(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b ) ax + b = t y = e t, t = ax + b dy dx = dy dt dt dx = d d dt et dx (ax + b) = et a = ae ax+b.. (.) f(x) dx x = g(t) x t, f(x) f(g(t)), dx dx dt dt f(x)dx = f(g(t)) dx dt dt. t a b x g(a) g(b) g(b) g(a) f(x)dx = b a f(g(t)) dx dt dt. 1

[ 1] e ax+b dx (a 0, b ). ax + b = t e ax+b = e t, x = t b a e ax+b dx = e t 1 a dt = 1 a eax+b. dx = dx dt dt = 1 dt a dt [ 2] 1 + t. 2 x = tan 1 t dx dt = 1 dx = dx 1 + t 2 dt dt = dt 1 + t = dx = x = tan 1 t. 2 dt 1 + t2 u vdx = uv uv dx, b u vdx = [uv] b a b a a uv dx. e a+b = e a e b, e a b = ea e b, eab = (e a ) b. log AB = log A + log B, log A B = log A log B, log Ak = k log A, A, B 0. a = e b b = log a e log a = a, log e b = b. (e x ) = e x, (log x ) = 1 x. 1 e x dx = e x, dx = log x, x y = e ax log x dx = x log x x. y 1 O a > 0 a < 0 x 2

sin 2 x + cos 2 x = 1, 1 + tan 2 x = 1 cos 2 x, cos(α + β) = cos α cos β sin α sin β, sin(α + β) = sin α cos β + cos α sin β ( (sin x) = sin x + π ) = cos x 2 ( (cos x) = cos x + π ) = sin x 2 (tan x) = 1 cos 2 x. ( cos xdx = sin x = cos sin xdx = cos x = sin sin 1 x = y sin y = x 1 < x < 1, π 2 < y < π 2. cos 1 x = y cos y = x 1 < x < 1, 0 < y < π. tan 1 x = y tan y = x < x <, π 2 < y < π 2. (sin 1 x) = x > 0 1 1 x 2, (cos 1 x) = x n m = m x n, x n 1 m = m, (n, m = 1, 2,... ) x n x 1 2 = x, x 1 2 = 1 x. (x α ) = αx α 1 (α; ). α = 1 2, ( ) 1 x = 2 x. x α dx = xα+1 (α 1; ). α + 1 dx x 1 dx = = log x. x 1, 1 x 2 (tan 1 x) = 1 1 + x. 2 x π 2 ). ( x π 2 ). 3

0 0.1 x, x 2 3x + 2 = 0 x,, x = 1 x = 2., ( x) ( y(x)), y (x) = 3y(x). x, y(x).,.. y(x) = Ce 3x (C )., ( x, y) ( u(x, y)) (u x (x, y), u y (x, y), ).. 0.2 (1) y (x) = 4x x, y = y(x). y = 2x 2 + C, C. (2) y (t) = 4y(t) t, y = y(t). y = Ce 4t, C. (3) u xx (x, y) + u yy (x, y) = 0. x y. u = u(x, y). u = log(x 2 + y 2 ). 4

0.3 0.3.1. t [s] 0 m [kg] mg t y(t) [m] t v(t) [m/s] t a(t) [m/s 2 ] y(t). (y, v.) mg v(t) = dy dt a(t) = d2 y dt 2. F (t) F (t) = ma(t) a(t). Newton. F F = mg y(t) my (t) = mg y (t) = g. y(t) = 1 2 gt2 + v(0)t + y(0) [m]., y(0), v(0) 0 y(t) = 1 2 gt2 [m/s] [ ]. [ ] t = 1, 5, 10, 20, 30. 5

,. k kv(t), F = mg kv(t). my (t) = mg ky (t) my (t) + ky (t) mg = 0. y = mg (( v(0) m ) ( ) ) 1 e k m t + t + y(0) [m] k g k., y(0), v(0) 0 y = mg ( t m )) (1 e k m t [m] k k. [ ]. t.. ( ) [mm] 0.02 0.15 0.5 1 2 3 [m/s] 0.01 2 0.5 2.2 4 6.2 7.8 6

0.3.2.. t [s], ky (t) m [kg], t y(t) [m]. (,.) 0 y(t), ky(t) [N]. k > 0. y(t) my (t) = ky(t). k y = C 1 cos m t + C 2 sin. k t m y = A sin (C 1, C 2, A, φ y(0), y (0) ) ( ) k m t + φ [ ]. 0.3.3 q q t I(t) L C I(t) I (t) + LCI(t) = 0. (.) I = C 1 cos LCt + C 2 sin LCt ( ) y = A sin LCt + φ (C 1, C 2, A, φ I(0), I (0) ). 7

0.3.4 t [h] ( ) y = y(t) [ ] u(t, t) = y(t + t) y(t) y(t). u(t, t) t t + t ( t ). u(t, t) t t u( t) (u(0) = 0 ) y(t + t) y(t) t = u( t) u(0) y(t) t. t 0 y(t) y (t) = u (0)y(t). u (0) = k y = y(0)e kt. k t = 0,. [ ].. ( ) 0 1630 1660 1800 1850 1900 1950 1970 2004 ( ) 2.5 4 4.45 8 10 16 22 36.2 63.3 8

. k ly(t) (l ) y (t) = (k ly(t))y(t) y = y(0) ( 1 l y(0)) e k kt + l y(0) k. [ ]. [ ],. 0.4 1... 2.,. 3.. 4.,. 9

1 1 x. x y = y(x) y (x), y (x),... F (x, y, y, y, ) = 0 y..,., y, y, y 1,., y, y, y, x., ( ).. 1.0 1 f(x, y) 2, y (x) = f(x, y) (1) 1. x x 0, y(x 0 ), y (x 0 ),. { y = f(x, y) (2) y(x 0 ) = a 0. (2). y = y(x) ( ) (x, y) f(x, y). (x, y) (1, f(x, y)) ( )., (x, y) f(x, y). (x, y). (x, y) f(x, y).,.,. x. 10

1.0 f(x, y), f y (x, y) xy D, (x 0, a 0 ) D. (x 0, a 0 ) (2). [ ] f(x, y), f y (x, y). [ ] (x 0, a 0 ). f(x, y) = y y 2, ( )..(. ) 2.5 2 2 1.5 1 1 0.5-1 -0.5 0.5 1-1 -0.5 0.5 1-0.5-1 -1 y = y y 2-1.5 y = y y 2. y = y 1 (x), y = y 2 (x) (2) 2. x > x 0. d dx (y 1(x) y 2 (x)) 2 = 2(y 1(x) y 2(x))(y 1 (x) y 2 (x)) = 2(f(x, y 1 ) f(x, y 2 ))(y 1 (x) y 2 (x)), f(x, y 1 ) f(x, y 2 ) = f y (x, ξ) (ξ y 1 (x) y 2 (x) ), f x (x, y) (x 0, a 0 ) C < 2C(y 1(x) y 2 (x)) 2 11

e 2Cx e 2Cx d dx (y 1(x) y 2 (x)) 2 2Ce 2Cx (y 1 (x) y 2 (x)) 2 < 0 = d ( e 2Cx (y 1 (x) y 2 (x)) 2) dx e 2Cx (y 1 (x) y 2 (x)) 2. e 2Cx (y 1 (x) y 2 (x)) 2 < e 2Cx 0 (y 1 (x 0 ) y 2 (x 0 )) 2 = 0 x > x 0 y 1 (x) = y 2 (x) (2). 1.1 1 ( ) a y (x) + ay(x) = 0 (3) 1. 1 (1) f(x, y) = ay. 1.1. (1.1) y(x) = Ce ax (C : ) (4). [ ] (4), (e ax ) = ae ax. (4). 1.0,. 1.2. a, x 0, y 0, y (x) + ay(x) = 0, (3) y(x 0 ) = y 0 (5) 12

y(x) = y 0 e a(x x 0) (6). [ ] (3) 1.1 y(x) = Ce ax (4),, (5) C. (4) x = x 0, y(x 0 ) = Ce ax 0 (5) y 0 = Ce ax 0 C = y 0 e ax 0. C (4) (6). 1.2 1 ( ) x, y(x), a, r(x). y (x) + ay(x) = r(x) (7) 1. f(x, y) = ay + r(x). (1) 1.3. r(x). (7) ( ) y(x) = r(x) e ax dx + C e ax (C : ) (8). [ ] (8) ( ( ) y (x) = r(x) e ax dx + C) e ax + r(x) e ax dx + C (e ax ) ( ) = (r(x) e ax ) e ax + r(x) e ax dx + C ( ae ax ) = r(x) a y(x) (8) (7). 13

(8), (7) 1.0,. [ 1.4] y (x) + 2y(x) = 2x. e 2x,. e 2x e 2x y (x) + 2e 2x y(x) = 2xe 2x. = ( e 2x y(x) ) e 2x y(x) = 2xe 2x dx + C. ( ) 1 2xe 2x dx = 2x 2 e2x, y = x 1 2 + Ce 2x. ( ) 1 2 2 e2x dx + C = xe 2x 1 2 e2x + C 1.3 1 y(x), f(x), r(x). y (x) + f(x)y(x) = r(x). (9) 1. r(x) 0, r(x) 0. (1) f(x, y) = f(x)y + r(x). 1.4. f(x). 1 y(x) = C e F (x) (C : ) (10). F (x) f(x). 1.5. f(x), r(x). 1 ( ) y(x) = r(x) e F (x) dx + C e F (x) (C : ) (11) 14

. F (x) f(x). e F (x) (10). ( [ 1.5 ] (11) (9). ) e F (x) = f(x) e F (x) (11) ( ( ) (e y (x) = r(x) e F (x) dx + C) e F (x) + r(x) e F (x) F (x) dx + C ) ( ) ( f(x) = r(x) e F (x) e F (x) + r(x) e F (x) dx + C ) e F (x) = r(x) f(x) y(x) (9). (11) (9) 1.0,. [ 1.8.] y (x) + 2 x y(x) = x, y(1) = 0. y 2x 2x dx = x 2 + C (C ) e x2. e x2 y (x) + 2xe x2 y(x) = xe x2 ( ). e x2 = 2xe x 2 = ( ) e x2 y(x) e x2 y(x) = ( x)e x2 dx + C C. x 2 = t ( x)e x2 dx = 1 2 ex2 + C C. e x2 y(x) = 1 2 + Ce x2 C,. x = 1 y(1) = 0 C = e 2 y(x) = e1 x2 1. 2 15,

= ( ) 1 (9). r(x) 0 (9) F (x) y(x) = C e (F (x) : f(x), C : ), C u(x) F (x) y(x) = u(x) e r(x) 0 (9).. 1.4 1 (7) r(x),,.,.. 1.6. ( ) 1 y (x) + f(x)y(x) = r(x). (9) F (x) y(x) = y p (x) + C e. y p (x) (9), F (x) f(x), C. C e F (x) (9) r(x) 0 y (x) + f(x)y(x) = 0 (12). [ ] y(x), y p (x) (9) y (x) + f(x)y(x) = r(x), (y p (x)) + f(x)y p (x) = r(x) 16

. (y(x) y p (x)) + f(x)(y(x) y p (x)) = 0, y(x) y p (x) (12). 1.4 F (x) y(x) y p (x) = C e. 1.6 (9) y p (12). f(x) a ( ) r(x),,,. [r(x) = n ] { n (a 0 ) y p (x) = n + 1 (a = 0 ). [ 1.10] y + 3y = x 2 1 (a). y p = αx 2 + βx + γ (a) y (y p ) + 3y p = (2αx + β) + 3(αx 2 + βx + γ) = 3αx 2 + (2α + 3β)x + β + γ = x 2 1. 3α = 1 2α + 3β = 0 β + γ = 1. α = 1 3, β = 2 9, γ = 7 27 y p = 1 3 x2 2 9 x 7 27., y + 3y = 0 (b) y = Ce 3x a y = 1 3 x2 2 9 x 7 27 + Ce 3x. 17

[r(x) = cos ωx, sin ωx ] y p (x) = α cos ωx + β sin ωx α, β. [ 1.12] y + 2y = cos x (a). y p = α cos x + β sin x (a) y (y p ) +2y p = ( α sin x+β cos x)+2(α cos x+β sin x) = (2α+β) cos x+(2β α) sin x = cos x. 2α + β = 1 α + 2β = 0. α = 2 5, β = 1 5 y p = 2 5 cos x + 1 sin x. 5, y + 2y = 0 (b) y = Ce 2x (a) y = 2 5 cos x + 1 5 sin x + Ce 2x. [r(x) = ke qx (q a) ] y p (x) = αe qx α. [ 1.13] y y = 2e 2x (a). y p = αe 2x (a) y (y p ) y p = (2α α)e 2x = 2e 2x. α = 2., y y = 0 (b) y = Ce x (a) y = 2e 2x + Ce x. 1.5, y (x) = f(x)g(y(x)) (f,g. ) (13), ( ). (13). 18

[ ] (Step 1) g(y 0 ) = 0 y 0., y(x) y 0 (13). (Step 2) y(x) = y 0. ((13) x y(x) y 0 x y(x) y 0.) (Step 2-1) (13) g(y(x)) : 1 dy g(y(x)) dx = f(x). (Step 2-2) x : 1 dy g(y(x)) dx dx = f(x) dx + C. (Step 2-3) x y dy g(y) = f(x) dx + C.. [ ] y (x) = y(x)(2 y(x)) (a). (a) (13) f(x) = 1, g(y) = y(2 y). y = 0, y = 2 (a). y 0, 2 y(y 2) 1 dy y(y 2) dx = 1. x 1 dy = y(y 2) dx dx = ( 1 = 2(y 2) 1 ) 2y = 1 2 log y 2 y = x + C. y y = 2Ce2x Ce 2x 1 1 dx = x + C. (C ). 1 y(y 2) dy dx = 1 2 log y 2 y + C.. y = 0 y = 2.. 19

1.6 ( ),. 1.6.1 ( ) y(x) y (x) = f x (f. ) [ ] u = y y u u, x y = (xu) = u + xu u = f(u) u x. [ ] y (x) = x (a). y(x) f(u) = 1 u u = 1 + u2 xu, u = ± C x 2 x x 2 + y 2 = C. 1.6.3 y (x) = f(ax + by(x) + c) (f, a, b, c. ) [ ] u = ax + by + c y u u,. [ ] y (x) = 1 x + y(x) (a). u = x + y u = 1 + 1 u u log u = x + C y log(x + y + 1) = C. 20

2 2 2.1 2 (2.1) y + ay + by = 0, a, b : 2.,.. 2.1.1 1 2 λ 1 = cos q 1 + i sin q 1, λ 2 = cos q 2 + i sin q 2 λ 1 λ 2 = (cos q 1 + i sin q 1 )(cos q 2 + i sin q 2 ) = cos(q 1 + q 2 ) + i sin(q 1 + q 2 ). λ 1 λ 2 λ 2 λ 1 q 1 q 2 q 1 O O O q 2. λ = p + iq (p, q ) e λ = e p (cos q + i sin q). e λ 1+λ 2 = e λ 1 e λ 2., λ = p + iq, x, x x e λx, d dx eλx = λ e λx, e λx dx = 1 λ eλx. λ 0., i. 21

2.1.2 C 1, C 2 y 1 (x), y 2 (x) (2.1) = C 1 y 1 (x) + C 2 y 2 (x) (2.1).. [ ] C 1 y 1 (x) + C 2 y 2 (x) (2.1) (C 1 y 1 (x) + C 2 y 2 (x)) + a(c 1 y 1 (x) + C 2 y 2 (x)) + b(c 1 y 1 (x) + C 2 y 2 (x)) = C 1 (y 1(x) + ay 1(x) + by 1 (x)) + C 2 (y 2(x) + ay 2(x) + by 2 (x)) = 0 (2.1). 2.1.3, a = OA, b = OB { a, b} ( ), O, A, B. k 1, k 2 k 1 a + k 2 b = 0 = k 1 = k 2 = 0., a = (a 1, a 2 ), b = (b 1, b 2 ) { a, b} a 1 b 1 a 2 b 2 0.. 1 {y 1 (x), y 2 (x),, y m (x)} k 1, k 2,, k m x k 1 y 1 (x) + k 2 y 2 (x) + + k m y m (x) = 0 = k 1 = k 2 = = k m = 0.. [ ] 1. {1, x, x 2,, x m }. 2. {e a 1x, e a 2x,, e amx }, (a 1,, a m. (2.1) 1. 22

2.1.4 2 (2.2) λ 2 + aλ + b = 0 (2.1),. a 2 4b D. [1] (2.1). [1-1] D > 0. λ 1, λ 2 ( 2 ) {e λ 1x, e λ 2x }. [1-2] D = 0. λ ( ) {e λx, x e λx }. [1-3] D < 0. λ 1, λ 2 ( 2 ) {e λ 1x, e λ 2x } [2] {y 1 (x), y 2 (x)} (2.1), (2.1) y(x) = k 1 y 1 (x) + k 2 y 2 (x), (k 1, k 2 )... A. [1] λ y = e λx (2.1). [2] λ, y = xe λx (2.1). [.] [1] y = e λx (2.1) (e λx ) + a(e λx ) + b(e λx ) = λ 2 e λx + aλe λx + be λx = (λ 2 + aλ + b)e λx = 0 (2.1). [2] λ λ = a 2, y = xeλx (2.1) (xe λx ) + a(xe λx ) + b(xe λx ) = (λ 2 + aλ + b)xe λx + (2λ + a)e λx = 0 23

y = xe λx. [ ] [1]. (2.5) { y + ay + by = 0, (1) y(0) = A, y (0) = B (2) 2 (2.1). (2). (2 2.) B. (2.5). y(0) = 0, y (0) = 0 y(x) 0. [.] y 1 (x), y 2 (x) (2.5). ( (y 1 y 2 ) 2 + (y 1 y 2) 2 ) = 2(y1 y 2 )(y 1 y 2) + 2(y 1 y 2)(y 1 y 2) = 2{(y 1 y 2 ) a(y 1 y 2) b(y 1 y 2 )}(y 1 y 2) ( ) < K (y 1 y 2 ) 2 + (y 1 y 2) 2 (K ), (y 1 (x) y 2 (x)) 2 + (y 1(x) y 2(x)) 2 < ekx ( (y 1 (0) y 2 (0)) 2 + (y 1(0) y 2(0)) 2 ). 0 y 1 (x) y 2 (x). [ ] C. {y 1 (x), y 2 (x)} {( ) ( )} y 1 (0) y 2 (0), y 1(0) y 2(0) [ ]., k 1 y 1 (x) + k 2 y 2 (x) 0, (k 1, k 2 ) (0, 0) k 1, k 2. k 1 y 1(x) + k 2 y 2(x) 0 ( ) ( ) ( ) y 1 (x) y 2 (x) 0 k 1 + k y 1(x) 2 = y 2(x) 0 x = 0.. 24

( ) ( ) ( ) y 1 (0) y 2 (0) 0 k 1 + k y 1(0) 2 =, (k y 2(0) 1, k 2 ) (0, 0) 0 k 1, k 2. Y (x) = k 1 y 1 (x) + k 2 y 2 (x), Y (0) = 0, Y (0) = 0 (2.1) B Y (x) 0. {y 1 (x), y 2 (x)}. [ ] [1-1] [1-3] D. {y 1 (x), y 2 (x)} (2.1) (2.1) y(x) k 1, k 2 y(x) = k 1 y 1 (x) + k 2 y 2 (x), (k 1, k 2 ). [ ] {( ) ( )} y 1 (0) y 2 (0), y 1(0) y 2(0) k 1, k 2. ( ) ( ) ( ) y 1 (0) y 2 (0) y(0) k 1 + k y 1(0) 2 = y 2(0) y (0) Y (x) = k 1 y 1 (x) + k 2 y 2 (x), Y (0) = y(0), Y (0) = y (0) (2.1) B Y (x) y(x). y(x) = k 1 y 1 (x) + k 2 y 2 (x). [ ] [2]. 25

2.1.5 D < 0, D < 0 y(x) = k 1 e λ 1x + k 2 e λ 2x, (k 1, k 2 ).,. a, b λ 1 = A+iB, λ 2 = A ib (A, B ). A, B {e Ax cos Bx, e Ax sin Bx} y = C 1 e Ax cos Bx + C 2 e Ax sin Bx (C 1, C 2 ).. [ ] 1 2 (eλ 1x + e λ 2x ) = e Ax cos Bx (= y 1 (x) ), 1 2i (eλ 1x e λ 2x ) = e Ax sin Bx (= y 2 (x) ) ( )2., y 1 (0) y 2 (0) y 1(0) y 2(0) = 1 0 D A B = B = 0 2. [2]. 26

2.3 2 (2.12) y + ay + by = r(x), r(x) : 0 2. 2.3.1 (2.12) y p (x) (2.12).. y = y p (x) + (2.1) [ ] y(x) (2.12). Y (x) = y(x) y p (x) Y (x)+ay (x)+by (x) = (y (x)+ay (x)+by(x)) (y p(x)+ay p(x)+by p (x)) = r(x) r(x) = 0 Y (x) (2.1). 2.3.2 [ 1] 2 y i (x) (i = 1, 2) y + ay + by = r i (x), i = 1, 2., C 1, C 2 Y = C 1 y 1 (x) + C 2 y 2 (x) Y + ay + by = C 1 r i (x) + C 2 r 2 (x). [ ]. = (C 1 y 1 (x) + C 2 y 2 (x)) + a(c 1 y 1 (x) + C 2 y 2 (x)) + b(c 1 y 1 (x) + C 2 y 2 (x)) = C 1 (y 1 + ay 1 + by 1 ) + C 2 (y 2 + ay 2 + by 2 ) = C 1 r 1 + C 2 r 2 = 27

[ 2] (i) λ, y p (x) = A e λx (a) y + ay + by = ke λx (b) A. (ii) λ, D 0, y p (x) = Ax e λx (c) (b) A. [ ] (i) (a) (b) = (A e λx ) + a(a e λx ) + b(a e λx ) = A(λ 2 + aλ + b)e λx λ 2 + aλ + b 0. (i) (c) (b) = (Ax e λx ) + a(ax e λx ) + b(ax e λx ) = A{(2λ + a)e λx + (λ 2 + aλ + b)xe λx } λ 2 + aλ + b = 0, 2λ + a = D 0. [ 3] y(x), r(x), Y (X) = Re y(x), Z(X) = Im y(x). y(x) y + ay + by = r(x), (a), Y (x), Z(x) Y + ay + by = Re r(x) Z + az + bz = Im r(x). [ ] y = Y + iz, r(x) = Re r(x) + iim r(x) (a) = (Y + iz) + a(y + iz) + b(y + iz) = (Y + ay + by ) + i (Z + az + bz) (a) = Re r(x) + iim r(x),. 28

k, k 1, k 2, α, β. (2.12) y p (x). 1. r(x) = k e λx, (λ ) (a) λ y p (x) = Ae λx. (b) λ, D = a 2 4b 0 y p (x) = Axe λx. 2. r(x) = k cos βx k sin βx, (a) ±iβ y p (x) = A cos βx + B sin βx. (b) ±iβ y p (x) = A x cos βx + B x sin βx. 3. r(x) = k e αx cos βx k e αx sin βx, (a) α ± iβ y p (x) = A e αx cos βx + B e αx sin βx. (b) D 0 α ± iβ y p (x) = A xe αx cos βx + B xe αx sin βx. 4. r(x) = n, (a) a 0, b 0 (b) a 0, b = 0 (c) a = 0, b = 0 y p = n. y p = n + 1. y p = n + 2. (2.12) A, B,. [ ] 1 [ 2]. 3 λ = α + iβ e αx cos βx = Ree λx, e αx sin βx = Ime λx [ 2, 3]. 4. 29

2.3.3,.. C. (i). Ids I B r P Ids db Φ = S Ids P db = k Ids r r 3. (Biot-Savard ) C, C B = k Ids r r 3 C, C S.. B S B n ds (n S ). Φ S C. Φ = LI (L )., ( ) Φ, C Φ E. E = dφ dt. (Lenz ). 2 E = L di dt. ( ). 30

(ii). +q E q E. ±q E E = q C (C ).. (iii). R E I. E = IR.. E = sin ωt, t I(t). L, C, R,,, E L, E C, E R, q. E L = LI : ( ) E C = q C : ( ) E C E R = IR : ( ) q q I = q, E L L C I R E R E L + E C + E R + E = 0 : ( ) E LI RI + q C = E = sin ωt E, I = q (P ) LI + RI + 1 I = ω cos ωt C. 31

(P ), (P ). (P 0 ) LI + RI + 1 C I = 0 Lλ 2 + Rλ + 1 C = 0. D ( D = R 2 4L ) < 0 C.. λ = R 2L ± Di, { e R 2L t cos Dt, e R 2L t sin Dt }. 2.3.1 (P ) I(t) = I 0 (t) + I p (t),. I 0 (t) = C 1 e R 2L t cos Dt + C 2 e R 2L t sin Dt, I p (t) (P ) ((P 0 ) ). I 0 (t) < ( C 1 + C 2 )e R 2L t 0 (t ) I 0 (t), I p (t). I p (t). Ĩ ( P ) LĨ + RĨ + 1 C Ĩ = ωeiωt. ( P ). ( P ) A Ĩ = Ae iωt ( Lω 2 + irω + 1 C ) e iωt = ωe iωt 32

A = ω Lω 2 + irω + 1 C = 1 Lω + ir + 1 Cω ( Z ). 1 Z = 1 π Z ei(φ ( P ) 2 ), φ tan φ = Ĩ = 1 Z eiωt = 1 π Z ei(ωt+φ 2 ) 1 Lω cω R., cos ωt = Ree iωt 2.3.2. [ 3] ReĨ I p = 1 Z cos(ωt + φ π 2 ) = 1 sin(ωt + φ) Z (P ). I = E/R. Z = ( 1 Cω Lω)2 + R 2., ω ω 1 LC. L = C = 1 ω I 1. 1 I_1 where L=C=R=1,omega=0.25,0.5,1 0.5 10 20 30 40-0.5-1 33

1 I_1 where L=C=R=1,omega=1,2,8 0.5 10 20 30 40-0.5-1 34

3.1 3.1.1 0 f(t) dt = F (t) = f(t) 0 f(t) dt = 3.1.2 f(t) (0, ) t, s. F (s) = 0 e st f(t) dt s, F (s) f(t) L(f(t)), L(f), L(f)(s),. (5.1) L(f)(s) = 0 e st f(t) dt. (5.1) L t f(t) s F (s),. f(t) :,. : M, α f(t) < Meαt, (t > 0), α < s s F (s), Laplace.. 3.1.3 L(f(t)) = F (s) t f(t), ( ). L 1 (F (s)) = f(t) L 1. 35

3.1.4 [ 1. ] f 1 (t), f 2 (t),f 1 (s), F 2 (s) c 1, c 2 L(c 1 f 1 (t) + c 2 f 2 (t)) = c 1 L(f 1 (t)) + c 2 L(f 2 (t)), L 1 (c 1 F 1 (s) + c 2 F 2 (s)) = c 1 L 1 (F 1 (s)) + c 2 L 1 (F 2 (s)). : [ 1..] a : L(e at ) = 1 s a (s > Re a ) L(1) = 1 s ( ) ( ) 1 1 L 1 = e at L 1 = 1 s a s [ 2..] a L(f(t)) = F (s) (s > α ) L(e at f(t)) = F (s a) (s > α+a ) : 36

[ 2..] n = 1, 2, : L(t n ) = n! s n+1, (s > 0 ) n! L(e at t n ) =, (s > a ) (s a) n+1 ( ) ( ) 1 L 1 = tn s n+1 n!, 1 L 1 = eat t n. (s a) n+1 n! [ 3. Heaviside.] { 0 (t < λ ) H λ (t) = 1 (t > λ ) λ > 0 Heaviside. : L(H λ ) = e λs s (s > 0.) 37

3.1.5 ( ) (I) f(t) L(f (t)) = s L(f(t)) f(0), (s ). (II) f(t) 2 L(f (t)) = s 2 L(f(t)) s f(0) f (0), (s ). : 38

3.2 3.2.1 { y (t) + y (t) 6y(t) = 6e t, (1) y(0) = 0, y (0) = 1 (2). 1 (1) L(y (t)) + L(y (t)) 6L(y(t)) = 6L(e t ). L(y (t)) = s 2 L(y(t)) s y(0) y (0), L(y (t)) = s L(y(t)) y(0),, L(e t ) = L(y(t)) = Y (s) (s 2 + s 6)Y (s) sy(0) y (0) y(0) = 2 (2) (s 2 + s 6)Y (s) =. Y. 3 Y (s) Y (s) = 39

.. = 1 1 2 s + 3 + 3 1 2 s 1 1 s 2 4 (3) y(t) = L 1 (Y (s)) = 1 2 e 3t + 3 2 et e 2t (1), (2). 5. (3) y (t) = 3 2 e 3t + 3 2 et 2 e 2t, y (t) = 9 2 e 3t + 3 2 et 4 e 2t. (1) (3) (1). (2). 40