: January 14, 28..,. A = (a ij ) ij m n. ( ). B m n, C n l. A = max{ a ij }. ij A + B A + B, AC n A C (1) 1. m n (A k ) k=1,... m n A, A k k, A. lim k A k = A. A k = (a (k) ij ) ij, A k = (a ij ) ij, i, j lim k a (k) ij = a ij lim k A k = A. lim k A k = A. A A k (k ) n A, A e A : e A := E + A + A2 2! + + Ak k! + =. e x. f(x) = k= a nx k x < r. A r. f(a) = a k A k A, P P 1 AP = diag(α 1,..., α n ) α i. k= k= A k k!. P 1 f(a)p = a k P 1 A k P = a k diag(α k 1,..., α k n) = diag(f(α 1 ),..., f(α n )) f(a) f(a) = P diag(f(α 1 ),..., f(α n )) P 1. ( 1.). : B : 1
R ( ) f 1 (t),..., f n (t). ( )c i (i = 1,..., n),, c 1 f 1 (t) + + c n f n (t) = = c 1 = = c n =.. W (f 1,..., f n )(t) := f 1 (t)...... f n (t) f 1(t)...... f n(t).... 1 (t)...... f n (n 1) (t) f (n 1).. ( W (f 1,..., f n ).) 3. 3 n. 1,2. 1. p(t), q(t), r(t) R. 3 d 3 f(t) 3 + p(t) d2 f(t) V.. + q(t) df(t) + r(t)f(t) = (2) (i) V.. (f, g V f + g V. ) (ii) a, b, c, f() = a, f () = b, f () = c f(t) V. (iii) V 3. 3 f 1 (t), f 2 (t), f 3 (t) V, f(t) V f(t) = c 1 f 1 (t) + c 2 f 2 (t) + c 3 f 3 (t) (c i ). 2. f(t), g(t), h(t) (2). f(t), g(t), h(t), t = t det W (f, g, h)(t ). t det W (f, g, h)(t ) t det W (f, g, h)(t). : B : 2
p(t), q(t), r(t). 3. p, q, r. 3 d 3 f(t) 3 + p d2 f(t) + q df(t) + rf(t) = (3) V. ϕ(x) = x 3 + px 2 + qx + r. (.). (i) ϕ(x), α, β, γ, V e αt, e βt, e γt 3. V = {c 1 e αt + c 2 e βt + c 3 e γt c i C}. (ii) ϕ(x) 2, α 2, β, V e αt, te αt, e βt 3. V = {c 1 e αt + c 2 te αt + c 3 e βt c i C}. (iii) ϕ(x) 3, α 3, V e αt, te αt, t 2 e αt 3. V = {c 1 e αt + c 2 te αt + c 3 t 2 e αt c i C}. 1. p, q, r. ϕ(t) 3,,. α β = a + bi β = a bi.,, e αt, e at cos bt = eβt + e βt, e at sin bt = eβt e βt 2 2i 3. e ix = cos x+i sin x,.. (2) d f(t) f (t) f (t). F (t) := := f(t) f (t) f (t) f (t) f (t) f (t) =, A(t) = 1 1 r(t) q(t) p(t) 1 1 r(t) q(t) p(t) f(t) f (t) f (t) : B : 3
, F (t) = A(t)F (t). (.) p, q, r A(t) = A. f (t) = cf(t) f(t) = e ct, F (t) = AF (t), F () = v, F (t) = e ta v. d (eta v) = A e ta v. (A A(t).) 4. v, v = t (f(), f (), f ()) (3) F (t) = e ta v. 4 3, 4 A, F (t) = P e t(p 1 AP ) P 1 v. 1.. 5. p(t), q(t), r(t), b(t) R. 3 d 3 f(t) 3 + p(t) d2 f(t) + q(t) df(t) + r(t)f(t) = b(t) (4) f (t). f (t) b(t) d 3 f(t) + p(t) d2 f(t) + q(t) df(t) + r(t)f(t) = 3. 2. b(t), (4),. Ax = b, x, x Ax =. (4).. (Examples) ( ) 1 2 1. 2 A = e A. 1 4 : B : 4
K115 1(. ) ( ) 2, 3 2 1, P = P 1 2 AP =. 1 1 3 P 1 e A P = P 1 n= A n n! P = n= (P 1 AP ) n n! = ( ) ( ) 1 2 n e 2 =. n! 3 n e 3 n= ( ) ( ) e A e 2 = P P 1 2e 2 e 3 2e 2 + 2e 3 =. e 3 e 2 e 3 e 2 + 2e 3 2. f(t) f() = 1, f () = 3. d 2 f(t) 3 df(t) + 2f(t) =. 1. 3( 2 ). ϕ(x) = x 2 3x+2 = (x 1)(x 2), 3( 2 ) c 1 e t +c 2 e 2t. ϕ(x) α = 1, 2 d 2 e αt 3 deαt + 2e αt = (α 2 3α + 2)e αt = f(t) = c 1 e t +c 2 e 2t. f() = 1 c 1 +c 2 = 1, f () = 3 c 1 + 2c 2 = 3. c 1 = 1, c 2 = 2. ( 1 (ii) 2 ) f(t) = e t + 2e 2t. 2.. A = ( ) ( ) f(t) f(t)., = e At 1 g(t) 3 ( 1 2 3 ) e At ( 1 3 ), d eat = Ae At ( ) ( ) f (t) f(t) = A. g (t) g(t) g(t) = f (t), g (t) = 2f(t) + 3g(t). f (t) 3f (t) + 2f(t) =. ( ) ( ) f() 1 = g() 3 ( ) f() = 1, f () = g() = 3. e At 1 1 e At 3, 1. (A.) : B : 5
3., ϕ(x) = x 2 3x + 2 α, β f (t) αf (t) = β(f (t) αf(t)). F 1 (t) = f (t) αf(t), F 1(t) = βf 1 (t). F 1 (t) = C 1 e βt. F 2 (t) = f (t) βf(t) F 2 (t) = C 2 e αt. F 1 (t), F 2 (t) f (t) F 2 (t) F 1 (t) = (α β)f(t) = C 2 e αt C 1 e βt. f(t) = c 1 e αt + c 2 e βt. 1. 3. 1, 2. 3. 3.. f(t) f() = 1, f () = 3 d 2 f(t) + df(t) + f(t) =. x 2 + x + 1 α = 1+ 3i β = 1 3i. ( 2 2 ) f(t) = c 1 e αt + c 2 e βt. f() = 1 c 1 + c 2 = 1. f () = 3 αc 1 + βc 2 = 3. c 1 = 1 7 2 6 3i, c2 = 1 + 7 2 6 3i. f(t) = eαt + e βt + 7 ( ) 3 e αt e βt 2 3 2i ( 3 = e 1 2 t e 2 it + e 3 ) ( 2 it + 7 3 3 1 2 3 e 2 t e 2 it e 3 ) 2 it 2i ( ) = e 1 2 t cos 3t/2 + 7 3 ( ) 2 t sin 3t/2. 1 3 e. 1 ( ) ( ) c 1 e 1 2 t cos 3t/2 + c 2 e 1 2 t sin 3t/2. 4. (1) d2 f(t). 2α df(t) + α 2 f(t) =. (2) d2 f(t) 2 df(t) + f(t) = t 2. (1) ϕ(x) = x 2 2αx + α 2 = (x α) 2. c 1 e αt + c 2 te αt., e αt 1. te αt d 2 (te αt ) 2α d(teαt ) + α 2 (te αt ) = ϕ (α)e αt + ϕ(α)te αt =. : B : 6
e αt, te αt ( 3), 1 c 1 e αt + c 2 te ( αt. ) 1 1 2. A =, α 2 2α v e At v. (2) f (t) f(t) f(t) f (t) α = 1 (1), f(t) = f (t) + c 1 e t + c 2 te t. f (t).. 1... 2 2. f (t) = at 2 + bt + c (2). 2a 2(2at + b) + (at 2 + bt + c) = t 2, a = 1, b = 4, c = 6. f (t) = t 2 + 4t + 6 (2). f(t) = t 2 + 4t + 6 + c 1 e t + c 2 te t. 2.. f(t) = g(t)h(t) g(t). (2) h(t) (g (t) 2g (t) + g(t))h(t) + 2(g (t) g(t))h (t) + g(t)h (t) = t 2. h(t) h (t) g(t) = e t., g(t) = e t e t h (t) = t 2, h (t) = e t t 2. h(t) e t (t 2 +4t+6). f (t) = g(t)h(t) = t 2 +4t+6. 3.. D = d, D2 = d2, (1 D) 2 = (1 D) 2 f(t) = (D 2 2D + 1)f(t) = t 2. 1 (1 D) 2 = ( 1 + D + D 2 + )2 = 1 + 2D + 3D 2 + f(t) = (1 + 2D + 3D 2 + )t 2 = t 2 + 4t + 6.,, t 2 +4t+6., (2). 4... ( ) G(t) = W (e t, te t e t te t ) = e t e t + te t ( ) ( ) ( ) f (t) t ( ) ( ) = G(t) G(t) 1 = e t 1 t t e t 1 + t t f (t) t 2 1 1 + t 1 1 t 2 ( ) ( ) ( ) = e t 1 t t ( ) t 3 e t t = e t 1 t 1 1 + t t 2 e t t3 e t. 1 1 + t t2 e t : B : 7
f (t) = e t t3 e t + te t t2 e t.. 4. 1. (2) t 2 e t, a n t n (2), a n. a n. (.) (2), a sin x + b cos x, (2) a, b. 2..,. 3.,. (2),. 4.,. (Problems) 1. (a) (1). (b) (1) e A. 2. A e A. ( ) 1 2 (1) 1 (2) ( ) 1 2 1 3. α, β, γ.. (1) e αt, e βt, e γt.. (2) e αt, te αt, e βt.. (3) e αt, te αt, t 2 e αt.. 4. f(t), g(t), h(t) 1 (2). (1) a(t), b(t), c(t) 3 1. a(t), b(t), c(t) a(t), b(t), c(t).. d a(t), b(t), c(t) = a (t), b(t), c(t) + a(t), b (t), c(t) + a(t), b(t), c (t).. : B : 8
(2) w(t) = det W (f, g, h) w (t) = p(t)w(t). w(t) = w(t )e R t t p(t). (3) 2. 5.. (1) { f (x) = 7f(x) 4g(x) g (x) = 12f(x) + 7g(x) (2) { f (x) = 75f(x) + 16g(x) g (x) = 36f(x) + 77g(x) 6. g(t). (1) f() = 1, f () = 4. d 2 f(t) + df(t) 2f(t) = g(t). (1) g(t) = (2) g(t) = t (3) g(t) = sin t 7.. d 3 f(t) 3 3 d2 f(t) + 4 df(t) 12f(t) = t. 8. A. (1). det(e ta) = exp ( Tr A n TrA A, : Tr A := i a ii. (..) :. (2) A ( n A n = ), n Tr A n =.,,., CT,.. n=1 n tn ) 9. α,. d 2 f df 2x dx2 dx f = αf. : B : 9
V = {ax 2 + bx + c a, b, c C}. ( α.) : F : d 2 dx 2x d 2 dx 1, F V V. F f = αf, F α. (. ). 1.,, : (1), 1/2, 1/2. (2), 2/3, 1/3. (3),. x %, y %. ( (3) x 1 + y 1 = 1.). (a) n x n %, y n %, ( ) ( ) ( ) x 1 2 n+1 x = 2 3 n y n+1 y n. (b) 365 x 365, y 365? (c) (x := lim n x n y := lim n y n, t (x, y ) (a) 1. ),,.,,, 1/3,, 1/4, 1/2, 1/4,,,, 1/3. 1 2 1 3 : B : 1
(Appendix) 1 4 f (t).,. 4 (2). 1 1 (i). (iii) (ii). (ii). (ii). f(t) (2), 3 F (t) = A(t)F (t) (5). (5) F (t) (2) f(t). (5). (2). f(t) (2).. (5) t v := F () = F (t) = v + f() f () f () A(t)F (t) (6). (. ) G(t) G() = v (2), F (t) G(t), v F (t) (6) F (t). t. (t <.) t < R R. A(t) t R A(t) M M. v = (6) (1) F (t) A(t)F (t) 3M F (t). (7) h(t) := e 3Mt t F (t). h(t). (7) h (t) = 3Me 3Mt F (t) + e 3Mt F (t). h(t) h() =. h(t) F (t) =. F (t), F (t). (2) (6) F (t). F (t) = (F n (t)) n F n+1 (t) = v + A(t)F n (t) (8) : B : 11
. F (t) = lim n F n (t), (6). (F n (t)) n. R t < R. A(t) M M F n+1 (t) F n (t) v 3Mt n n!. t. (t <.) n =. n = k 1, (8) F k+1 (t) F k (t) = A(t)(F k (t) F k 1 (t)) A(t)(F k (t) F k 1 (t)) 3M F k (t) F k 1 (t) (3M)k+1 k! n = k. m n t k = (3Mt)k+1 (k + 1)!. F m (t) F n (t) = (F m (t) F m 1 (t)) + (F m 1 (t) F m 2 (t)) + + (F n+1 (t) F n (t)) (F m (t) F m 1 (t)) + (F m 1 (t) F m 2 (t)) + + (F n+1 (t) F n (t)) m 1 k=n 3Mt k k! m 1 k=n 3MR k k! (m, n ). t R F m (t) F (t). F n(t) = A(t)F n 1 (t) F n(t), F (t) F (t) = A(t)F (t). F () = F n () = v. a(t), b(t), f (t) = a(t)f(t) + b(t), b(t) g (t) = a(t)g(t), f(t) = g(t) g 1 (t)b(t) t. (.) g(t) = R t t e a(t).. f(t) (4), 3, F (t) = A(t)F (t) + b(t) (9). b(t) = t (,, b(t)). (9) F (t) (4). (9). (2) g 1 (t), g 2 (t), g 3 (t). (p, q, r.) G(t) = W (g 1, g 2, g 3 )(t). 2 G(t) t. G (t) = A(t)G(t). F (t) = G(t) G(t) 1 b(t) t. (9). ( (A(t)B(t)) = A (t)b(t) + A(t)B (t).) : B : 12