III Kepler ( )

III 9 8 3....................................... 3.2 Kepler ( ).......................... 0 2 3 2.................................. 3 2.2......................................... 7 3 9 3.......................................... 9 3.2..................................... 22 3.3......................................... 28 4 32 4. p(0) 0.......................................... 32 4.2............................................ 35

2

. x y = y(x) F F (x, y, y,..., y (n) ) = 0 (.) ( ) n C n y(x) x (n) = f(t, x, x,..., x (n ) ) y 2 = log x + C (C ) y 2xyy = A, B x = Ae t + Be 2t 2 x 3x + 2x = 0 2 n n,, (I) dy dx = f(x)g(y) g(y) 0 dy = f(x) g(y) dx dy g(y) = f(x) dx. y = x( y 2 ) dy : y ± y 2 = x dx 2 log + y y = 2 x2 + C + y y = ±e2c e x 2 ±e 2C C y = Cex2 Ce x2 + (C 0 ) y =, y = ( C =, 0 ). 3 () 2xy = y (2) xyy x 2 = 3 (3) xy = y (4) (x )y + y = 0 (5) (cos 2 x)y + sin x cos 2 y = 0 (6) ( + x)y + x( y)y = 0 URL http://www.math.u-ryukyu.ac.jp/ sugiura/ [K] : [KM], : [Y] : 2 t x 3..9. : () y = C p x, (2) 2 y2 = 2 x2 + 3 log x + C, (3) y = Cx, (4) (x )(y ) = C, (5) tan y = cos x + C, y = π 2 + nπ (n Z), (6) xyex y = C 3

(II) dy ( y ) dx = f x u = y/x y = ux u x + u = y = f(u),.2 xy = y + x 2 + y 2 u = f(u) u x : x > 0 y = y ( y ) 2 x + + u = y/x u + xu = u + + u x 2. du + u 2 = dx x, i.e., log(u + + u 2 ) = log x + c. y y + x 2 + y 2 = kx 2. (k = e c > 0:.) x 2 + y 2 y x2 + y 2 y = /k y = ( kx 2 ), (k > 0: ) (.2) 2 k x < 0 y = y ( y ) 2 x + (.2) x x = 0 y < 0 0 (.2).2 4 () xyy = x 2 +y 2 (2) x + yy = 2y (3) (x+y)+(x y)y = 0 (4) y 2 +x 2 y = xyy (5) x cot y ( x y + xy = 0 (6) x cos y x + y sin y ) ( y y sin y x x x cos y ) xy = 0 x.3 dy ( ) ax + by + c dx = f (aβ αb 0) am + bn + c = 0, αm + βn + γ = 0 αx + βy + γ m, n x = s + m, y = t + n dt ds = dt/dx ds/dx = dy ( ) ( ) as + bt a + bt/s dx = f = f αs + βt α + βt/s 5 () (2x y + 3) (x 2y + 3)y = 0 (2) (x + y + ) 2 y = 2(y + 2) 2 (3) (3x + y 5) (x 3y 5)y = 0 (4) (5x 7y) (x 3y + 4)y = 0.4 ( ) a 6 () (x + y) 2 y = a 2 (x + y = u) (2) yy = (2e x y)e x (e x = s) 4.2 : () y 2 = 2x 2 (log x + C), (2) y x = Ce x y x, y = x, (3) y 2 2xy x 2 = C, (4) ye y/x = C, (5) cos y x = Cx, (6) xy cos y x = C 5.3 : () (y ) 2 (x + )(y ) + (x + ) 2 = C, (2) log y + 2 + 2 Arctan y+2 = C, y + 2 = 0, x (3) 3 2 log{(x 2)2 + (y + ) 2 } Arctan y+ x 2 = C, (4) (3y 5x + 0)2 = C(y x + ), y x + = 0 6.4 : () y a Arctan x+y = C, (2) (y + 2e a x ) 2 (y e x ) = C 4

(III) ( ) : q(t) = 0 x = p(t)x x = Ce R p(t) dt (.3) C t x = p(t)x + q(t) (.3) x = C(t)e R p(t) dt (.4) (.3) x = C (t)e R p(t) dt + C(t)e R p(t) dt p(t) = C (t)e R p(t) dt + p(t)x C (t) = q(t)e R p(t) dt (.4) (.3) x = e R { p(t) dt C + q(t)e R } p(t) dt dt (C: ).3 dx dt = x + cos t + t : dx dt = x + t x = C C(t). x = + t + t x = C (t) + t C(t) ( + t) 2 = C (t) + t x + t. C (t) = ( + t) cos t C(t) = ( + t) sin t + cos t + C.5 7 x = sin t + cos t + t + C + t (C: ) () x x = sin t (2) x 2x = e 3t (3) x + x = t 2 (4) x cos t + x sin t = (5) tx + x = t log t. (Bernoulli ) dx dt = p(t)x + q(t)xα (α 0, ) α = 0,, u = x α.4 t 2 x = tx + x 3 u = ( α)p(t)u + ( α)q(t) : x 0 u = x 3 = x 2 u = 2x 3 x t 2 u = 2tu 2 t 2 u = 2tu u = Ct 2. u = C(t)t 2 ( t 2 u = t 2 2C(t) ) t 3 + C (t) t 2 = 2tu + C (t). 7.5 : () x = 2 (sin t + cos t) + Cet, (2) x = e 3t + Ce 2t, (3) x = t 2 2t + 2 + Ce t, (4) x = sin t + C cos t, (5) x = 2 t log t 4 t + C t 5

C (t) = 2 C(t) = 2t + C x 2 = u = 2t + C t 2 x = 0 (C: ).6 8 () t 2 x = tx + x 2 (2) x + x t = 2x2 log t (3) x + x = tx 3 dx.2 (Riccati ) dt = p(t)x2 + q(t)x + r(t) Riccati x 0 (t) x = u + x 0 (t) u + x 0 = p(u + x 0 ) 2 + q(u + x 0 ) + r = pu 2 + (2px 0 + q)u + (px 0 2 + qx 0 + r) x 0 (t) Bernoulli (α = 2).5 x + t 2 t t 2 x + t 2 t 3 x2 = 0 : x = t x = u + t u + = t 2 (u + t) t t2 t 2 t 3 (u + t)2 = t u t 2 t 3 u2 + z = u 2 z z = t z + t 2 t3 z = + Ct t(t ) x = t x = z + t = t2 ( + C) (C ). + Ct.7 9 () x + t t 2t t x + x2 = 0 (2) x + e 2t ( + 52 et ) x + x 2 = 0 (IV) C - ϕ(x, y) ϕ P (x, y)dx + Q(x, y)dy = 0 (.5) ϕ/ x = P, ϕ/ y = Q (.6) dϕ = ϕ ϕ dx + dy = P dx + Qdy = 0 x y ϕ(x, y) = c (.6) ϕ (.5) ( ) 8.6 : () x = 9.7: () t C log t, x = 0, (2) x = t(c (log t) 2, x = 0, (3) ) x 2 = t + 2 + Ce2t, x = 0 x = 2 t2 t+c, x =, (2) t x 2 = 2 et 3 e t + Ce t 3 2 et, x = 2 et. () x =, (2) x = 2 et. 6

. Ω R 2 P, Q C - (.6) C 2 - ϕ P y = Q x 3 (x 0, y 0 ), (x 0, y), (x, y) Ω (.8) ϕ ϕ(x, y) = c (.5) : (.6) ϕ (.7) (.7) 3 (x 0, y 0 ), (x 0, y), (x, y) Ω ϕ(x, y) = x 0 ϕ/ x = P ϕ (x, y) = y x x 0 x 0 P (s, y) ds + y P y (s, y) ds + Q(x 0, y) = (.7) y 0 Q(x 0, t) dt (.8) x = Q(x, y) Q(x 0, y) + Q(x 0, y) = Q(x, y) x 0 Q x (s, y) ds + Q(x 0, y).6 (2x + y)dx + (x + 2y)dy = 0 : (2x + y)/ y = (x + 2y)/ x =. ϕ = x x 0 (2s + y) ds + y y 0 (x 0 + 2t) dt = x 2 + xy x 2 0 x 0 y + x 0 y + y 2 x 0 y 0 y 0 x 2 + xy + y 2 = c.8 () (2xy + 6x)dx + (x 2 )dy = 0 (2) (y sin x x)dx + (y 2 cos x)dy = 0 ( (3) log y + ) ( ) x dx + x y + 2y dy = 0 (4) ( + x ) x 2 + y 2 dx + ( + y ) x 2 + y 2 dy = 0 P dx + Qdy = 0 µ(x, y) (µp )dx + (µq)dy = 0 µ(x, y) 2 µ. (.7) (µp ) y = (µq) x x (y ) µ P, Q x, y µ = x m y n µ x (.9) Qµ (x) = (P y Q x )µ (P y Q x )/Q R Py Qx x µ = e Q dx (.9) (P y Q x )/P y µ = e R Py Qx P dy (.9) 0 (.7) (x 0, y 0 ) (x, y) C R C P dx + Qdy C C C, C D (C C D ) Green R C P dx + Qdy R C P dx + Qdy = R D ( Q x P )dxdy = 0 (.8) C 3 y (x 0, y 0 ), (x 0, y), (x, y).8 : () x 2 y+3x 2 y = C, (2) y cos x 2 x2 + 3 y3 = C, (3) x log y+log x +y 2 = C, (4) x+ 3 (x2 +y 2 ) 3 2 y = C 2 [K] (.9) 7

.7 ydx + x log xdy = 0 : P = y, Q = x log x P y Q x = log x (P y Q x )/Q = /x. µ(x) = e R x dx = x y dx + log xdy = 0 (.8) x ϕ = x x 0 y y s ds + log x 0 dt = y log x y 0 log x 0. y 0 y log x = c.8 ydx + x( + xy 2 )dy = 0 : µ = x m y n µ m, n (x m y n y) y = (n + )x m y n, (x m y n x( + xy 2 )) x = (m + )x m y n + (m + 2)x m+ y n+2 n + = m +, m + 2 = 0 m = n = 2 (.8) ϕ = x x 0 s 2 y ds + y y 0 x 0 t 2 ( + x 0 t 2 ) dt = xy + y + x 0 y 0 y 0. xy + y = c.9 3 () e y dx x(2xy + e y )dy = 0 (2) 3y 2 (x y) 2 dx + {sin y y cos y 3y 2 (x y) 2 }dy = 0 (3) ydx + 2x( + x 2 y 3 )dy = 0 (4) (y 2 + xy)dx x 2 dy = 0 (V) Clairaut ( ) y = x dy dx + f ( ) dy dx (.0) f(s) C 2 - f (s) 0 x y + xy + f (y )y y = (x + f (y ))y = 0 y = 0, x + f (y ) = 0 y = c x + c 2 c 2 = f(c ) y = cx + f(c) (.) 2 f (s) 0 f ψ(x) y = xψ(x) + f(ψ(x)) (.2) y(x) y(x) y (x) = ψ(x) c (.) (.2) (.2) (.) 4 (.0) 3.9 : µ () µ = x 2, ey x + y2 = c, (2) µ = y 2, (x y) 3 sin y = c, y (3) µ = x 3 y 5, 2x 2 y 4 + 2 y = c, (4) µ = x y 2, x + log x = c. ([K] ) 4 y p.80 8

. Clairaut.9 y = xy e y : y = y + xy e y y y (x e y ) = 0. x = e y y = 0. x = e y y = log x y = x log x x. y = 0 y = c (c ) y = cx e c.0 ([K] p.24. 5 ) () y = xy + y 2 (2) y = xy log y (VI) F (t, x, x, x ) = 0 (a) F (x, x, x ) = 0 : p = x d2 x dt 2 = dp dt = dp dx dx dt = p dp dx x p F (x, p, p dp dx ) = 0 (b) F (t, x, x ) = 0 : p = x p = x t p F (t, p, p ) = 0 (c) α γ F (λt, λ α x, λ α x, λ α 2 x ) = λ γ F (t,( x, x, x ) λ : ) t = e s, x = ye αs t > 0 x = dx ds ds dt = e s αs dy e ds + yαeαs = e ( (α )s d ds + α) y, x = dx ds ds dt = ( e(α 2)s d ds + α ) ( d ds + α) y, F (t, x, x, x ) = 0 e γs F (, y, ( d ds + α)y, ( d ds + α )( d ) ds + α)y = 0 (a) t < 0.0 (3) t > 0 () xx + 2x 2 = 0 (2) x = + x 2 (3) t 3 x + (x tx ) 2 = 0 : () p = x x = p dp dp dx xp dx + 2p2 = 0. p 0 p = C x. p = 0 x = C 2 x 2 3 x3 = C t + C 2 (2) p = x p = + p 2. arctan p = t + C, p = tan(t + C ). dx dt = tan(t + C ) x = log cos(t + C ) + C 2 (3) (λt) 3 λ α 2 x + (λ α x λtλ α x ) 2 = λ α+ t 3 x + λ 2α (x tx ) 2 α + = 2α, α =, γ = 2 (c) t = e s, x = ye s x = dy ds + y, x = e s ( d2 y ds 2 d 2 y ds + dy 2 ds +(dy ds )2 = 0 p = dy ds dp ds = p dp dy p 0 p = C e y. dy + dy ds ) p dp dy +p+p2 = 0 ds = C e y C e y = C 2 e s. C e x/t = C 2 /t. p = 0 y = C 2 x = C 2 t 5.0 : () y = cx + c 2, y = x 2 /4, (2) y = cx log c, y = + log x. 9

. (5), (6) t > 0 ([K] pp.26 29, [KM] p.30. 6 ) () 2xx x 2 = (2) ( + x)x + x 2 = 0 (3) x + tx 2 = 0 (4) x + x 3 = 0 (5) txx tx 2 + xx = 0 (6) tx + 2x = t 2 x 2 x 2 (d) 2 x (t) + a(t)x (t) + b(t)x = 0 x = ϕ(t) x = ϕ(t)y x y y. ( t 2 )x 2tx + 2x = 0 : x = t x = ty x = ty + y, x = ty + 2y ( 2 t( t 2 )y + 2( 2t 2 )y = 0 y = t + t + ) y. t + y y c ( = { t 2 (t 2 ) = c 2 t ) } t + t 2. { y = c 2 log t t + + } + c 2 x t { t x = c 2 log t } t + + + c 2 t..2 ([K] p.77. 7 ) () ( + t 2 )x tx + x = 0 (2) (t 2 + 3t + 4)x + (t 2 + t + )x (2t + 3)x = 0.2 Kepler ( ) Newton Kepler ( ) ( ) 8 Kepler (Kepler ) I. II. III. 6. : () y = C 4 (t + C 2 ) 2 +, (2) ( + x) C 2 = C t + C 2, (3) x = 0 x = C 2 x = 2 C = 0 x = 2 t + C 2, C > 0 x = 2 C arctan t C + C 2, C < 0 x = t 2 +C C log t C t+ C + C2, (4) x = ± 2(t C ) /2 + C 2, x = C 2, (5) x = C 2 t C+, (6) R tx du C e u +2u+ = log t + C 2, xt = C 2. 7.2 ( ) : () t, c ( + t 2 + t log(t + + t 2 )) + c 2 t, (2) e t, c (t 2 + t + 3) + c 2 e t. 8 0

( ) O = t (0, 0, 0) ( ) t x(t) = t (x (t), x 2 (t), x 3 (t)) 9 ẍ = k x 3 x (.3) ẋ x t ẍ = d2 x k = GM G dt2 M 0 x(0), ẋ(0) x(0) O x(0) ẋ(0) : x(0), ẋ(0) e = t (sin θ 2 cos θ, sin θ 2 sin θ, cos θ 2 ) cos θ 2 0 sin θ 2 cos θ sin θ 0 R = 0 0 sin θ cos θ 0 Re = t (0, 0, ) sin θ 2 0 cos θ 2 0 0 Rx(0) = t (,, 0), Rẋ(0) = t (,, 0) R SO(3) t R = R Rx(0) t(0, 0, ) = x(0) tr t (0, 0, ) = x(0) e = 0, ẋ(0) y = t (y (t), y 2 (t), y 3 (t)) = Rx (.3) : ÿ i = k y 3 y i, (i =, 2, 3) y(0) = t (ξ, ξ 2, 0), ẏ(0) = t (η, η 2, 0) (.4) y 3 (t) 0 (.4) y 3 (t) 0 d dt { 2 ( y 2 + y 2 k 2 ) } = 0 E y2 + y 2 2 E = 2 ( y 2 + y 2 k 2 ) y2 + y 2 2 (.5) 2 (.4) y = r cos ϕ, y 2 = r sin ϕ ( ) ( ) ( ) y cos ϕ sin ϕ = ṙ + r ϕ y 2 sin ϕ cos ϕ ( ) ( ) ( ) ÿ = ( r r ϕ 2 cos ϕ sin ϕ ) + (2ṙ ϕ + r ϕ) sin ϕ cos ϕ ÿ 2 (.4) r r ϕ 2 = k r 2 (.6) 2ṙ ϕ + r ϕ = 0 (.7) (.7) d dt (r2 ϕ) = 0 r 2 ϕ c (c 0) (.8) 9 : t (x, x 2, x 3 ) (x, x 2, x 3 )

Kepler II ( ) r 2 ϕ = c (.6) r = c2 r 3 k (.5) E r2 E = 2ṙ2 + c2 2r 2 k r (.9) ( ) ( ) Kepler I ρ = /r ρ ϕ (.8) dr dt = dρ dϕ dρ ρ 2 = c dϕ dt dϕ, d 2 r dt 2 = c d2 ρ dϕ dϕ 2 dt = c2 ρ 2 d2 ρ dϕ 2. (.6) d2 ρ dϕ 2 + ρ = k c 2 20 ρ(ϕ) = A cos ϕ + B sin ϕ + k c 2 = A 2 + B 2 cos(ϕ ϕ 0 ) + k c 2 A = r(t 0 ) k2 c, B = ṙ(t0) 2 c (ϕ(t 0 ) = 0 ) (r, ϕ) r = p + e cos(ϕ ϕ 0 ) (.20) p = c 2 /k, e = k A2 + B 2 = k (r(t0 ) ϕ(t 0 ) k 2 ) 2 + ṙ(t 0 ) 2 (e ) 0 < e < (.6) e = e > (Kepler I) Kepler III T (.20) (x + ae) 2 a 2 + y2 b 2 =, a = p e 2, b = p e 2, (.2) (.8) T 2 ct = 0 2 r2 ϕ dt = πab, T = 2πab/c (.2) T 2 = 4π2 a 2 b 2 c 2 = 4π2 k a3 T a.3 0 < e < (.20) (.2) e = e > (cf. (.2)).4 x(0) ẋ(0) e x(0) = r(0)e, ẋ(0) = ṙ(0)e, (r(0) > 0) R SO(3) y(t) = Rx(0) (.4) ξ > 0, ξ 2 = η 2 = 0 y 2 (t) 0, y 3 (t) 0 (.9) E = 2 y 2 k y 20 3 2

E = 0 y (0) < 0 t c y (t c ) = 0 y (0) > 0 E > 0 t y (t) ( ) y (0) 0 2 2. dx k dt = f k (t, x,, x n ) k =,..., n (2.) x k (a) = b k k =,..., n (2.2) x = t (x,..., x n ), b = t (b,..., b n ), f = t (f,..., f n ) (2.), (2.2) dx dt = f(t, x) x(a) = b (2.3) 2. n x (n) = f(t, x, x,..., x (n ) ) (2.) x k = x (k ), k =,..., n f k (t, x,..., x n ) = x k, k =,..., n ; f n (t, x,..., x n ) = f(t, x,..., x n ) 2. (Lipschitz ) Ω R R n (R n ) f(t, x) x Lipschitz L > 0 (t, x), (t, y) Ω f(t, x) f(t, y) L x y (2.4) x = (x,..., x n ) R n x = { n k= x k 2} 2. f(x) = x α, x R () α = R Lipschitz (2) α > Lipschitz R Lipschitz (3) 0 < α < 0 Lipschitz 2. f(t, x) x D R R n x C f(t, x) x D Lipschitz 2. f = t (f,..., f n ) I R n f dt n f dt I f dt = t ( I f dt,..., I f n dt) 2 : f dt = ( 2 ( f k dt) ) 2 ( 2 n max f k dt n f dt) I k I k I I I I 3

2. : D (t, x), (t, y) D 0 θ (t, θx + ( θ)y) D. { n L := max (t,x) D k= f x k (t, x) 2 } /2 f(t, x) f(t, y) = d f(t, θx + ( θ)y) dθ 0 dθ n n f (t, θx + ( θ)y) x k y k dθ nl x y x k 0 k= (2.3) f(t, x) D R R n (a, b) D R = { (t, x) ; t a r, x b ρ } D (2.5) r, ρ > 0 M = n max (t,x) R f(t, x), α = min{r, ρ M } 2.2 f(t, x) R Lipschitz (2.3) I = [a α, a + α] x(t; a, b) : x(t) = b + x(t) : {x m (t)} : x 0 (t) b, x m (t) = b + a f(s, x(s)) ds (2.6) a f(s, x m (s)) ds (2.7) α 2. x m (t) b M t a αm ρ (t, x m (t)) R (2.7) f(t, x) Lipschitz L (2.4) x m+ (t) x m (t) ( nl t a ) m ρ (2.8) m! m = 0 m m x m+ (t) x m (t) {f(s, x m (s)) f(s, x m (s))} ds nl x m (s) x m (s) ds a a ( nl) m (m )! ρ t s a m ds ( nl t a ) m ρ m! a ( nlα) m m=0 m! e nlα < m > m 2 m max x m (t) x m2 (t) t [a α,a+α] k=m 2 max x k+(t) x k (t) t [a α,a+α] m ( k=m 2 nlα) k ρ k! {x m (t)} Cauchy [a α, a + α] x(t) (2.7) m x(t) (2.6) ( (2.7) {x m } ) : x(t), y(t) (2.6) Lipschitz x(t) y(t) a {f(s, x(s)) f(s, y(s))} ds nl a x(s) y(s) ds. Gronwall (ϕ 0 ) x(t) y(t) 0 4

2.3 (Gronwall ) ϕ(t), ψ(t), w(t) [a, b] ψ(t) 0 w(t) ϕ(t) + w(t) ϕ(t) + a a ψ(s)w(s) ds (a t b) (2.9) ψ(s)ϕ(s)e R t s ψ(u) du ds (a t b) (2.0) : v(t) = a ψ(s)w(s) ds v (s) = ψ(s)w(s) (2.9) ψ(t) 0 v (t) ψ(t)ϕ(t) + ψ(t)v(t). 2 e R t a ψ(u)du d R t dt (v(t)e a ψ(u)du ) ψ(t)ϕ(t)e R t a ψ(u)du. a t v(a) = 0 v(t) a v(t)e R t a ψ(u)du a ψ(s)ϕ(s)e R s a ψ(u)du ds. ψ(s)ϕ(s)e R t s ψ(u)du ds (2.9) 2 (2.0) 2.4 2.2 f(t, x) (t, x) C r x(t) C r+ 2 : x(t) (2.6) 2.2 dx dt = cx, x(0) = (2.7) x m(t) x m (t) x(t) = e ct 2.3 ([KM] p.49, p.56. 22 ) () x = 2x, x(0) = (2) x = tx, x(0) = 3 (3) x = x + x 2, x 2 = x x 2, x (0) =, x 2 (0) = 2 2.4 x + x = 0. x(0) = 0, x (0) = x = y, y = x, x(0) = 0, y(0) = sin t, cos t ( ) ( ) 0 x ( ) ( ) m x(0) : A = = I + ta + t2 2 0 A2 + + tm m! Am y(0) y m 2.2 () 2.2 Lipschitz c 0 x(t) = (t c) 2, t c; x(t) = 0, t < c dx dt = 2 x /2, x(0) = 0 2. x /2 x = 0 Lipschitz 2 f(t, x) x(t) ϕ(t) t = a ϕ ϕ(t) = P c n(t a) n t = a ( C ) 22 2.3 : () ( + e 2t )/2, (2) 3e t2 /2, (3) x = e 2t, x 2 = ( 2 )e 2t hint: (3) x,m (t) = P m k=0 ( 2t) k /k!, x 2,m (t) = ( 2 ) P m k=0 ( 2t) k /k! ( ) (), (2) 5

(2) 2.2 f(t, x) 2.5 f(t, x) R (2.3) I = [a α, a + α] R, α (2.5) Ascoli-Arzelà 2.6 (Ascoli-Arzelà) compact K R n {f λ (x)} λ Λ (I), (II) {f λ (x)} λ Λ compact, ( ) (I) (II) sup sup f λ (x) < ( ) λ Λ x K lim sup sup δ 0 λ Λ x,y K: x y <δ f λ (x) f λ (y) = 0 ( ) : {x m } K (I) m N {f λ (x m )} λ Λ Bolzano- Weierstrass {λ k } Λ m N {f λk (x m )} k= {f λ k } k= Cauchy ε > 0 (II) δ > 0 sup sup λ Λ x,y K: x y <δ f λ (x) f λ (y) < ε 3 K compact M N K M m=b δ (x m ) B δ (x) = { y ; y x < δ} x,..., x M k 0 k, l k 0 m =,..., M f λk (x m ) f λl (x m ) < ε 3 x K x x m < δ m {,, M} f λk (x) f λl (x) f λk (x) f λk (x m ) + f λk (x m ) f λl (x m ) + f λl (x m ) f λl (x) < ε 2.5 : 2.2 (2.6) x(t) x m (t) 23 : t k = a + k mα, k = m,...,, 0,,..., m x m (a) = b x m (t) = x m (t k ) + (t t k )f(t k, x m (t k )), t k t t k+, k = 0,,..., m x m (t) = x m (t k ) + (t t k )f(t k, x m (t k )), t k t t k, k = 0,,..., m +. g m (t) = f(t k, x m (t k )) t k t < t k+ (k 0), t k < t t k (k 0) x m (t) = b + 2. M ((2.5) ) x m (t) x m (s) M t s, s, t [a α, a + α], a g m (s) ds (2.) s = a Ascoli-Arzelà {x m (t)} compact {x m (t)} x(t) 23 {x m } 6

{g m (t)} f(t, x(t)) f(t, x) compact R ε > 0 δ > 0 (s, x), (t, y) R : s t + x y < δ = f(s, x) f(t, y) < ε {x m (t)} x(t) N N m N x m (t) x(t) < δ/2 N max{n, 2(M+)α δ } m N t k t + x m (t k ) x(t) t k t + x m (t k ) x m (t) + x m (t) x(t) < ( + M) t k t + δ 2 < δ g m (t) f(t, x(t)) < ε (2.) m x(t) (2.6) 2.5 dx dt = cx, x(0) = x m(t) x m (t) x(t) = e ct α = 2.6 ([KM] p.62. 24 ) () x = 3x + 2, x(0) = 0 (2) x = x 2, x(0) = 2.2 D R R n ( ) f C(D; R n ) 2.2 (t, x) D (t, x) U f U x Lipschitz f Lipschitz Lip x (D; R n ) D Lipschitz 2. C (D; R n ) Lip x (D; R n ) f Lip x (D; R n ), (a 0, b 0 ) D dx dt = f(t, x) x(a 0) = b 0 (2.2) (a 0, b 0 ) f Lipschitz 2.2 α 0 > 0 I 0 := [a 0 α 0, a 0 +α 0 ] x(t; a 0, b 0 ) a = a 0 +α 0, b = x(a ; a 0, b 0 ) (a, b ) D x(a ) = b (2.2) 2.2 α > 0 I := [a α, a + α ] x(t; a, b ) x(t; a 0, b 0 ) = x(t; a, b ) t I 0 I x(t) = x(t; a 0, b 0 ) (t I 0 ); x(t) = x(t; a, b ) (t I ) x(t) (2.2) 2.7 f Lip x (D; R n ), (a 0, b 0 ) D (2.2) [a 0, a ) x(t) t m a b := lim m x(t m ) (a, b ) D (t, x(t)) (a, b ) (t a ) x(t) a : (a, b ) D ε > 0 R = {(t, x); t a ε, x b ε} D M ε := n sup R f(t, x) t m a x(t m ) b N m N = 0 < a t m < ε 2M ε, x(t m ) b < ε 2 24 2.6 : () 2(e 3t )/3, (2) 3e t 2 7

t N t < a x(t) (2.2) x(t) x(t N ) x(t) b x(t) x(t N ) + x(t N ) b < ε. t N f(s, x(s)) ds M ε (t t N ) < ε 2. 2.8 f Lip x (D; R n ) (2.2) (α, ω) t α or t ω (t, x(t)) D D compact [α, ω ] (α, ω) t (α, ω)\[α, ω ] (t, x(t)) D\ : ( ) 2 x(t), y(t) [a, ω ), [a, ω 2 ) (ω ω 2 ) t = inf{t > a x(t) y(t)} t ω, t < ω 2. ε > 0 x(t ) = y(t ) x(t) y(t) (t < t < t + ε) (t, x(t )) D (2.2) ( ) ω = ω < {t m } t m ω (t m, x(t m )) compact (t m, x(t m )) (ω, b ) b 2.7 I R, D = I R n (2.2) I 2.9 D = I R n, I = (t 0, t ) ( t 0 < t ) f Lip x (D; R n ) (a 0, b 0 ) D (2.2) x(t) f(t, x(t)) A x(t) + B A, B t x(t) (2.3) x(t) I : ( ) x(t) ω < t [a, ω) x(t) x(t) = b + a f(s, x(s)) ds x(t) b + n a f(s, x(s)) ds b + n a (A x(s) + B) ds. A = 0 x(t) b + nb(t a ) lim t ω x(t) < 2.7 A > 0 0 x(t) + A B b + B A + na Gronwall ( 2.3) x(t) + B A ( b + B A )e na(t a ). a ( x(s) + B A ) ds lim t ω x(t) < 2.7 ω = t 2.3 I p > dx dt = xp, x(0) = x(t) = { t/(p )} /(p ) (2.3) t p x(t) ( ) 8

3 3. dx dt = A(t)x + b(t), t I := (t 0, t ) (3.) ( t 0 < t ) A(t) n b(t) R n b 0, n A = (a ij ) A 3. Ax A x. : Schwarz n Ax 2 n 2 = a ij x j i= j= ( n A = a ij 2) /2 n i= i,j= ( n a ij 2)( n x j 2) = A 2 x 2. j= 3.2 ( ) (a, ξ) I R n x(a) = ξ (3.) x(t; a, b) : [τ 0, τ ] (t 0, t ) c = max τ0 t τ A(t), c 2 = max τ0 t τ b(t) 3. A(t)x + b(t) c x + c 2 2.9 3.. (3.) dx dt x(t; a, ξ) j= (3.2) = A(t)x, x(a) = ξ (3.3) V = { x( ; a, ξ) ; ξ R n } (3.4) 3.3 ( ) () V dim V = n (2) v,..., v n V v,..., v n V t I( t I) v (t),..., v n (t) R n : () ξ, ξ R n, α, α 2 R α x( ; a, ξ ) + α 2 x( ; a, ξ 2 ) = x( ; a, α ξ + α 2 ξ ) (3.5) V e = t (, 0,..., 0),..., e n = t (0,..., 0, ) ϕ k = x( ; a, e k ) ξ = t (ξ,..., ξ n ) (3.5) x( ; a, ξ) = n k= ξ kϕ k. ϕ,..., ϕ n V c,..., c n R c ϕ + + c n ϕ n = 0 t = a c e + + c n e n = 0 c = = c n = 0. ϕ,, ϕ n dim V = n (2) v (t),..., v n (t) R n v,..., v n dim V = n v,..., v n V v (t),..., v n (t) R n c,..., c n j c jv j (t) = 0 v( ) = x( ; t, v(t)) (3.5) 9

cj v j ( ) = x( ; t, c j v(t)) = x( ; t, 0) = 0 v,..., v n 3. 3.3 ϕ k n R(t, a) = (ϕ (t) ϕ n (t)) R(t, a) (3.3) (resolvent matrix) 3.3 x(t; a, ξ) = R(t, a)ξ 3.4 (0) R(t, t) = I ( ) () R(s, t) = R(s, u)r(u, t) (s, u, t I) (2) R(s, t) R(s, t) = R(t, s) (3) s R(s, t) = A(s)R(s, t), R(s, t) = R(s, t)a(t). t : (0). () ξ R n x(s; u, R(u, t)ξ), x(s; t, ξ) x = A(s)x u x(u; u, R(u, t)ξ) = R(u, t)ξ = x(u; t, ξ) x(s; u, R(u, t)ξ) = x(s; t, ξ). R(s, u)r(u, t)ξ = R(s, t)ξ (2) () s = t (0) I = R(t, t) = R(t, u)r(u, t) (3) R d ds ϕ k = A(s)ϕ k R(t, s)r(s, t) = I t R (t, s)r(s, t) + R(t, s) R (s, t) = O t t R R(s, t) = R(t, s) (t, s)r(s, t) = R(s, t)a(t)r(t, s)r(s, t) = R(s, t)a(t). t t 3.5 (Wronskian, ) (x (t) x n (t)) ( det U(t) = exp dx dt = A(t)x x,..., x n U(t) = s ) tr A(u) du det U(s) (3.6) A = (a ij ) tr A = a + + a nn A (trace) 3. n = 3 3.5 : y(t) := det U(t) y (t) = {tr A(t)}y(t) 3.2 A(t) t A(t) = A(t) x = A(t)x () x(t) x(0). (2) R(s, t). : () d dt x(t) 2 = 0, (2) s {t R(s, t)r(s, t)} = O 3.6 v,..., v n (3.3) n V (t) = (v (t) v n (t)) 3.3 (2) V (t) R(t, a) = V (t)v (a) : v k (t) = x(t; a, v k (a)) = R(t, a)v k (a) V (t) = R(t, a)v (a) 3..2 dx dt = A(t)x + b(t), x(a) = ξ (3.7) 20

x(t; a, ξ) A(t) (3.3) x 0 (t; a, ξ) V b = { x( ; a, ξ) ; ξ R n }; V 0 = { x 0 ( ; a, ξ) ; ξ R n } (3.8) 3.7 () x V b V b = { x 0 + x ; x 0 V 0 }. (2) ( ) R(s, t) (3.3) x(t; a, ξ) = R(t, a)ξ + : () ; x V b ˆx := x x d dt ˆx = A(t)x + b(t) (A(t)x + b(t)) = A(t)ˆx a R(t, s)b(s) ds. (3.9) ˆx V 0 (2) () ξ = 0 3.4 { d t } R(t, s)b(s) ds = R(t, t)b(t) + A(t)R(t, s)b(s) ds dt a = b(t) + A(t) a a R(t, s)b(s) ds 3. () ( ) mẍ = kx (ẍ = d2 x dt.) ( ) ( ) 2 ( ) ( ) ( ) x x = x 2 x d x 0 x = dt x 2 ω 2 0 x ( ) ( ) ( ) ( ) ( 2 ) x (0) 0 cos ωt ω sin ωt =,, x 2 (0) 0 ω sin ωt cos ωt ( ) cos ω(t s) ω sin ω(t s) R(t, s) = x(0) = a, x (0) = b ω sin ω(t s) cos ω(t s) ω = k/m x(t) = a cos ωt + b sin ωt ω (2) ( ) f(t) ẍ + ω 2 x = f(t), b(t) = t (0, f(t)) (3.9) : ( ) ( R(t, s)b(s) = ε sin ω(t s) cos νs = ε ω ω cos ω(t s) cos νs 2ω f(t) = ε cos νt sin(ω(t s) + νs) + sin(ω(t s) νs) ω cos(ω(t s) νs) + ω cos(ω(t s) + νs) ) (i) ω ν (ii) ω = ν x(t) = x (t) = ( [ R(t, 0) a b ) + 0 ] R(t, s)b(s) ds = a cos ωt + b ω sin ωt + ε ω 2 (cos νt cos ωt). ν2 x(t) = a cos ωt + b ε sin ωt + t sin ωt. ω 2ω (i) lim t x(t) = (resonance) 2

3.2 3.2. M n (C) n (3.2) A A M n (C) C n2 ( R 2n2 ) 3.8 A, B M n (C) c C (0) ca = c A () A + B A + B (2) AB A B. : (0) () A C n2 (2) Schwarz A = (a ij ), B = (b ij ) AB 2 = n n 2 a ik b kj i,j= k= n i,j= ( n a ik 2)( n b kj 2) = A 2 B 2. k= 3.9 A M n (C) m () S m (A) := k! Ak Cauchy C n2 k=0 e A (2) e ta (t R) t ( ) d dt eta = Ae ta. : () m > m 3.8 m S m (A) S m (A) = k=m + m k! Ak k=m + k= k! A k 0 as m, m. (2) () R > 0 t < R {S m (ta)} e ta d dt S m(a) = m k= m k! ktk A k = A l! tl A l = AS m (ta) l=0 d dt eta = Ae ta 2 l = k 3.0 A M n (C) dx dt x(t) = e (t a)a ξ = Ax, x(a) = ξ (3.0) : x(t) = e (t a)a ξ (3.0) 3.9 (2) 3. 3.0 A t x = A(t)x e R t s A(u) du A(t) s A(u) du 25 3. A, B M n (C) () (a) (c) : (a) AB = BA (b) e ta e sb = e sb e ta (s, t R) (c) e ta e tb = e t(a+b) (t R). (2) e A (e A ) = e A. (3) P P e A P = e P AP. (4) det e A = e tr A. 25 2 2 A(t) = a(t)i + b(t)c (C ) cf. [K] p.0. 22

: () (a) = (c): m S m (ta)s m (tb) S m (t(a + B)) k,l m:k+l m+ t k+l A k B l k!l! m2 C 2m 2([m/2]!) 2 0 C = max{ ta, tb, } (c) = (a): t 2 t = 0 A 2 + 2AB + B 2 = (A + B) 2 (a) = (b): S m (ta)s m (sb) = S m (sb)s m (ta) m (b) = (a): s, t s = t = 0 (2) A A () (c) e A e A = e A A = I. (3) P A k P = (P AP ) k, k N, P S m (A)P = S m (P AP ) m (4) 3.5 e ta : (a) A : A = λ 0... 0 λ n e ta = k=0 Ak = t k k! Ak = e tλ 0... 0 e tλ n λ k 0... 0 λ k n (b) J Jordan : t k = t k /k!, λ 0 0 0 λ. J =...... 0 e tj = e tλ λ 0 0 λ t 0 t t 2 t n 0 t 0 t........ t2 t 0 t 0 0 t 0 (c) A P Jordan 3. (3) : P AP = J 0... 0 J r P e ta P = e tp AP = e tj 0... 0 e tjr Jordan 3.2 A M n (C) Φ(λ) = det(λi A) = (λ λ ) m (λ λ r ) m r (3.) 26 λ,..., λ r A m,..., m r P j G λj = {x C; (A λ j I)x = 0} : () P,..., P r P 2 i = P i ; P i P j = O (i j); P + + P r = I (2) l j m j l j (λ j I A) lj P j O, (λ j I A) lj P j = O. (3) N = A r j= λ jp j N N = O A l = = l r = 26 23

(4) P j A A A P j Lagrange : k r:k j P j = (A λ ki) k r:k j (λ (j =,..., r) (3.2) j λ k ) A P j : Φ(λ) = h (λ) (λ λ ) m + + h r(λ) (λ λ r ) m r (3.3) Φ = h (λ)g (λ) + + h r (λ)g r (λ). g j (λ) = Φ(λ)/(λ λ j ) m j 27 P j = h j (A)g j (A) (3.4) 3.2 P,..., P r A A = r j= λ jp j + N (3.5) λ j P j A, N A Jordan, N = O A A P j A (A ) 3.3 3.2 A P,..., P r e ta = r e λjt{ I + t! (A λ ji) + + tm j (m j )! (A λ ji) mj } P j (3.6) j= : e ta = e ta (P + + P r ) e ta P j = e t(a λ ji+λ j I) P j = e tλ ji e t(a λ ji) P j = e tλ j e t(a λ ji) P j (A λ j I) l j P j = O e ta P j = e tλ l j j k=0 tk k! (A λ ji) k P j 28 ( ) 0 3. () A =. i, i 0 P = A + ii i + i = 2 ( i i ) (, P 2 = A ii i i = 2 (P = P 2 ) e it = cos t + i sin t ( ) e ta = e it P + e it cos t sin t P 2 = sin t cos t i i 27 f(λ) = λ n + a λ n + + a n λ + a n f(a) = A n + a A n + + a n A + a n I 28 (3.6) (A λ j I) k = O (k l j ) l j ) 24

0 (2) A = 0. A (2 ), 2 0 (3.2) P = A 2I 2 = 3 2 2 2, P 2 = A + I 2 + = 3 e ta = e t P + e 2t P 2 0 (3) A = 2. A λ = (2 ), λ 2 = 2 A P, P 2 (λ ) 2 (λ 2) = λ (λ ) 2 + λ 2 = λ(λ 2) + (λ ) 2 0 0 P = A(A 2I) = 0 0, P 2 = (A I) 2 = 0 0 0 0 0 P 2 λ 2 = 2 l 2 = m 2 = (A I)P = O e ta = e t {I + t(a I)}P + e 2t P 2 = e t 0 0 0 3.3 x(0) = x 0 ( ) ( ) () x 3 4 = x. (2) x 2 5 = x. 2 (3) x = (5) x = (7) x = 2 4 2 4 0 6 3 7 2 5 3 6 0 0 0 2 0 0 0 0 0 0 0 0 0 + te t + e 2t 0 0 0. x. (4) x = x. (6) x = x. (8) x = 2 2 6 2 4 3 3 2 2 8 4 0 4 3 28 x. x. x. : { (2) ( ) ±i. i, i ( )} P, P 2 P = P 2 0 2 5 x(t) = cos t + sin t x 0. (8) 0 2 25

x(t) = { e t 5 4 6 4 0 0 0 4 cos 2t + 5 6 4 0 5 0 4 4 sin 2t + 0 6 56 24 } 5 50 20 x 0. 4 44 56 3.4 (cf. 3.7) x(0) = t (0, 0) ([KM] p.89.) () ( ) ( ) ( ) ( ) x 3 e 2t = x +. (2) x 4 = x + e 3t 0. 2 0 e t 5 (3) ( ) ( ) ( ) ( ) x 2 = x + e t cos t. (4) x 8 9 3 = x + sin t. 2 0 sin t 4 4 2 3.2.2 3.2 3.4 (Cayley-Hamilton) (3.) Φ(λ) Φ(A) = O : A = (a ij ) e = t (, 0,..., 0),..., e n = t (0,..., 0, ) Ae j = a j e + + a nj e n a j e + + (a jj A)e j + + a nj e n = 0, j n. a I A a 2 I a n I a 2 I a 22 I A a n2 I...... a n I a 2n I a nn I A e e 2. e n = 0 0. 0 (3.7) ij ã ij ã ij = a ij I, i j, ã jj = a jj I A (ã ij ) (A) (A) A (A) (3.7) Φ(A) O O. O Φ(A).......... O O O Φ(A) e e 2. e n 0 = 0. 0 29 Φ(A)e j = 0 ( j m) Φ(A) = O n A (3.) C n G λj = { x ; (λ j I A) m j x = 0 } λ j : 3.5 G λ G λr = C n ( ) 3.6 λ f (λ),..., f r (λ) 2 E j := { x ; f j (A)x = O } (j =,..., r); E 0 := { x ; f (A) f r (A)x = O } E 0 = E E r 29 A B BA = (det A)I 26

: r = 2 ( ) 2 h, h 2 f (λ)h (λ) + f 2 (λ)h 2 (λ) 30 λ A f (A)h (A) + f 2 (A)h 2 (A) = I. (3.8) x x = f 2 (A)h 2 (A)x, x 2 = f (A)h (A)x x = x + x 2 f (A)x = h 2 (A){f (A)f 2 (A)x} = 0, f 2 (A)x 2 = h (A){f (A)f 2 (A)x} = 0 x E, x 2 E 2 x E 0 x = x + x 2 = y + y 2, x, y E, x 2, y 2 E 2 z := x y = y 2 x 2 E E 2 (3.8) z = h (A)f (A)z + h 2 (A)f 2 (A)z = 0. 3.5 : 3.4 Φ(A) = O (3.) 3.6 3.2 : () 3.6 3.5 P i P j = O (i j), P + + P r = I Pi 2 = P i (P + + P r ) = P i (4) (2) (λ j I A) m j P j = 0 ϕ(λ) = (λ λ ) l (λ λ r ) l r ϕ(a) = O ϕ(a) = ϕ(a){p + + P r } ϕ(a)p j = { ki) l j k r:k j } (A λ j I) l j P j = O ϕ(a) = O ( ϕ ) (3) N = A r j= λ jp j = r j= (A λ ji)p j A P,..., P r P i P j = O (i j) N n = r j= (A λ ji) n P j = O N N = O l =... = l r = A T T AT = D (D ) AT = T D T = (l l n ) l j λ k Al j = λ k l j λ F λ = { x ; (λi A)x = 0 } l,..., l n F λ F λr F λ F λr = C n F λj P j (A λ j I)P j = O N = r j= (A λ ji)p j = O l = = l r = 3.6 F λ F λr = C n l F λj Al = λ j l C l,..., l n F λ,..., F λr T = (l l n ) T AT (3.2) (3.3) Φ(λ) ϕ(λ) h j (A)g j (A) (3.2) (3.3) /ϕ(λ) = a /(λ λ ) + + a r /(λ λ r ) = a g (λ) + + a r g r (λ), (g j (λ) := ϕ(λ)/(λ λ j )) λ = λ j a j g j (λ j ) = P j = g j (A)/g j (λ j ) (3.2) 30 27

3.3 n d n x dt n + a (t) dn x dt n + + a n (t) dx dt + a n(t)x = f(t) (3.9) f(t) f(t) 0 d n x dt n + a (t) dn x dt n + + a n (t) dx dt + a n(t)x = 0 (3.20) (3.9) a (t),..., a n (t) I 2. x k = x (k ), k =,..., n 0 0 0 0 x x. 0 0 0... d x 2 dt. = A x 2., A =.......... 0. x n x n...... 0 0 0 0 a n a n a 2 a (3.2) x(t) (3.20) t (x(t), x (t),, x (n ) (t)) (3.2) t (x (t), x 2 (t),, x n (t)) (3.2) x (t) C n - (3.20) 3.3 3.7 (3.9) n x (t),..., x n (t) 3.7 I C n x (t),..., x n (t) Wronskian ( ) W (t) := x (t) x n (t) x (t) x n(t) W (t) 0 x (n ) (t) x (n ) n (t) : c x (t) + + c n x n (t) = 0 k c x (k) (t) + + c nx n (k) (t) = 0 Wronskian 0 c = = c n = 0 3.2 (3.9) n x (t),..., x n (t) Wronskian W (t) 0 (3.2) tr A(t) = a (t) 3.5 W (t) = e R t s a(u) du W (s) t I W (t) 0 t I W (t) 0 3.3. (3.20) a,..., a n (3.2) Φ(λ) := det(λi A) = λ n + a λ n + + a n λ + a n (3.) Φ(λ) = (λ λ ) m (λ λ r ) m r (λ,..., λ r ) e(λ) = t (, λ, λ 2,..., λ n ), ˆp(λ) = (0,..., 0, Φ(λ)) Ae(λ) = λe(λ) ˆp(λ) 28

k Ae (k) (λ) = λe (k) (λ) + ke (k ) (λ) ˆp (k) (λ) ˆp (k) (λ j ) = 0, (0 k m j ) e j,k = e (k) (λ j )/k!, ( j r; 0 k m j ) Ae j,0 = λ j e j,0, Ae j,k = λ j e j,k + e j,k, P = (e,0 e,m e r,0 e,mr ) P AP = J 3 λ 0 0 J m (λ ) 0 0. 0 J m2 (λ 2 )... 0 λ. J =, J....... m (λ) =...... 0 0 λ 0 0 J mr (λ r ) 0 0 λ J m (λ) Jordan m 3. (c) e ta = P e tj P (3.2) n e tλ, te tλ,, t m e tλ ; e tλ 2, te tλ 2,, t m2 e tλ 2 ; ; e tλ r, te tλ r,, t mr e tλ r (3.22) (3.20) 32 3.2 () x + x x x = 0. (2) x + 2x + 5x = 0. (3) x + 2x + x = 0. : () D 3 + D 2 D = (D + ) 2 (D ) x = (c e t + c 2 te t ) + c 3 e t. (2) D 2 + 2D + 5 = (D + + 2i)(D + 2i) x = c e ( +2i)t + c 2 e ( 2i)t = c 3 e t cos 2t + c 4 e t sin 2t. c 3 = c + c 2, c 4 = i(c c 2 ) (3) D 4 + 2D 2 + = (D 2 + ) 2 = (D i) 2 (D + i) 2 x = c e it + c 2 te it + c 3 e it + c 4 te it = (c 5 + tc 6 ) cos t + (c 7 + tc 8 ) sin t. c 5 = c + c 3, c 6 = c 2 + c 4, c 7 = i(c c 3 ), c 8 = i(c 2 c 4 ) 3.3.2 (3.9) a,..., a n 3.7 (3.9) x (t) (3.20) x(t) x (t) + x(t) (3.9) x (t) 3.8 t 0 n f(t n ) dt n dt 2 dt = t 0 t 0 t 0 (t s) n (n )! : n = n n + t 0 = n τ f(s) ds (τ s) n f(t n+ ) dt n+ dt n dt = f(s) dsdτ t 0 t 0 t 0 t 0 (n )! (τ s) n [ ] (τ s) n t (t s) n f(s) dτds = f(s) dτds = f(s) ds. t 0 s (n )! t 0 n! n! 3 det P = Q k l (λ k λ l ) 0 (cf. Vandermonde ) 32 (3.22) (D λ ) m (D λ r) m r x = 0 n (3.20) D = d dt τ=s t 0 29

2 () : (D a) n x = f(t) (3.23) D = d dt e at Leipniz D n (e at x) = n k=0 ( ) n ( a) k e at D n k x = e at (D a) n x = e at f(t) k n 3.8 e at x = x = n j= t 0 (t s) n (n )! c j t j e at + t 0 e as f(s) ds + n c j t j. j= (t s) n e a(t s) f(s) ds (n )! (3.23) f(t) 0 (D a) n x = f(t) : (2) : x = t (D a) n f(t) = t 0 (t s) n e a(t s) f(s) ds (3.24) (n )! P (λ) := λ n + a λ n + + a n λ + a n = (λ λ ) m (λ λ r ) mr, λ k λ j (k j) P (λ) = h (λ) (λ λ ) + + h r(λ) (3.25) m (λ λ r ) mr x(t) = P (D) f(t) = h (D) (D λ ) m f(t) + + h r(d) (D λ r ) m r f(t) (3.26) x(t) P (D)x = f(t) f k (t) = (D λ k ) m f(t) = k t 0 (t s) mk e λ(t s) f(s) ds (m k )! () (D λ k ) m k f k (t) = f(t) P k (λ) := P (λ)/(λ λ k ) m k (P k (λ) ) P (D)x(t) = P (D){h (D)f + + h r (D)f r } = h (D)P (D)(D λ ) m f + + h r (D)P r (D)(D λ r ) mr f r = {h (D)P (D) + + h r (D)P r (D)}f (3.25) h (λ)p (λ) + + h r (λ)p r (λ) = (3.26) P (D)x = f(t) (3) : f(t) 30

(a) P (D)x = c (c ) P (0) = a n 0 Dx = = D n x = 0 x = c/a n P (0) = 0 P (D) = D p Q(D), Q(0) 0 D p x = c/q(0), x = (c/q(0)) t p /p! (b) ( D)x = ct m ( D)( + D + + D m ) = D m+ t m ( D)( + D + + D m )t m = t m. x = c( + D + + D m )t m (c) P (D)x = ct m P (0) 0 P (λ) = P (0)( λq(λ)) (Q(λ) ) (b) ( DQ(D))( + DQ(D) + D 2 Q(D) 2 + + D m Q(D) m )t m = ( D m+ Q(D) m+ )t m = t m x = (c/p (0))( + DQ(D) + D 2 Q(D) 2 + + D m Q(D) m )t m P (0) = 0 P (D) = D p Q(D) (Q Q(0) 0 ) Q(D)D p x = ct m D p x p (d) P (D)x = t m e at (D+a)(e at x) = D(e at x)+ae at x = e at x p (D+a) k (e at x) = e at x (k) e at P (D)x = P (D + a)(e at x) = t m (c) (e) f(t) k =, 2 x k (t) P (D)x = f k (t) x (t) + x 2 (t) P (D)x = f (t) + f 2 (t) P (D)(x + x 2 ) = P (D)x + P (D)x 2 = f (t) + f 2 (t) 3.3 () x + 3x 2x = 2t 3 t (2) x x + x = t 3 (3) x 3x + 2x = te t (4) x + 2x = t cos t (5) x + x + x = e t + t : () (D 2 + 3D 2)x = 2t 3 t 2( 3 2 D 2 D2 )x = 2t 3 t. x = 3 2 D ( t3 + 2 D2 2 t) ( ( 3 = + 2 D + ( 3 2 D2) + 2 D + 2 D2) 2 ( 3 + 2 D + 3) 2 D2) ( t 3 + 2 t) ( = + 3 2 D + 4 D2 + 39 8 D3) ( t 3 + 2 t) = t3 9 2 t2 6t 57 2. a = ( 3 + 7)/2, a 2 = ( 3 7)/2 c e at + c 2 e a2t c e a t + c 2 e a 2t t 3 9 2 t2 6t 57 2. (2) (D 3 D 2 + D)x = ( D + D 2 )(Dx) = t 3 Dx = D + D 2 t3 = { + (D D 2 ) + (D D 2 ) 2 + (D D 2 ) 3 }t 3 = ( + D D 3 )t 3 = t 3 + 3t 2 6. x = t 4 /4 + t 3 6t. λ 3 λ 2 + λ = 0 λ = 0, ( ± 3i)/2 x = c + c 2 e t/2 cos( 3t/2) + c 3 e t/2 sin( 3t/2) + t 4 /4 + t 3 6t. (3) e t (D 2 3D + 2)x = ((D ) 2 3(D ) + 2)(e t x) = (D 2 5D + 6)(e t x) = t ( ) e t x= t 5D 6 + D2 6 = { + 5D 6 D2 6 } t 6 = t 6 + 5 36. 6 3

λ 2 3λ + 2 = 0 λ =, 2 x = c e t + c 2 e 2t + (t/6 + 5/36)e t. (4) cos t = Re e it (D 2 + 2)x = te it e it (D 2 + 2)x = ((D + i) 2 + 2)(e it x) = (D 2 + 2iD + )(e it x), e it x = + 2iD + D 2 t = ( (2iD + D2 ))t = t 2i, x = e it (t 2i) = t cos t + 2 sin t + i(t sin t 2 cos t). λ 2 + 2 = 0 λ = ± 2i x = c cos 2t + c 2 sin 2t + t cos t + 2 sin t. (5) (D 2 +D+)x = e t e t (D 2 +D+)x = {(D+) 2 +D++}(e t x) = (D 2 +3D+3)(e t x) = e t x = +D+ 3 D2 3 = 3. (D2 + D + )x = t x = +D+D t = t 2 3 et + t λ 2 + λ + = 0 λ = ± 3i 2 3 3 x = c e t/2 cos 2 t + c 2e t/2 sin 2 t + 3 et + t. 3.5 ([K] p.77. 33 ) () x + x = t 2 (2) x 3x + 2x = te 2t (3) x + 2x + x = sin t (4) x + x = t 2 2t (5) x 2x + 4x = e t sin t (6) x + x = 2 cos 2 t t + e t 4 2 p(t)x + q(t)x + r(t)x = 0 (4.) x(t) = c n(t a) n p(t), q(t), r(t) x = a a = 0 x = 0 34 4. p(0) 0 p(0) 0 p(t) p q(t), r(t) : q(t) = q m t m, r(t) = m=0 r m t m ( t < r ) (4.2) m=0 x(t) = m=0 c mt m t x (t) = mc m t m = (m + )c m+ t m, x (t) = m= m=0 m(m )c m t m 2 = m=2 (m + )(m + 2)c m+2 t m 33 3.5 ( ) : () t 2, (2) ( 2 t2 t)e 2t, (3) 8 t2 sin t, (4) 3 t3 2t 2 + 4t, m=0 (5) tet 20 (3 cos t + sin t), (6) 6 sin 2t + t 2 t2 + 2 et. 34 [Ku] : ( ) 32

(4.): p(t) (m + )(m + 2)c m+2 t m + m=0 q m t m m=0 m=0 2 3 [ m q m k (k + )c k+ ]t m, m=0 k=0 (m + )c m+ t m + r m t m m=0 [ m r m k c k ]t m, m=0 t m {c m } k=0 k=0 m=0 c m t m = 0 m m (m + )(m + 2)c m+2 + q m k (k + )c k+ + r m k c k = 0 (4.3) {c m } c 0, c (c 0 = x(0), c = x (0) ) (4.2) t < r (4.3) m c mt m t < r 4. x + q(t)x + r(t)x = 0 q(t), r(t) t a < r x(t) x a < r : x(t) = k=0 c m (x a) m. (4.4) m=0 c 0 = x(a), c = x (a) c m (m 2) (4.4) c 0, c : a = 0 0 < ρ < r m q mρ m, m r mρ m M > 0 (4.3) q m ρ m M, r m ρ m M (m = 0,,...) m (m + )(m + 2) c m+2 Mρ m [(k + ) c k+ + c k ]ρ k k=0 {A m } A 0 = c 0, A = c m (m + )(m + 2)A m+2 = Mρ m [(k + )A k+ + A k ]ρ k + MA m+ ρ (4.5) c m A m k=0 (m + 2)(m + 3)A m+3 ρ m = Mρ m [(k + )A k+ + A k ]ρ k + M[(m + 2)A m+2 + A m+ ]ρ + MA m+2 ρ 2 k=0 = (m + )(m + 2)A m+2 MA m+ ρ + M[(m + 2)A m+2 + A m+ ]ρ + MA m+2 ρ 2 = [(m + )(m + 2) + M(m + 2)ρ + Mρ 2 ]A m+2. A m+3 (m + )(m + 2) + M(m + 2)ρ + Mρ2 = A m+2 (m + 2)(m + 3)ρ ρ, m m A mt m t < ρ m c mt m t < r 33

4. x + ω 2 x = 0 x(t) = m=0 c mt m m=0 (m + )(m + 2)c m+2t m + ω 2 m=0 c mt m = 0 t m ω 2 c m+2 = (m + )(m + 2) c m c 2n = ( ω 2 ) n c 0 /(2n)!, c 2n+ = ( ω 2 ) n c /(2n + )! x(t) = c 0 cos ωt + c sin ωt 4.2 (Legendre ) ( t 2 )x 2tx + α(α + )x = 0 (α ) t < : q(t) = 2t t 2 = ( 2)t 2m+, r(t) = m=0 α(α + ) t 2 = α(α + )t 2m. x(t) = m=0 c mt m ( ) m=0 [(m + 2)(m + )c m+2 m(m )c m 2mc m + α(α + )c m ]t m = 0 m=0 c m (m + 2)(m + )c m+2 + (α + m + )(α m)c m = 0. c 2n = l 2n c 0, c 2n+ = l 2n+ c n (α + 2n )(α + 2n 3) (α + )α(α 2) (α 2n + 2) l 2n = ( ), (2n)! n (α + 2n)(α + 2n 2) (α + 2)(α )(α 3) (α 2n + ) l 2n+ = ( ) (2n)! x (t) = l 2nt 2n, x 2 (t) = l 2n+t 2n+ x(t) = c 0 x (t) + c x 2 (t) α α 2n > α c 2n = 0 α = 0, 2, 4 x (t), 3x 2, 0x + (35/3)x 4 α 2n + > α c 2n+ = 0 α =, 3, 5 x 2 (t) x, x (5/3)x 3, x (4/3)x 3 + (2/5)x 5 Legendre α α P α (t) d α 2 α α! P α () = Legendre P α (t) = dt (t 2 ) α α (cf. p.43. p.7 22, p.27 2 ) 4. ([Ku] p.26. 35 ) () x tx + 2x = 0 (2) x + tx = 0 (3) x + tx + x = 0 (4) x t 2 x = 0 (5) ( t 2 )x 4tx 2x = 0 4.2 (n!) 2 (2n)! (2t)2n (2n+)! (n!) ( t 2 2 )2n+ ( t 2 )x 5tx 4x = 0 ([Ku] p.26.) (2n 3)!! (2n+)! t2n+, (2) x =, x 2 = t + P ( ) n (2n )!! n= t (2n+)!! 2n+, 35 4. : () x = t 2, x 2 = t 6 t3 P n=2 (3) x = P ( ) n (2n)!! t2n, x 2 = P ( ) n (2n+)!! t2n+, (4) x = + P t 4n n=, x q 4n q 2 (t) = t + P 4n n= q 4n+k = (4n + k)(4n 4 + k) (4 + k) (k =, 0, ) (5) x (t) = /( t 2 ), x 2 (t) = t/( t 2 ) t 4n+ q 4n q 4n+, 34

4.3 Chebyshev ( t 2 )x tx + α 2 x = 0 ([Ku] p.26. 36 ) () (2) α α (3) α = 0,, 2, 3 (4) t = cos s 4.2 (4.) a (t a) 2 x + q(t)(t a)x + r(t)x = 0 (q(t), r(t) a ) (4.6) a (4.) Frobenius a = 0 : t 2 x + tq(t)x + r(t)x = 0. (4.7) q(t), r(t) t = 0 (4.2) x(t) = t ρ c n t n = c n t n+ρ (c 0 0) (4.8) n c nt n t < r (4.7) (ρ + n)(ρ + n )c n t ρ+n + q k t k (ρ + m)c m t ρ+m + r k t k k=0 m=0 2 3 [ n (ρ + k)q n k c k ]t n+ρ, k=0 k=0 [ n r n k c k ]t n+ρ, k=0 t n+ρ c n m=0 c m t ρ+m = 0 ρ(ρ ) + q 0 ρ + r 0 = 0 (4.9) n {(ρ + n)(ρ + n ) + (ρ + n)q 0 + r 0 }c n = [(ρ + k)q n k + r n k ]c k (n ) (4.0) (4.9) (4.6) 2 ρ, ρ 2 4.2 (4.6) (i) ρ ρ 2 : x (t) = t ρ (ii) ρ ρ 2 = l (l ) : x (t) = t ρ k=0 c n (ρ )t n, x 2 (t) = t ρ2 c n (ρ 2 )t n (c 0 (ρ ) 0, c 0 (ρ 2 ) 0) c n (ρ )t n, x 2 (t) = cx (t) log t + t ρ2 c n (ρ 2 )t n. 36 4.3 : () x (t) = + P ( α 2 )(2 2 α 2 ) ((2n 2) 2 α 2 ) n= t (2n)! 2n, x 2 (t) = t + P ( α 2 ) ((2n ) 2 α 2 ) n= t (2n)! 2n+ (4) y(s) = x(cos s) d 2 y/ds 2 + α 2 y = 0. 35

(ii) c 0 (ρ ) 0 l = 0 c 0, c 0 (ρ 2 ) = 0 l c 0 (ρ 2 ) 0 c 0 (i), (ii) c 0 (ρ ), c 0 (ρ 2 ), c x q(t), r(t) t < r : ( ) (i) ρ ρ 2 : (4.0) I(ρ) = ρ(ρ ) + q 0 ρ + r 0, J k,n (ρ) = (ρ + k)q n k + r n k (4.) n I(ρ + n)c n (ρ) = J k,n (ρ)c k (ρ) (n =, 2,...) (4.2) k=0 c n ρ c n (ρ) I(ρ + n) 0, n =, 2,..., (4.2) {c n (ρ)} n c 0 (ρ) ρ ρ 2 I(ρ + n), I(ρ 2 + n) (n ) 0 ρ = ρ, ρ 2 (4.2) c n (ρ) (4.7) x(t; ρ) = c n (ρ)t n+ρ (ρ = ρ, ρ 2 ) (4.3) t < r (ii) ρ ρ 2 = l (l ) : (i) c 0 (ρ ) = (4.2) c n (ρ ) (4.3) x(t; ρ ) l = 0 x(t; ρ ) l I(ρ 2 + l) = I(ρ ) = 0 (4.2) c k (ρ 2 ) ρ ρ 2 c 0 (ρ) = (l = 0), c 0 (ρ) = ρ ρ 2 (l ) (4.4) (4.2) c n (ρ) (n ) l l I(ρ + l)c l (ρ) = J k,l (ρ)c k (ρ) I(ρ + l) = (ρ ρ 2 )(ρ + l ρ 2 ) c k (ρ) (k = 0,,, l ) ρ ρ 2 ρ ρ 2 c l (ρ) c n (ρ) (4.3) x(t; ρ) t 2 2 t 2 x(t; ρ) + q(t)t t x(t; ρ) + r(t)x(t; ρ) = c 0(ρ)I(ρ)t ρ = c 0 (ρ)(ρ ρ )(ρ ρ 2 )t ρ ρ t ρ ( ) [ t 2 2 ] t 2 ρ x(t; ρ) + q(t)t [ ] [ ] t ρ x(t; ρ) + r(t) ρ x(t; ρ) = (ρ ρ 2 )K(ρ, t)t ρ l = 0 K(ρ, t) = [2+(ρ ρ ) log t], l K(ρ, t) = [3ρ ρ 2ρ 2 +(ρ ρ )(ρ ρ 2 ) log t] ρ ρ 2 ( ) 0 [ ρ x(t; ρ)] ρ=ρ 2 (4.7) [ ρ x(t; ρ) ] ρ=ρ 2 = k=0 c n (ρ 2 )t n+ρ 2 log t + c n(ρ 2 )t n+ρ 2 = x(t; ρ 2 ) log t + c n(ρ 2 )t n+ρ 2. l c 0 (ρ) = ρ ρ 2 c 0 (ρ 2 ) = c (ρ 2 ) = = c l (ρ 2 ) = 0. n= c n(ρ 2 )t n+ρ2 t l+ρ2 = t ρ t ρ x(t; ρ ) c 0 (4.4) c 0(ρ) = 0 (l = 0), c 0(ρ) = (l ) 36

4.3 (Bessel ) t 2 x + tx + (t 2 α 2 )x = 0 (α 0) t = 0 ρ(ρ ) + ρ α 2 = 0 ρ = α, ρ 2 = α 4.2 x(t; ρ ) = t α c nt n (c 0 0) c = 0, 0 c 0 t α + [(α + ) 2 α 2 ]c t α+ + {[(α + n) 2 α 2 ]c n + c n 2 }t n+α = 0 n=2 [(α + n) 2 α 2 ]c n + c n 2 = 0 (n = 2, 3,...) (4.5) (α + n) 2 α 2 = n(2α + n) 0 (n 2) c = 0 c 3 = c 5 = = 0 37 c c 0 = 2 α Γ(α+) (4.5) c 2n = ( )n 2 α 2n c n!γ(+α+n). c = Bessel J α (t) = α Bessel J 0 (t) = ( ) n ( t ) 2n, J/2 (n!) 2 (t) = 2 ( ) n t ) α+2n (4.6) n!γ( + α + n)( 2 2 πt sin t, J /2(t) = 2 cos t. (4.7) πt J /2 (t) n!γ(+ 2 +n)22n+ = (2n)!! (n+ 2 )(n 2 ) 2 Γ( 2 )2n+ = (2n)!!(2n+)!! π = (2n + )! π sin t Taylor J /2 (t) ρ 2 = α ρ ρ 2 = 2α 4.2 (i) (4.6) J α (t) J α (t), J α (t) 38 2α 2α J α (t), J α (t) ( 4.2 (ii) c = 0 ) α : 4.2 (ii) x(t; ρ) (4.3) c n (ρ) (4.0) c (ρ) = 0, (ρ + n + α)(ρ + n α)c n (ρ) + c n 2 (ρ) = 0, n = 2, 3,..., (4.8) α = 0 c 0 (ρ) = (4.8) c 2n (ρ) = 0, c 2n (ρ) = ( ) n [(ρ + 2)(ρ + 4) (ρ + 2n)] 2 ρ c 2n(ρ) = 2c 2n (ρ) n k= ρ+2k. ρ 0 x (t) = J 0 (t) x 2 (t) = J 0 (t) log t n= ( ) n H ( n t ) 2n (n!) 2 (4.9) 2 H n = + 2 + + n (4.9) 2 0 Bessel 37 Γ(s) = R 0 e u u s, s > 0. Γ(s + ) = sγ(s) s, 2,... Γ(s) 38 α Γ( α + n) 37

α = m ( ) c 0 (ρ) = ( ) m (ρ + m)(ρ + m 2) (ρ + 2 m) (4.8) c 2n (ρ) c 2n (ρ) = ( ) n+m (ρ + m) m n k= (ρ + m 2k) 0 (ρ + m + 2l) n k=m+ n l= m l= n (ρ + m + 2l) n < m = m ( ) n+m n (ρ m + 2k) l= (ρ + m + 2l) n > m c 2n(ρ) ρ m x (t) = J m (t) x 2 (t) = 2 ( [ t m m 2) + (m n )! ( t n! 2 n=m+ ) 2n H ( m t 2m + m! 2) (4.20) c 2(n+m) ( m) = ( ) n m (H n m + H n ) ( ] t 2n + J m (t) log t (4.20) (n m)! n! 2) ( )n (n+m)! n! (cf. (4.6) α = m) (4.20) 2 m Bessel 4.4 4.2 ([Ku] p.37. 39 ) 2 2n+m () 2tx + x + tx = 0 (2) t 2 x + 3tx + ( + t)x = 0 (3) t 2 x + tx + 2tx = 0 (4) t 2 x + 2tx + tx = 0 (5) t 2 x + 5tx + 3( + t)x = 0 4.5 Legendre ( t 2 )x 2tx + α(α + )x = 0 (α ) ([Ku] p.37. 40 ) () x =, Legendre (2) x = (3) Legendre x = (t ) ρ c n(t ) n 4.6 () t 2 x + ( λ 2 β 2 t 2β + 4 α2 β 2) x = 0 ±α Bessel J α (t), J α (t) (cf. (4.6)) tj α (λt β ), tj α (λt β ) (2) Airy x tx = 0 x = t [ c J /3 ( 2 3 t3/2 ) + c 2 J /3 ( 2 3 t3/2 ) ] ([Ku] p.45.) 39 4.4 : (c 0, H 0 = 0) () x (t) = P (2) x (t) = t P ( ) n (n!) 2 t n P, x 2 (t) = x (t) log t 2 ( ) n 2 n H n (n!) 2 t n P, (4) x (t) = P 2 (5) x (t) = t P 40 P 4.5 : (3) x(t) = + ( ) n 8 n n!γ(n+ 3 4 ) t2n, x 2 (t) = t 2 ( ) n H n (n!) 2 t n P, (3) x (t) = ( ) n n!(n+)! tn, x 2 (t) = x (t) log t + t ˆ P ( ) n 3 n n!(n+2)! tn, x 2 (t) = 9x (t) log t t 3ˆ P + 3t + n= n= H n+h n 2 ( ) n!(n 2)! n 3 n t n P ( ) n 8 n n!γ(n+ 5 4 ) t2n, ( ) n 2 n (n!) 2 t n, x 2 (t) = x (t) log t ( ) n (H n +H n ) t n, n!(n )! n=2 (α+n) 2n 2 n (n!) 2 (t ) n ( t < 2 ), (a) n = a(a ) (a n + ). 38