1 I p2/30
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- さみ いんそん
- 5 years ago
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Transcription
1 I I p1/30
2 1 I p2/30
3 1 ( ) I p3/30
4 1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) I p4/30
5 1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) g(y) = f()d I p4/30
6 1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) g(y) = f()d G(y) = g(y) (g(y), y = y() (1) I p4/30
7 1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) g(y) = f()d G(y) = g(y) (g(y), y = y() (1), d d {G(y())} = dg d = g(y()) () = f() d I p4/30
8 1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) g(y) = f()d G(y) = g(y) (g(y), y = y() (1), d d {G(y())} = dg d = g(y()) () = f() d G(y) = f()d g(y) = f()d I p4/30
9 1 ( ) [ ] y = «y I p5/30
10 1 ( ) [ ] y = «y [ ] y = «d I p5/30
11 1 ( ) [ ] y = «y [ ] y = «d log y = + 2log + c (c ) I p5/30
12 1 ( ) [ ] y = «y [ ] y = «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C I p5/30
13 1 ( ) [ ] y = «y [ ] y = «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C y = C 2 e (C 0 ) I p5/30
14 1 ( ) [ ] y = «y [ ] y = «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C y = C 2 e (C 0 ) y 0 (y 0), C = 0 I p5/30
15 1 ( ) [ ] y = «y [ ] y = «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C y = C 2 e (C 0 ) y 0 (y 0), C = 0 y = C 2 e (C ) I p5/30
16 1 ( ) [ ] 2 y = ( 1)y I p6/30
17 1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d I p6/30
18 1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d 1, y = 1 «2 d y = Ce 1 (C 0) I p6/30
19 1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d 1, y = 1 «2 d y = Ce 1 (C 0) y 0, C = 0 I p6/30
20 1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d 1, y = 1 «2 d y = Ce 1 (C 0) y 0, C = 0 y = Ce 1 (C ) I p6/30
21 2 I p7/30
22 2 (u = y ), y = y() d = f y (2) I p8/30
23 2 (u = y ), y = y() d = f y (2), u() = y() y() = u(), I p8/30
24 2 (u = y ), y = y() d = f y (2), u() = y() y() = u(),, d = du d + u I p8/30
25 2 (u = y ), y = y() d = f y (2), u() = y() y() = u(),, d = du d + u du d + u = f(u) du d = f(u) u u I p8/30
26 2 (u = y ), y = y() d = f y (2), u() = y() y() = u(),, d = du d + u du d + u = f(u) du d = f(u) u u u = u() y() = u() I p8/30
27 2 (u = y ) [ ] 2yy = 2 + y 2 I p9/30
28 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y 2 I p9/30
29 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u 2 I p9/30
30 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2 I p9/30
31 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2u d u 2 1 du = log u 2 1 = log + c (c ) 2 I p9/30
32 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2u d u 2 1 du = log u 2 1 = log + c (c ) (u 2 1) = C (C 0 ) 2 I p9/30
33 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2u d u 2 1 du = log u 2 1 = log + c (c ) (u 2 1) = C (C 0 ) u() = y y2 2 = C 2 I p9/30
34 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2u d u 2 1 du = log u 2 1 = log + c (c ) (u 2 1) = C (C 0 ) u() = y y2 2 = C y = ±, C = 0 2 I p9/30
35 2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u du d + u = 1 + u2 du 2u d = 1 u2 2u u 2u d u 2 1 du = log u 2 1 = log + c (c ) (u 2 1) = C (C 0 ) u() = y y2 2 = C y = ±, C = 0 y 2 2 = C 2 (C ) I p9/30
36 2 (u = y ) [ ] ( 2y)y = 2 y I p10/30
37 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y I p10/30
38 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u I p10/30
39 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u I p10/30
40 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u 1 2u d 1 u + u 2 du = 2 log 1 u + u 2 = 2 log + c (c ) I p10/30
41 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u 1 2u d 1 u + u 2 du = 2 log 1 u + u 2 = 2 log + c (c ) (u 2 u + 1) 2 = C (C 0 ) I p10/30
42 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u 1 2u d 1 u + u 2 du = 2 log 1 u + u 2 = 2 log + c (c ) (u 2 u + 1) 2 = C (C 0 ) u() = y 2 y + y 2 = C I p10/30
43 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u 1 2u d 1 u + u 2 du = 2 log 1 u + u 2 = 2 log + c (c ) (u 2 u + 1) 2 = C (C 0 ) u() = y 2 y + y 2 = C 2 y + y 2 = 0, C = 0 I p10/30
44 2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y u() = y() y() = u(), d = du d + u du d + u = 2 u du u + u2 = u d (1 2u) u 1 2u d 1 u + u 2 du = 2 log 1 u + u 2 = 2 log + c (c ) (u 2 u + 1) 2 = C (C 0 ) u() = y 2 y + y 2 = C 2 y + y 2 = 0, C = 0 2 y + y 2 = C (C ) I p10/30
45 3 I p11/30
46 3 y y = y(), d = f(), I p12/30
47 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 I p12/30
48 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3) I p12/30
49 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) I p12/30
50 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y I p12/30
51 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u(, y()) = C, I p12/30
52 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u u(, y()) = C, + u = M(, y) + N(, y) y d d = 0 I p12/30
53 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u u(, y()) = C, + u = M(, y) + N(, y) y d d = 0 u(, y) = C y, u((y), y) = C y I p12/30
54 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u u(, y()) = C, + u = M(, y) + N(, y) y d d = 0 u(, y) = C y, u((y), y) = C y u((y), y) = C y, I p12/30
55 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u u(, y()) = C, + u = M(, y) + N(, y) y d d = 0 u(, y) = C y, u((y), y) = C y u d u((y), y) = C y, + u y = M(, y)d + N(, y) = 0 I p12/30
56 3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 M(, y)d + N(, y) = 0 (3), y u(, y), M(, y) = u u, N(, y) = y, u(, y) = C (3) u(, y) = C, u(, y()) = C y u u(, y()) = C, + u = M(, y) + N(, y) y d d = 0 u(, y) = C y, u((y), y) = C y u d u((y), y) = C y, + u y, u(, y) = C (3) = M(, y)d + N(, y) = 0 I p12/30
57 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, I p13/30
58 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y M(, y) N(, y) = y (u ) = 2 u y, ( ) I p13/30
59 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, M(, y) N(, y) = ( ) y (u ) ( ), (3) I p13/30
60 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, M(, y) N(, y) = ( ) y (u ) ( ), (3) [u ] I p13/30
61 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, M(, y) N(, y) = ( ) y (u ) ( ), (3) [u ] u = M(, y) u(, y) = M(, y)d + k(y), (k(y) y ) I p13/30
62 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, M(, y) N(, y) = ( ) y (u ) ( ), (3) [u ] u = M(, y) u(, y) = M(, y)d + k(y), (k(y) y ) y u y = y Md + dk k(y) u(, y) = N(, y) I p13/30
63 3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, M(, y) N(, y) = ( ) y (u ) ( ), (3) [u ] u = M(, y) u(, y) = M(, y)d + k(y), y u y = y Md + dk k(y) u(, y) [ ] u(, y) = (k(y) y ) = N(, y) N(, y) + l(), u = M(, y) l() I p13/30
64 3 [ ] 2sin 3yd cos 3y = 0 I p14/30
65 3 [ ] 2sin 3yd cos 3y = 0 [ ] y (2 sin3y) = 6 cos3y = (32 cos3y) I p14/30
66 3 [ ] 2sin 3yd cos 3y = 0 [ ] (2 sin3y) = 6 cos3y = y (32 cos3y) u(, y) = 2 sin3y d + k(y) = 2 sin 3y + k(y) I p14/30
67 3 [ ] 2sin 3yd cos 3y = 0 [ ] (2 sin3y) = 6 cos3y = y (32 cos3y) u(, y) = 2 sin3y d + k(y) = 2 sin 3y + k(y) y `2 sin3y + k(y) = 3 2 cos3y + dk y = 32 cos3y I p14/30
68 3 [ ] 2sin 3yd cos 3y = 0 [ ] (2 sin3y) = 6 cos3y = y (32 cos3y) u(, y) = 2 sin3y d + k(y) = 2 sin 3y + k(y) y `2 sin3y + k(y) = 3 2 cos3y + dk y = 32 cos3y dk = 0 I p14/30
69 3 [ ] 2sin 3yd cos 3y = 0 [ ] (2 sin3y) = 6 cos3y = y (32 cos3y) u(, y) = 2 sin3y d + k(y) = 2 sin 3y + k(y) y `2 sin3y + k(y) = 3 2 cos3y + dk y = 32 cos3y dk = 0 k = C ( ) I p14/30
70 3 [ ] 2sin 3yd cos 3y = 0 [ ] (2 sin3y) = 6 cos3y = y (32 cos3y) u(, y) = 2 sin3y d + k(y) = 2 sin 3y + k(y) y `2 sin3y + k(y) = 3 2 cos3y + dk y = 32 cos3y dk = 0 k = C ( ) 2 sin3y = C I p14/30
71 3 [ ] 2 + y 2 = C I p15/30
72 3 [ ] [ ] 2 + y 2 = C u(, y) = 2 + y 2 I p15/30
73 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y I p15/30
74 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) I p15/30
75 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 I p15/30
76 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), I p15/30
77 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), u(, y) = y d + k(y) = y + k(y) I p15/30
78 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), u(, y) = y d + k(y) = y + k(y) y dk (y + k(y)) = + y = I p15/30
79 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), u(, y) = y d + k(y) = y + k(y) y dk = 0 y (y + k(y)) = + dk = I p15/30
80 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), u(, y) = y d + k(y) = y + k(y) y y dk = 0 k = C ( ) (y + k(y)) = + dk = I p15/30
81 3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y 2d + 2y = 0 ( 2 d + y = 0 ) [ ] yd + = 0 [ ] y (y) = 1 = (), u(, y) = y d + k(y) = y + k(y) y (y + k(y)) = + dk = y dk = 0 k = C ( ) y = C I p15/30
82 4 I p16/30
83 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 I p17/30
84 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), I p17/30
85 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), [ ] ( ) ( ) I p17/30
86 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), [ ] ( ) ( ) [ ] F(, y) = 1 2 yd + = 0, I p17/30
87 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), [ ] ( ) ( ) [ ] F(, y) = 1 2 yd + = 0, [ ] F(, y) y 2 d + 1 = 0, I p17/30
88 4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), [ ] ( ) ( ) [ ] F(, y) = 1 2 yd + = 0, [ ] F(, y) y 2 d + 1 = 0 y, = C I p17/30
89 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, I p18/30
90 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y), y 2 + y 2 d y 2 = 0 I p18/30
91 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y), y y 2 + y 2 y 2 + y 2 d + «= y2 2 ( 2 + y 2 ) 2 = 2 + y 2 = y 2 «I p18/30
92 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y), y y 2 + y 2 y u(, y) = 2 d + k(y) + y2 y 2 + y 2 d + «= y2 2 ( 2 + y 2 ) 2 = 2 + y 2 = y 2 «I p18/30
93 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) 1 + y I p18/30
94 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) = tan y + k(y) y I p18/30
95 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) = tan y + k(y) y y tan 1 y «y + k(y) = y 2 y 2 + dk = 2 + y 2 I p18/30
96 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) = tan y + k(y) y y tan 1 y «y + k(y) = dk = y 2 y 2 + dk = 2 + y 2 I p18/30
97 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) = tan y + k(y) y y tan 1 y «y + k(y) = dk = 0 k = C ( ) y 2 y 2 + dk = 2 + y 2 I p18/30
98 4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d y 2 = 0, y «y 2 + y 2 = y2 2 ( 2 + y 2 ) 2 = «2 + y 2 y u(, y) = 2 + y 2 d + k(y) = 1 1 y 2 d + k(y) = tan y + k(y) y y tan 1 y «y + k(y) = y 2 y 2 + dk = dk = 0 k = C ( ) tan 1 y = C 2 + y 2 I p18/30
99 4 [ ], F I p19/30
100 4 [ ], F (i) F = F() ( ), F = F(y) (y ) I p19/30
101 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n I p19/30
102 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) I p19/30
103 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, I p19/30
104 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), I p19/30
105 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), F(, y) = m y n, I p19/30
106 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), F(, y) = m y n, y (m+1 y n+1 + m y n+2 ) = (m+2 y n m+1 y n+1 ) I p19/30
107 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), F(, y) = m y n, y (m+1 y n+1 + m y n+2 ) = (m+2 y n m+1 y n+1 ) m = 2, n = 1 I p19/30
108 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), F(, y) = m y n, y (m+1 y n+1 + m y n+2 ) = (m+2 y n m+1 y n+1 ) m = 2, n = 1 F(, y) = 1 2 y 1 + y «1 2 d + y 1 «= 0 I p19/30
109 4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n (iii) F = F( + y), F = F( y), F = F(y) [ ] (y + y 2 )d + ( 2 y) = 0, [ ] y (y + y2 ) = + 2y 2 y = (2 y), F(, y) = m y n, y (m+1 y n+1 + m y n+2 ) = (m+2 y n m+1 y n+1 ) m = 2, n = 1 F(, y) = 1 2 y 1 + y «1 2 d + y 1 «= 0, y = Ce y (C 0) I p19/30
110 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), I p20/30
111 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), I p20/30
112 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), I p20/30
113 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), F(4 2 y 3 2y) = F(12 2 y 2 2) = F (3 3 y 2 ) + F(9 2 y 2 1) = F(3 3 y 2 ) y I p20/30
114 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), F(4 2 y 3 2y) = F(12 2 y 2 2) = F (3 3 y 2 ) + F(9 2 y 2 1) = F(3 3 y 2 ) y df d = 1 F I p20/30
115 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), F(4 2 y 3 2y) = F(12 2 y 2 2) = F (3 3 y 2 ) + F(9 2 y 2 1) = F(3 3 y 2 ) y df d = 1 F df F = d I p20/30
116 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), F(4 2 y 3 2y) = F(12 2 y 2 2) = F (3 3 y 2 ) + F(9 2 y 2 1) = F(3 3 y 2 ) y df d = 1 df d F F = F() = (4 3 y 3 2y)d + (3 4 y 2 2 ) = 0 I p20/30
117 4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y y 2 1 = (33 y 2 ), F = F(), F(4 2 y 3 2y) = F(12 2 y 2 2) = F (3 3 y 2 ) + F(9 2 y 2 1) = F(3 3 y 2 ) y df d = 1 df d F F = F() = (4 3 y 3 2y)d + (3 4 y 2 2 ) = 0, 4 y 3 2 y = C I p20/30
118 5 ( ) I p21/30
119 5 ( ), y = y() + p()y = q() (4) d I p22/30
120 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ) I p22/30
121 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) I p22/30
122 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ) I p22/30
123 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d}, C = C() (4) (C ) I p22/30
124 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} I p22/30
125 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} I p22/30
126 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} = dc d ep{ p()d} = q() I p22/30
127 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} = dc d ep{ p()d} = q() dc = q() ep{ p()d} d I p22/30
128 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} = dc d ep{ p()d} = q() dc = q() ep{ p()d} C() = q() ep{ p()d}d, d I p22/30
129 5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) ( ) y = p()d y = C ep{ p()d} (C ), C = C() (4) dc + p()y = d d ep{ p()d} Cp() ep{ p()d} + p()c ep{ p()d} = dc d ep{ p()d} = q() dc = q() ep{ p()d} C() = q() ep{ p()d}d, (4) d «y = q() ep{ p()d}d ep{ p()d} I p22/30
130 5 ( ) [ ] y y = 4 (5) I p23/30
131 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 I p23/30
132 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 log y = log( 2 + 1) + c y = C I p23/30
133 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 log y = log( 2 + 1) + c y = C y = C() (5) I p23/30
134 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 log y = log( 2 + 1) + c y = C y = C() (5) C = 4 ie C = 4( 2 + 1) I p23/30
135 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 log y = log( 2 + 1) + c y = C y = C() (5) C = 4 ie C = 4( 2 + 1) C() = c (c ) I p23/30
136 5 ( ) [ ] y y = 4 (5) [ ] d y = 0 log y = log( 2 + 1) + c y = C y = C() (5) C = 4 ie C = 4( 2 + 1) C() = c (c ) y = c (c ) I p23/30
137 5 ( ) [ ] y 1 y = cos (6) I p24/30
138 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 I p24/30
139 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C I p24/30
140 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C y = C() (6) I p24/30
141 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C y = C() (6) C = cos ie C = cos I p24/30
142 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C y = C() (6) C = cos ie C = cos C() = sin + c (c ) I p24/30
143 5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C y = C() (6) C = cos ie C = cos C() = sin + c (c ) y = sin + c (c ) I p24/30
144 6 I p25/30
145 6 [ ] I p26/30
146 6 [ ] y = f(a + by + c) u = a + by + c I p26/30
147 6 [ ] y = f(a + by + c) u = a + by + c [ ] y = ( + y) 2 I p26/30
148 6 [ ] y = f(a + by + c) u = a + by + c [ ] y = ( + y) 2 [ ] y = tan( + c) (c ) I p26/30
149 6 [ ] I p27/30
150 6 [ ] y = y + f(y ) I p27/30
151 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 I p27/30
152 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) I p27/30
153 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) y = C + f(c) ( ) I p27/30
154 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) y = C + f(c) ( ) + f (p) = 0, p I p27/30
155 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) y = C + f(c) ( ) + f (p) = 0, p = f (p) y = p + f(p) ( ) I p27/30
156 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) y = C + f(c) ( ) + f (p) = 0, p [ ] = f (p) y = p + f(p) ( ) y = y (y ) 2 I p27/30
157 6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 p = 0, p = C ( ) y = C + f(c) ( ) + f (p) = 0, p [ ] = f (p) y = p + f(p) ( ) y = y (y ) 2 [ ] y = C C2 y = I p27/30
158 6 [ ] I p28/30
159 6 [ ] y + p()y = q()y α (α 0,1) u = y (1 α), (1 α)y α (1 α)y α y + (1 α)p()y (1 α) = (1 α)q(), : I p28/30
160 6 [ ] y + p()y = q()y α (α 0,1) u = y (1 α), (1 α)y α (1 α)y α y + (1 α)p()y (1 α) = (1 α)q(), : u + (1 α)p()u = (1 α)q() I p28/30
161 6 [ ] y + p()y = q()y α (α 0,1) u = y (1 α), (1 α)y α (1 α)y α y + (1 α)p()y (1 α) = (1 α)q(), : u + (1 α)p()u = (1 α)q() [ ] y + y = 4( + 1) y I p28/30
162 6 [ ] y + p()y = q()y α (α 0,1) u = y (1 α), (1 α)y α (1 α)y α y + (1 α)p()y (1 α) = (1 α)q(), : [ ] y + y = u + (1 α)p()u = (1 α)q() 4( + 1) y [ ] y 2 = ce (c ) I p28/30
163 6 [ ] I p29/30
164 6 [ ] y = p()y 2 + q()y + r() I p29/30
165 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u I p29/30
166 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u I p29/30
167 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u u 2 y 1()u 2 u = p()(y 1 ()u + 1) 2 + q() `y 1 ()u 2 + u + r()u 2 I p29/30
168 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u u 2 y 1()u 2 u = p()(y 1 ()u + 1) 2 + q() `y 1 ()u 2 + u + r()u 2 u u + (2p()y 1 () + q()) u = p() I p29/30
169 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u u 2 y 1()u 2 u = p()(y 1 ()u + 1) 2 + q() `y 1 ()u 2 + u + r()u 2 u u + (2p()y 1 () + q()) u = p() u y = y 1 () + 1 u() I p29/30
170 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u u 2 y 1()u 2 u = p()(y 1 ()u + 1) 2 + q() `y 1 ()u 2 + u + r()u 2 u u + (2p()y 1 () + q()) u = p() u y = y 1 () + 1 u() [ ] y = y 2 (2 1)y + 1 I p29/30
171 6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u y 1() u u 2 = p() y 1 () + 1 «2 + q() y 1 () + 1 «+ r() u u u 2 y 1()u 2 u = p()(y 1 ()u + 1) 2 + q() `y 1 ()u 2 + u + r()u 2 u u + (2p()y 1 () + q()) u = p() u y = y 1 () + 1 u() [ ] y = y 2 (2 1)y + 1 [ ] y = 1 y = Ce (C ) I p29/30
172 6 [ ] I p30/30
173 6 [ ] y, F(, y, y ) = 0, y = p p I p30/30
174 6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 I p30/30
175 6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 [ ] y = C 1 log + C 2 (C 1, C 2 ) I p30/30
176 6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 [ ] y = C 1 log + C 2 (C 1, C 2 ) y, F(y, y, y ) = 0, y = p y = dp p, y p I p30/30
177 6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 [ ] y = C 1 log + C 2 (C 1, C 2 ) y, F(y, y, y ) = 0, y = p y = dp p, y p [ ] y + (y ) 2 = 0 I p30/30
178 6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 [ ] y = C 1 log + C 2 (C 1, C 2 ) y, F(y, y, y ) = 0, y = p y = dp p, y p [ ] y + (y ) 2 = 0 [ ] y = log(c 1 + C 2 ) (C 1, C 2 ) I p30/30
1 1 x y = y(x) y, y,..., y (n) : n y F (x, y, y,..., y (n) ) = 0 n F (x, y, y ) = 0 1 y(x) y y = G(x, y) y, y y + p(x)y = q(x) 1 p(x) q(
1 1 y = y() y, y,..., y (n) : n y F (, y, y,..., y (n) ) = 0 n F (, y, y ) = 0 1 y() 1.1 1 y y = G(, y) 1.1.1 1 y, y y + p()y = q() 1 p() q() (q() = 0) y + p()y = 0 y y + py = 0 y y = p (log y) = p log
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![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
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3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4
Chap9.dvi
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3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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no35.dvi
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
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dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp
+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy
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