1 I p2/30

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Transcription:

I I p1/30

1 I p2/30

1 ( ) I p3/30

1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) I p4/30

1 ( ), y = y() d = f() g(y) ( g(y) = f()d) (1) g(y) = f()d I p4/30

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

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

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

1 ( ) [ ] y = 1 + 2 «y I p5/30

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «d I p5/30

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «d log y = + 2log + c (c ) I p5/30

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C I p5/30

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «d log y = + 2log + c (c ) y = e c 2 e, ±e c = C y = C 2 e (C 0 ) I p5/30

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «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

1 ( ) [ ] y = 1 + 2 «y [ ] y = 1 + 2 «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

1 ( ) [ ] 2 y = ( 1)y I p6/30

1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d I p6/30

1 ( ) [ ] 2 y = ( 1)y 1 [ ] y = 1 «2 d 1, y = 1 «2 d y = Ce 1 (C 0) I p6/30

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

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

2 I p7/30

2 (u = y ), y = y() d = f y (2) I p8/30

2 (u = y ), y = y() d = f y (2), u() = y() y() = u(), I p8/30

2 (u = y ), y = y() d = f y (2), u() = y() y() = u(),, d = du d + u I p8/30

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

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

2 (u = y ) [ ] 2yy = 2 + y 2 I p9/30

2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y 2 I p9/30

2 (u = y ) [ ] 2yy = 2 + y 2 [ ] d = 2 + y 2 2y = 1 + ` y 2 ` y u() = y() y() = u(), d = du d + u 2 I p9/30

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

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

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

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

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

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

2 (u = y ) [ ] ( 2y)y = 2 y I p10/30

2 (u = y ) [ ] ( 2y)y = 2 y [ ] d = 2 y 2y = 2 ` y 1 2 ` y I p10/30

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

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 = 21 1 2u d (1 2u) u I p10/30

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 = 21 1 2u 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

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 = 21 1 2u 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

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 = 21 1 2u 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

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 = 21 1 2u 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

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 = 21 1 2u 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

3 I p11/30

3 y y = y(), d = f(), I p12/30

3 y y = y(), d = f(), (, y) M(, y) + N(, y) d = 0 M(, y)d, + N(, y) = 0 I p12/30

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

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

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

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

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

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

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

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

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

3 [ ] M(, y) = u u, N(, y) = y, M(, y) y = 2 u N(, y), y = 2 u y, I p13/30

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

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

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

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

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

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

3 [ ] 2sin 3yd + 3 2 cos 3y = 0 I p14/30

3 [ ] 2sin 3yd + 3 2 cos 3y = 0 [ ] y (2 sin3y) = 6 cos3y = (32 cos3y) I p14/30

3 [ ] 2sin 3yd + 3 2 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

3 [ ] 2sin 3yd + 3 2 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

3 [ ] 2sin 3yd + 3 2 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

3 [ ] 2sin 3yd + 3 2 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

3 [ ] 2sin 3yd + 3 2 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

3 [ ] 2 + y 2 = C I p15/30

3 [ ] [ ] 2 + y 2 = C u(, y) = 2 + y 2 I p15/30

3 [ ] 2 + y 2 = C [ ] u(, y) = 2 + y 2 u u d + = 2d + 2y y I p15/30

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

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

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

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

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

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

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

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

4 I p16/30

4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 I p17/30

4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), I p17/30

4 P(, y)d + Q(, y) = 0, F(, y), F(, y)p(, y)d + F(, y)q(, y) = 0 F(, y), [ ] ( ) ( ) I p17/30

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

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

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

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y), y 2 + y 2 d + 2 + y 2 = 0 I p18/30

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 = 0 2 + y 2 «I p18/30

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 = 0 2 + y 2 «I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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 1 1 + y + k(y) y I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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 1 1 + y + k(y) y y tan 1 y «y + k(y) = 1 1 + y 2 y 2 + dk = 2 + y 2 I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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 1 1 + y + k(y) y y tan 1 y «y + k(y) = dk = 0 1 1 + y 2 y 2 + dk = 2 + y 2 I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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 1 1 + y + k(y) y y tan 1 y «y + k(y) = dk = 0 k = C ( ) 1 1 + y 2 y 2 + dk = 2 + y 2 I p18/30

4 [ ] F(, y) = 1 2 +y 2 yd + = 0, [ ] F(, y) y 2 + y 2 d + 2 + 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 1 1 + y + k(y) y y tan 1 y «y + k(y) = 1 1 + y 2 y 2 + dk = dk = 0 k = C ( ) tan 1 y = C 2 + y 2 I p18/30

4 [ ], F I p19/30

4 [ ], F (i) F = F() ( ), F = F(y) (y ) I p19/30

4 [ ], F (i) F = F() ( ), F = F(y) (y ) (ii) F = m y n I p19/30

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

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

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

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

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

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

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

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

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), I p20/30

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 y 2 1 = (33 y 2 ), I p20/30

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 y 2 1 = (33 y 2 ), F = F(), I p20/30

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 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

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 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

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 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

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 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

4 [ ] (4 2 y 3 2y)d + (3 3 y 2 ) = 0 F = F(), [ ] y (42 y 3 2y) = 12 2 y 2 2 9 2 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

5 ( ) I p21/30

5 ( ), y = y() + p()y = q() (4) d I p22/30

5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ) I p22/30

5 ( ), y = y() + p()y = q() (4) d, d + p()y = 0 ( ), (4) I p22/30

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

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

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

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

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

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

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

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

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 log y = log( 2 + 1) + c y = C 2 + 1 I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 log y = log( 2 + 1) + c y = C 2 + 1 y = C() 2 + 1 (5) I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 log y = log( 2 + 1) + c y = C 2 + 1 y = C() 2 + 1 (5) C 2 + 1 = 4 ie C = 4( 2 + 1) I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 log y = log( 2 + 1) + c y = C 2 + 1 y = C() 2 + 1 (5) C 2 + 1 = 4 ie C = 4( 2 + 1) C() = 4 + 2 2 + c (c ) I p23/30

5 ( ) [ ] y + 2 2 + 1 y = 4 (5) [ ] d + 2 2 + 1 y = 0 log y = log( 2 + 1) + c y = C 2 + 1 y = C() 2 + 1 (5) C 2 + 1 = 4 ie C = 4( 2 + 1) C() = 4 + 2 2 + c (c ) y = 4 + 2 2 + c 2 + 1 (c ) I p23/30

5 ( ) [ ] y 1 y = cos (6) I p24/30

5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 I p24/30

5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C I p24/30

5 ( ) [ ] y 1 y = cos (6) [ ] d 1 y = 0 log y = log + c y = C y = C() (6) I p24/30

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

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

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

6 I p25/30

6 [ ] I p26/30

6 [ ] y = f(a + by + c) u = a + by + c I p26/30

6 [ ] y = f(a + by + c) u = a + by + c [ ] y = ( + y) 2 I p26/30

6 [ ] y = f(a + by + c) u = a + by + c [ ] y = ( + y) 2 [ ] y = tan( + c) (c ) I p26/30

6 [ ] I p27/30

6 [ ] y = y + f(y ) I p27/30

6 [ ] y = y + f(y ), p = y y = p + f(p) y = p + p + f (p)p ( + f (p))p = 0 I p27/30

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

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

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

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

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 + 1 2 (y ) 2 I p27/30

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 + 1 2 (y ) 2 [ ] y = C + 1 2 C2 y = 1 2 2 I p27/30

6 [ ] I p28/30

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

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

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

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 2 + 4 2 (c ) I p28/30

6 [ ] I p29/30

6 [ ] y = p()y 2 + q()y + r() I p29/30

6 [ ] y = p()y 2 + q()y + r() y = y 1 () y = y 1 () + 1 u I p29/30

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

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

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

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

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

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 = 1 + 1 1 + Ce (C ) I p29/30

6 [ ] I p30/30

6 [ ] y, F(, y, y ) = 0, y = p p I p30/30

6 [ ] y, F(, y, y ) = 0, y = p p [ ] y + y = 0 I p30/30

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

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

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

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

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