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i Leibniz ydy = y 2 /2 1675 11 11 [6] 100 Bernoulli Riccati 19 Fuchs 2 1-5 1-7 4 18 x [5] 3 2
ii y y + f (y) = 0, f (y) = 9 100 (y y 5/3 ) t x = 5 4 log(t4 6t 2 + 4C 1 t 3) + C 2 y = (t 3 3t + C 1 ) 3/2 (t 4 6t 2 + 4C 1 t 3) 9/8 C 1, C 2 [5] 4 1-1 d 2 y dx 2 + ax2 y = 0, a > 0 (1-1-24) y = x [ ( ) ( )] a a AJ 1/4 2 x2 + BY 1/4 2 x2 A, B (1-1-25) 1/4 1-1 (1-1-24) x + x + global analysis local analysis
iii 1 2 1-1 1-2 2 A 5-6 3 4 5 1 4 2-3 5-3 5-5 2010 3
iv 1 1 1-1... 2 1-2......... 8 1-3... 13 1-4................................ 18 1-4-1.... 18 1-4-2 I.... 20 1-4-3 II... 23 1-5.... 25 1-6 1 28 1-7.................... 37 1................... 42 2 43 2-1............................. 44 2-2.................... 51 2-3... 53 2-4................... 63 2-5........................... 75 2................... 79 3 81 3-1... 81 3-2... 83 3-3...................... 88
v 3-4.................... 93 3-5 I... 101 3-6 II... 111 3-6-1... 111 3-6-2................................ 113 3................................. 116 4 118 4-1................................ 119 4-2............................. 121 4-3 p-................................... 124 4-4 c-................................... 128 4-5.................... 130 4-6............................. 136 4................................. 139 5 140 5-1... 140 5-1-1 1 1.................... 140 5-1-2 1.................... 143 5-1-3...................... 145 5-2... 147 5-3................... 148 5-4 1 x =+... 153 5-5............... 157 5-6............................. 159 5................................. 166 167 187 189 190
1 1 2 4 1-1 [5] 2 1-2 1-3 x + x 1 1-4 1-5 Painlevé 1-7
2 1 1-1 f (x) g(x) f (x) g(x) lim x x 0 f (x) g(x) = 0 (1-1-1) x x 0 x x 0 f (x) g(x) g(x) f (x) f (x) g(x) x x 0 (1-1-2a) g(x) f (x) x x 0 (1-1-2b) Landauo f (x) = o(g(x)) x x 0 (1-1-3) f (x) g(x) f (x) g(x) lim x x 0 f (x) g(x) = 1 (1-1-4) f (x) g(x) f (x) x x 0 (1-1-5) x x 0 f (x) g(x) f (x) g(x) x x 0 (1-1-6a) g(x) f (x) x x 0 (1-1-6b) *1 f (x) g(x) x = x 0 x K 1 o O
1-1 3 f (x) g(x) K (1-1-7) x x 0 f (x) g(x) O f (x) = O(g(x)) x x 0 (1-1-8) {ϕ m (x)} m ϕ m+1 (x) = o(ϕ m (x)) x x 0 (1-1-9) asymptotic sequences {ϕ m (x)} a m ϕ m (x) k m=0 k y(x) a m ϕ m (x) ϕ k (x) x x 0 (1-1-10) m=0 a m ϕ m (x) y(x) asymptotic series m=0 y(x) a m ϕ m (x) x x 0 (1-1-11) m=0 {(x x 0 ) m } a m (x x 0 ) m y(x) a m = y(m) (x 0 ) m=0 m! k y(x) m=0 a m (x x 0 ) m = y(k+1) (x 0 + θ(x x 0 )) (x x 0 ) k+1 (x x 0 ) k (k + 1)! (0 <θ<1) x x 0 (1-1-12) a m (x x 0 ) m y(x) m=0
4 1 y(x) a m (x x 0 ) m x x 0 (1-1-13) m=0 x + {x m } y(x) a 0 + a 1 x + a 2 x + = a 2 m x m (1-1-14) m=0 leading term leading behavior controlling factor f (x) a m ϕ m (x), g(x) b m ϕ m (x) x x 0 (1-1-15) m=0 m=0 cf(x) + dg(x) (ca m + db m )ϕ m (x) c, d x x 0 (1-1-16) m=0 f (x) a n (x x 0 ) n, g(x) b n (x x 0 ) n n=0 f (x)g(x) h n (x x 0 ) n, n h n = a m b n m x x 0 (1-1-17) n=0 m=0 n=0 x 0 x + {x m } y(x) a 0 + a 1 x + a 2 x + = a 2 m x m (1-1-18) x ( y(x) a 0 a 1 x m=0 ) dt a 2 x + a 3 2x 2 + a 4 3x 3 + = m=2 a m x m+1 (m 1) (1-1-19) f (x) g(x) x x 0 (1-1-6 ),
1-1 5 x x f (t)dt g(t) dt x x 0 (1-1-20) f (x) f (x) f (x) f (x) 4 5 2 3 1-1 3 y = tan x y = x y 1 y = tan x y = x y = sin x 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10º 30º x 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1-1
6 1 y = sin x sin x x, tan x x x 0 (1-1-21) 1-1 x = 0.17 10 3 sin x x, x 0 x 0.52(π/6) 30 4% tan x x, x 0 x = 0.52(π/6) 30 sin x x = x3 6 + O(x5 ) x 0 (1-1-22) tan x x = x3 3 + O(x5 ) x 0 (1-1-23) O(x 3 ) 1/6 1/3 d 2 y dx + 2 ax2 y = 0, a > 0 (1-1-24) y = [ ( ) ( )] a a x AJ 1/4 2 x2 + BY 1/4 2 x2 A, B (1-1-25) x + 1-4 x + (1-1-24) y(x) = exp(q(x)) (1-1-26) [5] 5 exp(q(x)) d 2 [ ] 2 Q(x) dq(x) + + ax 2 = 0 (1-1-27) dx 2 dx (1-1-24) (1-1-27) 1 2
1-1 7 d 2 Q dx 2 ( ) 2 dq (1-1-28) dx 1 2 (1-1-27) 1 (1-1-28) dq(x) ±i ax dx x + (1-1-29) Q(x) ± i a 2 x2 x + (1-1-30) Q(x) = ± i a 2 x2 + R(x), x 2 R(x) x + (1-1-31) (1-1-27) ±i ( a + d2 R dx + ±i ax + dr ) 2 2 dx + ax2 = 0 (1-1-32) [5] 2 (1-1-31) 2x R (x), 2 R (x) x + (1-1-33) R(x) (1-1-32) (1-1-33) (1-1-32) i a 2i ax dr dx x + (1-1-34) R(x) 1 log x x + (1-1-35) 2 R(x) (1-1-33) x + [ y(x) exp ± i a 2 x2 1 ] 2 log x = 1 [ exp ± i ] a x 2 x2 x + (1-1-24) y(x) 1 [ ] ax 2 ax 2 exp A sin + B cos A, B x + (1-1-36) x 2 2
118 4 d 2 r dt = (4-0-1) 2 r = ( x(t),y(t) ) r 0 (x 0,y 0 ) u 0 y y 0 = (x x 0 )tanθ (x x 0) 2 2v 2 0 cos2 θ (4-0-2) r 0 (x 0,y 0 ) v 0 θ (4-0-2)
4-1 119 1000 m 3 2 h = 2 9.8 [m/s 2 ] 1000 [m] 140 [m/s] 500 [km/h] ICBM 2 (4-0-2) (4-0-1) 6400 km d 2 r dt = GM e 2 r (4-0-3) 2 GM r e r 1-7 1 4-1 1-7 Brook Taylor 1
120 4 ( ) ( ) 2 1 + x 2 2 dy = 4y 2 (y 1) (4-1-1) dx [6] 40 1715 y = u 2 (1 + x 2 ) (4-1-2) (4-1-1) ( ) 2 du (1 + x 2 ) 2xu du dx dx + u2 1 = 0 (4-1-3) (4-1-3) 1 d 2 [ u (1 + x 2 ) du ] dx 2 dx xu = 0 (4-1-4) (4-1-4) A d2 u dx 2 = 0 du dx = c c (4-1-5a) (4-1-5b) ( B 1 + x 2) du xu = 0 (4-1-6) dx A (4-1-5b) (4-1-3) u c 2 ( 1 + x 2) 2cxu + u 2 1 = 0 (4-1-7) (4-1-2) (4-1-7) y y = (1 + x 2 ) ( cx ± 1 c 2 ) 2 (4-1-8) 4-1 B (4-1-6) (4-1-3) ( ) 1 + x 2 u 2 = 0 (4-1-9a) (4-1-2) (4-1-9a) y = 1 (4-1-9b)
4-2 121 y 20 15 10 5 y=1 10 8 6 4 2 0 2 4 6 8 x 10 4-1 4-1 (4-1-9b) (4-1-8) c 1-7 II 4-2 1 ( F x,y, dy ) = 0 (4-2-1) dx p dy dx *1 (4-2-1) x,y,p (4-2-2) 1 2 (x,y) z(x,y) p = z(x,y), q = z(x,y) x y, r = 2 z(x,y) x 2, s = 2 z(x,y) x y, t = 2 z(x,y) y 2
122 4 F(x,y,p) = 0 (4-2-3) F(x,y,p) x,y 1 p (x 0,y 0 ) (x,y) p F(x 0,y 0, p) = 0 (4-2-4) p 0 (4-2-3) (x 0,y 0 ) (x,y) = (x 0,y 0 ) p = p 0 p = f (x,y) (4-2-5) f (x,y) (x 0,y 0 ) 1 f (x,y) (x 0,y 0 ) (5-3-2a) dy = f (x,y) (4-2-6) dx (x,y) = (x 0,y 0 ) (4-2-4) p 0 f (x,y) p (x 0,y 0 ) p 0 m ] ] ] F(x 0,y 0, p) = 2 F(x 0,y 0, p) = = m 1 F(x 0,y 0, p) = 0 p p=p 0 p 2 p=p 0 p m 1 p=p 0 m ] F(x 0,y 0, p) p m 0 (4-2-7) p=p 0 (4-2-3) x = x 0 y = y 0 p = p 0 ] ] F(x,y,p) F(x,y,p) + p 0 x x=x 0,y=y 0 p=p 0 y x x=x 0,y=y 0 p=p 0 ( p)m m ] F(x,y,p) m! p m x 0 (4-2-8) x=x 0,y=y 0 p=p 0 x = x x 0, y = y y 0, p = p p 0 (4-2-9) ] ] F(x,y,p) F(x,y,p) + p 0 x x=x 0,y=y 0 p=p 0 y x=x 0,y=y 0 p=p 0 0 (4-2-10)
4-2 123 (4-2-8) p A( x) 1/m A x 0 (4-2-11a) y p 0 x + B( x) 1+1/m B x 0 (4-2-11b) x = x 0 y = y 0 y y 0 + p 0 (x x 0 ) + B(x x 0 ) 1+1/m x 0 (4-2-12) (x 0,y 0 ) (x,y) = (x 0,y 0 ) m 2 (x,y) (x 0,y 0 ) p (4-2-4) 2 (4-2-3) F(x,y,p) = x x 0 + y y 0 C(p p 0 ) 2 = 0 (4-2-13) C (x 0,y 0, p 0 ) (4-2-8) (1 + p 0 ) x C( p) 2 x 0 (4-2-14) p ± (1 + p 0 ) x/c x 0 (4-2-15a) y p 0 x ± 2 3 1 + p0 C ( x)3/2 x 0 (4-2-15b) (4-2-15b) y y 0 + p 0 (x x 0 ) ± 2 1 + p0 3 C (x x 0) 3/2 x 0 (4-2-15c) (x,y) (x 0,y 0 ) (x 0,y 0, p 0 ) 4-2 (4-2-12) (x 0,y 0 ) cusps
140 5 1 2 4 5-1 [5] B. 5-1-1 1 1 A. separation of variables P(y) dy = Q(x) (5-1-1) dx y x P(t) dt = Q(t) dt + C C (5-1-2) P 1 (y)q 1 (x) dy dx = P 2(y)Q 2 (x) (5-1-3)
5-1 141 y P 1 (t) x P 2 (t) dt = Q 2 (t) dt + C C (5-1-4) Q 1 (t) P 2 (y)q 1 (x) = 0 1 1 2xy dy dx = (x + 1)(y2 1) (5-1-5) x + 1 dx = 2y dy (5-1-6) x y 2 1 x + log x = log y 2 1 + C y = ± 1 ± x exp(x C) C (5-1-7) (5-1-6) = 0 x = 0 y = ±1 (5-1-8) (5-1-5) y = ±1 (5-1-7) C + x = 0 B. P(x,y) dy = Q(x,y) (5-1-9) dx P(x,y), Q(x,y) x y n P(x,y) = x n P(1,v), Q(x,y) = x n Q(1,v) (5-1-10) v = y x (5-1-11) y v (5-1-11) (5-1-9) xp(1,v) dv = [Q(1,v) vp(1,v)] dx (5-1-12) P(1,v) dx dv = Q(1,v) vp(1,v) x (5-1-13)
142 5 (x 2 xy) dy dx = (y2 + xy) (5-1-14) (5-1-11) x(1 v) dv = [ (v 2 + v) v(1 v) ] dx 1 v dx dv = 2v2 x 1 ( log v + 1 ) = log x + C C 2 v (5-1-15) log xy + x y = C (5-1-16) C. Bernoulli equations 1695 James Bernoulli dy dx = a(x)y(x) + b(x)y(x)n n 1 (5-1-17) z(x) = y(x) 1 n (5-1-18) dz = (1 n)[a(x)z(x) + b(x)] (5-1-19) dx y(x) 1 n = Ce F(x) + (1 n)e F(x) x e F(t) b(t) dt, F(x) (1 n) x a(t) dt D. Riccati equations C (5-1-20) 1712 dy dx = a(x)y(x)2 + b(x)y(x) + c(x) (5-1-21)
5-1 143 1 dz y(x) = a(x)z(x) dx (5-1-22) 2 d 2 [ z a ] dx (x) dz 2 a(x) + b(x) + a(x)c(x)z(x) = 0 (5-1-23) dx (5-1-23) (5-1-21) (5-1-21) y 1 (x) y(x) = y 1 (x) + z(x) (5-1-24) (n = 2) dz dx = [ 2a(x)y 1 (x) + b(x) ] z(x) + a(x)z(x) 2 (5-1-25) E. exact equations P(x,y) + Q(x,y) dy dx = 0, f (x,y) = C, P(x,y) y = Q(x,y) x P(x,y) = f (x,y), Q(x,y) = f (x,y) x y (5-1-26) C (5-1-27) 5-1-2 1 1 ( F x,y, dy ) [ ] n [ ] n 1 [ ] 2 dy dy dy + q n 1(x,y) + + q 2 (x,y) + q 1 (x,y) dy dx dx dx dx dx + q 0 (x,y) = 0 (5-1-28) q k (x,y)(k = 0, 1,...,n 1) (x,y) A. 1 (5-1-28) dy/dx n
144 5 [ ][ ] [ ] dy dy dy dx r 1(x,y) dx r 2(x,y) dx r n(x,y) = 0 (5-1-29) dy dx r k(x,y) = 0 (5-1-30) f k (x,y,c) = 0 C (5-1-31) (5-1-29) f 1 (x,y,c) f 2 (x,y,c) f n (x,y,c) = 0 (5-1-32) 1 ( ) 2 dy 2 dy dx dx + 1 y2 = 0 (5-1-33) ( )( ) dy dy dx + y 1 dx y 1 = 0 (5-1-34) (Ce x y + 1)(Ce x y 1) = 0 C (5-1-35) B. 1 (5-1-28) (x,y) k ( y ) F(x,y,p) x k G x, p = 0 (5-1-36) ( y ) p = f x (5-1-37) (5-1-28) (5-1-9, 10) C. 2 (x,y) x,y 1
( ) ( ) dy dy y = f x + g dx dx x p = f (p) + [ xf (p) + g (p) ] dp dx 5-1 145 p x dx dp f (p) p f (p) x (5-1-38) (5-1-39) g (p) p f (p) = 0 (5-1-40) y = x dy dx 1 dy/dx x ( x + 1 p 2 ) dp dx = 0 (1-7-13 ) (1-7-14 ) 5-1-3 A. d n y = f (x) (5-1-41) dxn y = x x 0 dx 1 x1 x 0 dx 2 xn 1 n 1 (x x 0 ) k f (x n )dx n + C k x 0 k! k=0 C 0, C 1,...,C n 1 (5-1-42) y = 1 (n 1)! x n 1 (x t) n 1 (x x 0 ) k f (t)dt + C k x 0 k! k=0 (5-1-43) B. y k
190 c- 129 p- 124 45 91, 97, 177 81 83, 177 83 38 104 165 2 88 88 26 143 19 87, 104 41, 136, 145 18 25 47 148 85 48 55 26 14 49 82 19 165 22 22 47 126 88, 177 89, 96 89, 95 126 20 2 2 3 3 4 4 3 125 123 104 14 154 9 86, 115 92, 100, 113 14
191 84 51 119 107 141 38 14 47 48 47 82 160 162 33 33, 153 147 22 86, 102 44 20 39 142 140 126 84 22, 102 120, 141 90 91, 99 91, 97 2 142 46 82 18 45 P 154 19
2010 Printed in Japan ISBN978-4-627-07651-8