2 R x0 ty(t)dt;(4) R x0 (x t 1)y(t)dt = x;(5) R x0 y(t)dt+ R x0 (x t)(2ty(t) +y2(t)dt = x.3. a20 f(x)a33 R1 a98a24a99a24a55 a6 f(x) 6 0; f(x+ 1) f(x). a20a127a48 a6= 0.(1) a142a1a143 a65a24a66dydx +ay = f(x):a113 a120 a101a129a65a24a66 a103a6a245 a103 a43 a220 1-a144a1a145a67a24a
19、108(2) a142a1a143a137a1a138a1a139a65a24a66dydx +ay = f(x)y2:a113 a120 a101a129a65a24a66 a103a6a245 a103 a43 a220a201a225a50 1-a144a1a145a67a24a692.6 a146a3a147a72a76a72a77a43 a220a204a118a24a35a24a37 a78 a54a24a50a24a35a49a37a24a65a49a66M(x;y)dx+N(x;y)dy = 0; (2.19)a79a24a102 a17a24a34a196a35a24a47a
20、24a48 U(x;y),a104a24a105dU(x;y) = M(x;y)dx+N(x;y)dy;a148a24a238Ux(x;y) = M(x;y); Uy(x;y) = N(x;y); (2.20)a106a80 a81a150a149a152a151a25 a26 ,a18a154a153a152a155a86a25 a26 .a34 a100a152a156a152a26 a78 a61a24a55 a65a32a66 (2.19)a196 a202 a204a118 dU(x;y) 0. a33a107U(x;y) Ca132a107a65a24a66 (2.19)a50a2
21、4a134a24a54a24a135a24a67a24a55a129a208a1a157 Ca107a24a125a24a126a24a127a48a24a69a61a12a66a49a50a49a52a12a112a12a158a160a159a89a161a12a162 a78a151a79 (2.19) a50a49a65a49a66a107a51a107a12a163a198a49a65a151a66a49a50a151a43 a220a12a164a106a55a202a134a165a198 (2.19)a107a1a163a198a24a65a24a66a24a112a24a93
22、a24a94a120a47a49a48 U(x;y)a50a24a43a1a156a114a68a24a6911a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20a73a1a74 2.3a75a24a187a24a173 M(x;y) a166 N(x;y) a167a1a82a1a168a1a169a1a170 G : a 0a90 y 0:12a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a
23、17a16a14a18a14a19a14a5a14a20a198 a45a213 (x0;y0) = (0; 1),a128a1a50a24a53a24a54 (2.22)a103U =Z x0syds+Z y11tdt =x22 y + ln( y); y 0a90 y 0;U =Z x0( s)ds+Z y1x22 +1tdt = x22 y + ln( y); y 0:2. a58 (2.20)a50a1a202a24a43a24a54 Ux = M a159a102a203a24a55 a103U(x;y) ZM(x;y)dx+(y) = x22 y +(y); (2.25)a88a3
24、8 (y)a204a52a24a69 a198a139a195 (2.20) a202a1a7a24a54 Uy = N,a103x22 +0(y) = x22 +1y;a2380(y) = 1y;a58 a247(y) = lnjyj;a100a1a191a1a205a1a206a24a119a24a36a24a37a127a48a24a55a151a207a122a81a49a245a1a207a114a159a41a43 a220 U(x;y)a238a196a69a113a194a128a1a50 (2.25)a250a105U = x22 y + lnjyj:a206a24a196a
25、24a202a58 (2.20)a50a1a202a1a7a24a54 Uy = Na159a102a203a24a101U(x;y) ZN(x;y)dy + (x) = x22 y + lnjyj+ (x);a88a38 (x)a204a52a24a69 a198a139a195 (2.20)a50a1a202a24a43a24a54 Ux = M,a103xy + 0(x) = xy;13a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20a2380(x) = 0;
26、a221a1a60a196a24a213 (x) = 0,a33a107a206a105a24a133a97a1a66a24a50a211a1a53a54 U(x;y).3. a58 (2.24) a159a102a203a1a208 a16a11a1a199a202a114a159a120a47a49a48 U(x;y). a81a24a208a24a55a237a79a196a252 (2.24)a50a209a24a240a203a24a204a118xydx+x22 +1ydy =xydx+ x22 dy+ 1ydy = dx22 y + lnjyj:a58 a247a1a210 a1
27、59U x22 y + lnjyj:a211 a124a24a55a129a51a1a72a24a135a24a223a1a212a24a43 a220a196a1a197a49a235a49a196a114a159a122a65a24a66 (2.24)a50a24a134a24a54a24a135a24a67a24a101U x22 y + lnjyj = C: a226 a2271. a67a24a61a24a228a24a65a24a66a24a101(1) dydx = 2xy+3x2y2x2+2x3y ;(2) dydx = 2x(1 yex2)ex2 ;(3) dydx = 1+exye xy (xy 1);(4) dydx = 2x cos2 y1+x2 sin 2y.14
