Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mph.cs.msu.ru/stud/2009_ODU-1-RazgulinDenisov.pdf Äàòà èçìåíåíèÿ: Tue Nov 24 15:40:53 2009 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 22:47:52 2016 Êîäèðîâêà: IBM-866

. ..

.. , ..



1



2009 .


" " , . .. " " . c . .., 2009 . c .. , .. , 2009 .




3



1 1.1 . . . . . . . . 1.2 , . . . . . . 1.2.1 . . . . . . . . . . . 1.2.2 . . . . . . . . . . . . 1.3 , . . . . . . 1.4 . . . . . . . . . . . . . . . . . . . 1.4.1 . . . . . . . . . . 1.4.2 . . . . . . . 1.4.3 . . . . . . . . . . . . . 2 2.1 , . . . . . . . . . . . . . . . 2.1.1 . . . . . . . 2.1.2 - . . . . . . . . . . . . . 2.1.3 . . . . . . . . . . . . . . . . . . . 2.1.4 . 2.1.5 . . . . . . . . . . . . . . . . . . . . . . 2.2 , . . . . . . . . . . . . . 2.2.1 . . . . . . . . . 2.2.2 . . . . . . . . . . . . . . . . . . . . . .

.

7 7

. 10 . 10 . 12 . 13 . . . . 15 17 19 22 25 . . . . . 25 25 27 29 30

. 31 . 36 . 36 . 39


4 2.2.3 2.2.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n- . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 . . . . . . . . . . . . . . . . 2.3.3 . . . . . . . . . 2.3.4 n- . . . . . . . . . . . . . 2.3.5 n- . 2.3.6 n- . . . . ( ) 41 42

2.3

45 45 46 48 52 54 55 55

2.4

3 3.1 n- . . . . . . . . . . 3.2 n- . . . . . . . . . . . . . . . . . . . . . . . . 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 . . . . . . . . . . . 3.4.1 . . . . . . . . . . . . . . . . . 3.4.2 . 3.4.3

61

61 65 67 67

69 71 71 72 74


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 . . . . . . . . . . n . . . . . . . . . . . . . . . . . 3.5.1 . . . . . . . . . . . . . . . . 3.5.2 - . . . . . . . . 3.4.4 3.4.5

5 . 75

. 77

. 81 . 83 . 83 . 87

3.5

4 4.1 . . . . . . . . . . . . 4.1.1 . . . . . . . . . . . . 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 . . . . . . . . . . . . . . . . . . 4.3.2 . . . 4.3.3 , . . . . . . . . . . . . . . 4.4 . . . . . . . . . . . . . . . . . . 4.4.1 , . .

89 89 89 90 93 93

95 96 96 97 99

101 102


6 4.4.2 4.4.3 . . . . .

, 103 . . . . . . . . . . . . . 106

A 108 A.1 . . . . . . . . . . . . . . . . . 108 A.2 . . . . 109 B 112 B.1 - . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.2 . . . . . . . . . . . . . . . . 114 B.3 . . . 116 B.4 , . . . . . . 117 B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 122


1.1.

7

1
1.1.
, . . 1.1.1. y (t) , y (t) + (y (t))2 - et y (t) = 1 + t, a t b.

1.1.2. u(t, x) , utt (t, x) + ut (t, x) = (t2 + x)u(t, x), a t b, c x d.

1.1.3. u(t, x) , ut (t, x) - uxx (t, x) + u(t, x) = 0, a t b, c x d.

, , . , , . , 1.1.1 1.1.2, , 1.1.3 í . . , , . y (t) F (t, y (t), y (t)) = 0, t [a, b],


8

1.

F (t, y , p) í . n- y (t) F (t, y (t), y (t), . . . , y
( n)

(t)) = 0,

t [a, b],

F (t, y , p1 , . . . , pn ) í n + 2 . n- , , y
( n)

(t) = F (t, y (t), y (t), . . . , y

(n-1)

(t)),

t [a, b],

(1.1)

F (t, y , p1 , . . . , pn-1 ) í n + 1 . . fi (t, y1 , y2 , . . . , yn ), i = 1, 2, . . . , n. y1 (t), . . . , yn (t) y1 (t) = f1 (t, y1 (t), y2 (t), . . . , yn (t)), t [a, b], y2 (t) = f2 (t, y1 (t), y2 (t), . . . , yn (t)), t [a, b], (1.2) ... yn (t) = fn (t, y1 (t), y2 (t), . . . , yn (t)), t [a, b]. (1.1) (1.2). , y (t) (1.1). y1 (t) = y (t), y2 (t) = y (t), ... y
n-1

(t) = y

(n-2)

(t),

yn (t) = y

(n-1)

(t).

y1 (t), = y1 (t) y2 (t) = ... yn-1 (t) = = yn (t)

. . . , yn (t) y2 (t), y3 (t), t [a, b], t [a, b], (1.3) yn (t), t [a, b], F (t, y1 (t), y2 (t), . . . , yn (t)), t [a, b].

. y1 (t), . . . , yn (t) (1.3), y (t) = y1 (t) (1.1).


1.2.

9

. 1.1. 1.1.4: í (), í ().

(1.1) (1.2) . . (y1 (t), y2 (t), . . . , yn (t)) (1.2) n + 1 (t, y1 , y2 , . . . , yn ). (t, y1 (t), y2 (t), . . . , yn (t)) . (y1 , y2 , . . . , yn ) , (y1 (t), y2 (t), . . . , yn (t)) í . 1.1.4.

y1 (t) = -y2 (t), t [0, 4 ], y2 (t) = y1 (t), t [0, 4 ]

y1 (t) = cos t, y2 (t) = sin t. (t, y1 , y2 ) , , í (. . 1.1).


10

1.

1.2. ,
. . 1.2.1. , x. , f (t), t. t x(t). , d2 x = f (t). dt2 (1.4)

, f (t) x(t). (1.4)
t

x(t) =
t0 t0

f ()dd + c1 + c2 t,

(1.5)

t0 - , c1 c2 í . (1.5) , (1.4) x(t). . , x(t) t0 , x0 = x(t0 ) v0 = x (t0 ). c1 = x0 , c2 = v0 x(t) . (1.4) . , ,


1.2.

11

, x(t) x (t), , x(t), d2 x = f (t, x(t), x (t)), dt2 f (t, x, p) í . . - r(t) = (x(t), y (t), z (t)). ï , , . - ïï f (t, r(t), r (t)) = (f1 (t, r(t), r (t)), f2 (t, r(t), r (t)), f3 (t, r(t), r (t))). ï ï ï ï ï ï ï r(t) ï d2 r ï ïï = f (t, r(t), r (t)). ï dt2 , x(t), y (t), z (t) d2 x = f1 (t, x(t), y (t), z (t), x (t), y (t), z (t)), dt2 d2 y = f2 (t, x(t), y (t), z (t), x (t), y (t), z (t)), dt2 d2 z = f3 (t, x(t), y (t), z (t), x (t), y (t), z (t)), dt2 fi (t, x, y , z , u, v , w), i = 1, 2, 3 í . . u(t) = x (t), v (t) = y (t), w(t) = z (t).

x(t), y (t),


12 z (t), u(t), v (t) w(t) x y z u v w (t) (t) (t) (t) (t) (t) = = = = = =

1.

u(t), v (t), w(t), f1 (t, x(t), y (t), z (t), u(t), v (t), w(t)), f2 (t, x(t), y (t), z (t), u(t), v (t), w(t)), f3 (t, x(t), y (t), z (t), u(t), v (t), w(t)).

, t0 , x(t0 ), y (t0 ), z (t0 ), u(t0 ), v (t0 ) w(t0 ). 1.2.2. . . . , , t u(t). u(t) , . u(t), du = au(t) - bu(t), dt (1.6)

a í , b í . , u(t). (1.6) u(t) = C exp (a - b)t , C í . , u0 = u(t0 ).


1.3. (1.6) u(t) = u0 exp (a - b)(t - t0 ) .

13

, : . u(t), v (t). , , . , , . : u (t) = au(t) - bu(t)v (t), a b í . , , , . : v (t) = cu(t)v (t) - dv (t), c d í . , u(t) v (t) u (t) = au(t) - bu(t)v (t), v (t) = cu(t)v (t) - dv (t). t0 u0 = u(t0 ) v0 = v (t0 ).

1.3. ,
, y (t) = f (t, y (t)), (1.7)


14

1.

. 1.2. y (t) = f (t, y (t)).

f (t, y ) D (t, y ). (1.7). 1.3.1. y (t) (1.7) [a, b], : 1. y (t) C 1 [a, b]; 2. (t, y (t)) D t [a, b]; 3. y (t) = f (t, y (t)) t [a, b] . C n [a, b] n N n [a, b] , C [a, b] . y (t) í (1.7) [a, b]. (t, y (t)), t [a, b]. . , . (t0 , y (t0 )) (1, f (t0 , y (t0 )) (. . 1.2). (1.7) C y (t, C ), , . 1.3.1. y (t) =
3

y 2 (t).

(1.8)


1.4.

15

. 1.3. y0 (t) = 0.

y (t) = (t - C )3 , 27 (1.9)

C í . (1.8) y0 (t) = 0. , (1.9) C . (1.7) , , (1.7). , . 1.3.1 y0 (t) = 0 , (t0 , 0) (t - t0 )3 (. , y (t, t0 ) = 27 . 1.3).

1.4.



16

1.

. 1.4. 1.4.1 : y1 (t) = - C 2 - t2 .

C 2 - t2 y2 (t) =

t y , , y t. , , . , y = y (t), t = t(y ) , , t = ( ), y = ( ), í . . 1.4.1. y (t) = - t . y (t) (1.10)

[-C + , C - ] 0 < < C y1 (t) = C 2 - t2 , y2 (t) = - C 2 - t2 . , [-C, C ], t C t -C . (t, y1 (t)) , (t, y2 (t)) í (. . 1.4). , (1.10) C (-C, 0) , (C, 0). , y = y (t). , .


1.4. 1.4.1.

17

( ) M (t, y )dt + N (t, y )dy = 0. (1.11)

, M (t, y ) N (t, y ) D R2 |M (t, y )| + |N (t, y )| > 0, (t, y ) D. (1.12)

(1.11) (1.7), (1.11) M (t, y ) = f (t, y ), N (t, y ) = -1. (1.11). , . 1.4.1. t = ( ), y = ( ) (1.11) [1 , 2 ], : 1. ( ), ( ) [1 , 2 ] | ( )| + | ( )| > 0, [1 , 2 ]; 2. (( ), ( )) D, [1 , 2 ]; 3. t = ( ), y = ( ) (1.11) , M (( ), ( )) ( ) + N (( ), ( )) ( ) = 0, [1 , 2 ]. (1.13)

t = ( ), y = ( ) í (1.11). (t, y ) , t = ( ), y = ( ), [1 , 2 ]. 1 , ( ) = 0, ( ) = 0 0 (1 , 2 ). , , = -1 (t) = -1 (y ) , , (1.11) y = (-1 (t))


18

1.

t0 = (0 ), t = ( -1 (y )) y0 = (0 ). (1.10). 1.4.2. (1.10) tdt + y dy = 0. t = C cos , y = C sin , [0, 2 ] , C . , (1.10), - . , , [0, 2 ], y = y (t) t = t(y ), . , . . (t, y , c) (t, y ) D c, C0 . 1.4.2. (t, y , c) = 0 (1.11) D, c C0 (1.11). , (1.11), (1.11) t = ( ), y = ( ), D, c C0 , (( ), ( ), c) 0. ~ ~ , , , . . 1.4.3. tdt + y dy = 0 t2 + y 2 - c = 0. C0 .


1.4.

19

1.4.4. y (t) = 3 y 2 (t) 1.3.1 , y > 0, y- (t - C )3 = 0. 27

R2 , , y0 (t) 0 C . 1.4.2. , . 1.4.3. (1.11) D, D V (t, y ) V (t, y ) V (t, y ) , + >0 t y M (t, y ) = V (t, y ) , t N (t, y ) = V (t, y ) , y (t, y ) D. (1.14)

1.4.1. (1.11) D V (t, y ) = C. (1.15)

. 1.4.2 , (1.15) . (1.15) (t0 , y0 ) D C0 = V (t0 , y0 ). (1.12) (1.14) : V (t0 , y0 ) = M (t0 , y0 ) = 0, t V (t0 , y0 ) = N (t0 , y0 ) = 0. y

. t0


20 y = g (t) , y0 = g (t0 ) V (t, g (t)) = C0

1.

(1.16)

. (1.16), dC0 = 0 = dV (t, g (t)) = V (t, g (t)) V (t, y ) dt + dg (t) = t y = M (t, g (t))dt + N (t, g (t))g (t)dt,

t = t y = g (t) (1.11). , (1.15) (1.11). , (1.15) (1.11). t = ( ), y = ( ), [1 , 2 ] í (1.11) , (( ), ( )) D [1 , 2 ]. , C , V (( ), ( )) = C, [1 , 2 ].

1.14 [1 , 2 ] d V (( ), ( )) = M (( ), ( )) ( ) + N (( ), ( )) ( ). d ( ), ( ) í (1.11), (1.13), d V (( ), ( )) = 0, d , V (( ), ( )) = C, [1 , 2 ], [1 , 2 ].

(1.15) í (1.11). 1.4.1. 1.4.1 , (t0 , y0 ) D (1.11), (1.14).


1.4. 1.4.2. a(t, y ) = (M (t, y ), N (t, y )), (1.14) : a(t, y ) = gradV (t, y ).

21

, (1.11) , . 1.4.2. M (t, y ), N (t, y ) D , , (1.12). , (1.11) D, , M (t, y ) N (t, y ) = , y t (t, y ) D. (1.17)

. . (1.11) . V (t, y ) , (1.14). y , t, 2 V (t, y ) M (t, y ) = , y t y N (t, y ) 2 V (t, y ) = , t y t

(1.17). . (1.17).
t y

V (t, y ) =
t0

M ( , y )d +
y
0

N (t0 , )d ,

(t0 , y0 ) í D. V (t, y ) t, = M (t, y ). y t


22 (1.17),
t

1.

V (t, y ) = y
t0

M ( , y ) d + N (t0 , y ) = y
t

=
t0

N ( , y ) d + N (t0 , y ) = N (t, y ). t

, V (t, y ) 1.4.3 (1.11) . 1.4.3. 1.4.4. D ² = ²(t, y ) = 0 , ²(t, y )M (t, y )dt + ²(t, y )N (t, y )dy = 0 (1.18)

. 1.4.3. M dt + N dy = 0 D (t, y ) = C , (t, y ) D (t, y ) (t, y ) + > 0, t y (t, y ) D.

D. . 1.4.1 1.4.1 D . (( ), ( )) í . (( ), ( )) C . 0 = dC = ( ) + ( ) d . t y , (1.13): M (( ), ( )) ( ) + N (( ), ( )) ( ) = 0, | ( )| + | ( )| > 0.


1.4. , t M ( ) 0 = y ( ) 0 N

23

. , N M . t y

, , - M = 0, N = 0, = 0, t = 0. y (t, y ) (t, y ) y t ²(t, y ) = = 0. M (t, y ) N (t, y ) ²M = , t ²N = , y

²(t, y ) , (1.18) V = (t, y ). 1.4.3. . , ²(t, y ) , V (t, y ) , dV = ²M dt + ²N dy . f (V ), f (s) í , f (s) = 0, f (V )dV = d f (V )dV = ²f (V )M dt + ²f (V )N dy .

²1 (t, y ) = ²(t, y )f (V (t, y )) í .


24

1.

, (1.18) , ²(t, y )M (t, y ) = ²(t, y )N (t, y ) , y t . ² M N ² -M =² - . (1.19) N t y y t . , . , (1.19) . 1. N 1 M - = g (t) í t, N y t ² = ²(t). (1.19) ² (t) = ²(t)g (t) ²(t) = exp{ g (t)dt}. 2. 1 M N - = h(y ) í y , M y t ² = ²(y ). (1.19) ² (y ) = -²(y )h(y ) ²(y ) = exp{- h(y )dy }.


2.1.

25

2
2.1. ,
f (t, y ) = {(t, y ) : |t - t0 | T, |y - y0 | A}.

[t0 - T , t0 + T ] y (t) = f (t, y (t)) y (t0 ) = y0 . (2.2) y (t), (2.1) (2.2). . [t1 , t2 ] , t0 - T t1 < t2 t0 + T , t0 [t1 , t2 ]. 2.1.1. y (t) ï ï (2.1), (2.2) [t1 , t2 ], : y (t) C 1 [t1 , t2 ], |y (t) - y0 | A ï t [t1 , t2 ], y (t) (2.1) t [t1 , t2 ] ï (2.2). 2.1.1. , (2.1), (2.2) . [t0 - T , t0 + T ] y (t)
t

(2.1)

y (t) = y0 +
t0

f ( , y ( ))d .

(2.3)


26

2.

, y (t) . 2.1.1. y (t) (2.1), ï (2.2) [t1 , t2 ] , y (t) C [t1 , t2 ], ï |y (t) - y0 | A t [t1 , t2 ] y (t) (2.3) ï ï t [t1 , t2 ]. . y (t) ï (2.1), (2.2) [t1 , t2 ]. 2.1.1 , y (t) C [t1 , t2 ], |y (t) - y0 | A t [t1 , t2 ]. , ï ï y (t) (2.3) t [t1 , t2 ]. ï (2.1) t0 t,
t t

y ( )d = ï
t0 t0

f ( , y ( ))d , ï

t [t1 , t2 ].

(2.2),
t

y (t) = y0 + ï
t0

f ( , y ( ))d , ï

t [t1 , t2 ].

, y (t) ï (2.3) t [t1 , t2 ]. y (t) , y (t) C [t1 , t2 ], |y (t) - y0 | ï ï ï A t [t1 , t2 ] y (t) (2.3) t [t1 , t2 ], ï
t

y (t) = y0 + ï
t0

f ( , y ( ))d , ï

t [t1 , t2 ].

(2.4)

, y (t) ï (2.1), (2.2). (2.4) t = t0 , , y (0) = y0 . ï (2.2) . y (t) [t1 , t2 ], ï
t

y (t) = y0 + ï
t0

f ( , y ( ))d ï


2.1.

27

[t1 , t2 ] t f ( , y ( )) C [t1 , t2 ]. ï , y (t) [t1 , t2 ]. ï (2.4), , y (t) (2.1), 2.1.1 ï . 2.1.2. - (2.1), (2.2). , -. 2.1.2. z (t) C [a, b] ,
t

0

z (t)

c+d
t0

z ( )d ,

t [a, b],

(2.5)

c , d , t í [a, b]. z (t) ced|
t-t0 |

0

,

t [a, b].

(2.6)

. t
t

t0 . t [t0 , b]. c + dp(t),

p(t) =
t0

z ( )d ,

p (t) = z (t) 0, p(t0 ) = 0. (2.5) , p (t) t [t0 , b]. e-d(t-t0 ) , p (t)e-d(
t-t0 )

ce-d(

t-t0 )

+ dp(t)e-d(

t-t0 )

,

t [t0 , b].

d p(t)e-d( dt
t t-t0 )

ce-d(

t-t0 )

,

t [t0 , b].

t0 t , p(t)e-d(
t-t0 )

- p(t0 )

c
t0

e-d(

-t0 )

d =

c 1 - e-d( d

t-t0 )

,

t [t0 , b].


28 , p(t0 ) = 0, dp(t) z (t) c + dp(t) c + ced(
t-t0 )

2. ced(
t-t0 )

- c. , , t [t0 , b]

- c = ced(

t-t0 )

(2.6) t [t0 , b] . (2.6) t [a, t0 ]. (2.5)
t t0

0

z (t)

c-d
t0

z ( )d = c + d
t

z ( )d ,

t [a, t0 ].


t0

q (t) =
t

z ( )d ,

t [a, t0 ].

q (t) = -z (t) 0, q (t0 ) = 0. (2.5) , -q (t) c + dq (t), t [a, t0 ]. e-d(t0 -t) , -q (t)e-d(
t0 -t)

ce-d(

t0 -t)

+ dq (t)e-d(

t0 -t)

,

t [a, t0 ].

- d q (t)e-d( dt
t0 -t)

ce-d(

t0 -t)

,

t [a, t0 ].

t t0 ,
t0 -d(t0 -t)

q (t)e

- q (t0 )

c
t

e-d(

t0 - )

d =

c 1 - e-d( d

t0 -t)

,

t [a, t0 ].

, dq (t) z (t) c + dq (t)

ced(

t0 -t)

- c.
t0 -t)

c + ced(

- c = ced|

t-t0 |

,

t [a, t0 ]

(2.6) t [a, t0 ] , 2.1.2.


2.1. 2.1.3.

29

. 2.1.2. f (t, y ), , y , |f (t, y1 ) - f (t, y2 )| L|y1 - y2 |, (t, y1 ), (t, y2 ) ,

L í . 2.1.1. f (t, y ) fy (t, y ) , f (t, y ) y . , fy (t, y ) , L , |fy (t, y )| L, (t, y ) .

, |f (t, y1 ) - f (t, y2 )| = |fy (t, )(y1 - y2 )| L|y1 - y2 |, (t, y1 ), (t, y2 ) .

2.1.2. f (t, y ) y , . , , f (t, y ) = (t-t0 )|y -y0 |. , y = y0 , t = t0 , (t, y1 ), (t, y2 ) |f (t, y1 ) - f (t, y2 )| = |t - t0 | § |y1 - y0 | - |y2 - y0 | T |y1 - y2 |.

2.1.3. f (t, y ) y , . , , y, 0 y 1; f (y ) = - |y |, -1 y 0. , [-1, 1]. , . , . L, | y 1 - y 2 | L|y1 - y2 |, y1 , y2 [-1, 1]. y1 > 0, y2 = 0. y .
1 2 L2 y1 , , 0 < y1 < L-2 ,


30

2.

2.1.4. (2.1), (2.2). 2.1.1. f (t, y ) y . y1 (t), y2 (t) í (2.1), (2.2) [t1 , t2 ], y1 (t) = y2 (t) t [t1 , t2 ]. . y1 (t) y2 (t) í (2.1), (2.2), 2.1.1 , (2.3).
t

y1 (t) = y0 +
t0 t

f ( , y1 ( ))d ,

t [t1 , t2 ],

y2 (t) = y0 +
t0

f ( , y2 ( ))d ,

t [t1 , t2 ].

,
t t

|y1 (t) - y2 (t)| =
t0 t

f ( , y1 ( ))d -
t0

f ( , y2 ( ))d

|f ( , y1 ( )) - f ( , y2 ( ))|d ,
t0

t [t1 , t2 ].

,
t

|y1 (t) - y2 (t)|

L
t0

|y1 ( ) - y2 ( )|d ,

t [t1 , t2 ].

z (t) = |y1 (t) - y2 (t)|,
t

0

z (t)

L
t0

z ( )d ,

t [t1 , t2 ].


2.1.

31

- 2.1.2 c = 0 d = L, z (t) = 0, t [t1 , t2 ]. , y1 (t) = y2 (t), t [t1 , t2 ] 2.1.1 . 2.1.4. , (2.1), (2.2) . , f (y ) = y, 0 y y 1, 0,

- |y |, -1

y (t) = f (y (t)), y (0) = 0 y1 (t) = 0, y2 (t) = t2 /4, -t2 /4, 0 t 2, - 2 t 0.

2.1.5. . , [t0 - T , t0 + T ], , , . . 2.1.2. f (t, y ) , y |f (t, y )| M, (t, y ) .

[t0 - h, t0 + h], h = min T , A , M

y (t) , y (t) C 1 [t0 -h, t0 +h], |y (t)-y0 | A, t [t0 - h, t0 + h], y (t) = f (t, y (t)), t [t0 - h, t0 + h], y (t0 ) = y0 . (2.7) (2.8)


32

2.

. 2.1.1 , y (t) C [t0 - h, t0 + h] , |y (t) - y0 | A, t [t0 - h, t0 + h],
t

y (t) = y0 +
t0

f ( , y ( ))d ,

t [t0 - h, t0 + h].

(2.9)

, . yk (t), k = 0, 1, 2, . . . , y0 (t) = y0 ,
t

y

k+1

(t) = y0 +
t0

f ( , yk ( ))d ,

t [t0 - h, t0 + h], k = 0, 1, 2, . . . . (2.10)

, , k = 0, 1, 2, . . . yk (t) C [t0 - h, t0 + h], |yk (t) - y0 | A, t [t0 - h, t0 + h].

k = 0 , y0 (t) = y0 . k = m. ym (t) C [t0 - h, t0 + h], ,
t

|ym (t) - y0 |

A,

t [t0 - h, t0 + h].

y

m+1

(t) = y0 +
t0

f ( , ym ( ))d ,

t [t0 - h, t0 + h]

(2.11)

, ym+1 (t) C [t0 -h, t0 +h] |ym+1 (t)-y0 | A, t [t0 -h, t0 +h]. , |ym (t) - y0 | A, t [t0 - h, t0 + h], f (t, ym (t)) [t0 - h, t0 + h]. , (2.11), t [t0 - h, t0 + h]. , ym+1 (t) C [t0 - h, t0 + h].


2.1.
t

33

|y

m+1

(t) - y0 | =
t0

f ( , ym ( ))d
t

t

|f ( , ym ( ))|d
t0 t0

M d

Mh

M§

A = A, M

t [t0 -h, t0 +h].

|ym+1 (t) - y0 | A, t [t0 - h, t0 + h]. , yk (t) C [t0 -h, t0 +h] |yk (t)-y0 | A, t [t0 -h, t0 +h], k = 0, 1, 2, . . . . , , t [t0 - h, t0 + h] |y
k+1

(t) - yk (t)|

ALk

|t - t0 |k , k!

k = 0, 1, 2, . . . .

(2.12)

k = 0
t

|y1 (t) - y0 (t)| = y0 +
t0

f ( , y0 )d - y
t

0

f ( , y0 )d
t0

Mh

A,

t [t0 - h, t0 + h],

k = 0 (2.12) . (2.12) k = m - 1. , k = m.
t t

|y

m+1

(t) - ym (t)| = y0 +
t0 t

f ( , ym ( ))d - y0 -
t0

f ( , y

m-1

( ))d

|f ( , ym ( )) - f ( , y
t0

m-1

( ))|d ,

t [t0 - h, t0 + h].


34

2.

(2.12) k = m - 1,
t

|y

m+1

(t) - ym (t)|
t

L
t0

|ym ( ) - y

m-1

( )|d

L
t0

ALm-1

|t - t0 |m | - t0 |m-1 d = ALm , (m - 1)! m!

t [t0 - h, t0 + h].

, (2.12) k = m, k N. yk (t)
k

yk (t) = y0 +
n=1

(yn (t) - y

n-1

(t)),

n = 1, 2, . . .

yk (t) [t0 - h, t0 + h]


(yn (t) - y
n=1

n-1

(t))

(2.13)

[t0 - h, t0 + h]. (2.13) [t0 - h, t0 + h]. (2.12) , |yn (t) - y
n-1

(t)|


ALn

-1

hn-1 = cn , (n - 1)!

t [t0 - h, t0 + h],

n = 1, 2, . . .


n=1

cn . ,

(2.13) [t0 - h, t0 + h]. , yk (t) [t0 - h, t0 + h] y (t). yk (t) [t0 - h, t0 + h], y (t) , y (t) C [t0 - h, t0 + h]. , |y (t) - y0 | A, t [t0 - h, t0 + h]. , |yk (t) - y0 | A, t [t0 - h, t0 + h], k = 0, 1, 2, . . . .


2.1.

35

. 2.1. .

k t [t0 - h, t0 + h], , |y (t) - y0 | A, t [t0 - h, t0 + h]. , y (t) (2.9). [t0 - h, t0 + h] yk (t) y (t) > 0 k0 ( ) , k k0 ( ) |yk (t) - y (t)| < t [t0 - h, t0 + h]. > 0 () = Lh k0 = k0 ( ()) , k k0 |f ( , yk ( )) - f ( , y ( ))| L|yk ( ) - y ( )| < , h [t0 - h, t0 + h].


t t

f ( , yk ( ))d -
t0 t0

f ( , y ( ))d <

|t - t0 | h

,

t [t0 - h, t0 + h],

(2.10) k t [t0 - h, t0 + h]. , y (t) (2.9). , , y (t) C [t0 -h, t0 +h], |y (t)-y0 | A, t [t0 - h, t0 + h] (2.9). , y (t) [t0 - h, t0 + h] 2.1.2 . , [t0 - T , t0 + T ], -


36

2.

A } M (. . 2.1). , , (t, y (t)) , |y (t) - y0 | A, t [t0 - h, t0 + h]. , f (t, y ) M A L. h = min{T , } M A, , , A M . , , f (t, y ) . 2.1.1. a > 0 [t0 - h, t0 + h], h = min{T , y (t) = a(y (t)2 + 1), y (0) = 0.

f (t, y ) = a(y 2 + 1) t y . y (t) = tg(at) [-h1 , h1 ], - , . 2a 2a

2.2. ,
2.2.1. , F (t, y (t), y (t)) = 0. (2.14)

, F (t, y , p) D (t0 , y0 , y0 ) R3 : D = {(t, y , p) : |t - t0 | a, |y - y0 | b, |p - y0 | c}, (2.15)

a, b, c í . 2.2.1. y (t) (2.14) [t1 , t2 ], :


2.2. , y 37 1. y (t) [t1 , t2 ]; 2. (t, y (t), y (t)) D t [t1 , t2 ]; 3. [t1 , t2 ] (2.14). (2.14) , F (t, y , p) = p - f (t, y ), f (t, y ) ( ) (t0 , y0 ). F (t, y (t), y (t)) = 0, y (t0 ) = y0 . (2.16)

, : (y (t)) - (t + y (t))y (t) + ty (t) = 0.
2 2

(2.17)

p - (t + y )p + ty = 0 p1 = t, p2 = y , , : y (t) = t, y1 (t) = t2 + C1 , 2 y2 (t) = C2 exp{t}, C1 , C2 R. y (t) = y (t).

2.2.1. (2.17) y (0) = 1 (. . 2.2): t2 + 1, y2 (t) = exp{t}. (2.18) 2 (2.17) c y (0) = 0 (. . 2.2-): y1 (t) = t2 , y2 (t) = 0, 2 y1 (t), t < 0, y2 (t), t < 0, y4 (t) = y2 (t), t 0, y1 (t), t 0. y1 (t) =

y3 (t) =

(2.19)


38

2.

.

.

.

.

. 2.2. 2.2.1, 2.2.2: .

, (2.16). . , . F (t, y (t), y (t)) = 0, y (t0 ) = y0 , y (t0 ) = y0 . (2.20)

2.2.2. (2.17) y (0) = 1, y (0) = 0, (t0 , y0 , y0 ) = (0, 1, 0), F (0, 1, 0) = 0, F (0, 1, 0) = -1 = 0, p (2.21)

t2 + 1. 2 (2.17) y (0) = 1, y (0) = 1, y (t) = (t0 , y0 , y0 ) = (0, 1, 1), F (0, 1, 1) = 0, F (0, 1, 1) = 1 = 0, p (2.22)

y (t) = exp{t}. (2.17) y (0) = 1, y (0) = y0 , y0 {0; 1}, (t0 , y0 , y0 ) = (0, 1, y0 ), F (t0 , y0 , y0 ) = 0, (2.23)


2.2. , y 39 . (2.17) y (0) = 0, y (0) = 0, (t0 , y0 , y0 ) = (0, 0, 0), F (0, 0, 0) = 0, F (0, 0, 0) = 0, p (2.24)

(2.19). (2.20): 1. (t0 , y0 , y0 ) R3 ; F (t0 , y0 , y0 ) = 0; 2. y (t0 ) = y0 , y (t0 ) = y0 F (t0 , y0 , y0 ) = 0. p 2.2.2. 2.2.1. F (t, y , p) D, (2.15), : 1. F (t0 , y0 , y0 ) = 0; F (t, y , p) 2. F (t, y , p), , y F (t0 , y0 , y0 ) 3. = 0. p (2.25) F (t, y , p) D; (2.26) p (2.27)

h > 0 , [t0 - h, t0 + h] (2.20). . (t0 , y0 , y0 ) F (t, y , p) = 0. (2.28) (2.25)-(2.27) , 0 (t0 , y0 ),


40 p = f (t, f (t, y ) = y

2. y ), 0 - F (t, y , f (t, y ))/ y , F (t, y , f (t, y ))/ p (2.29)

(2.28). , y0 = f (t0 , y0 ). (2.30) 0 (2.14) y (t) = f (t, y (t)), , (2.20) y (t) = f (t, y (t)), y (t0 ) = y0 . (2.31)

, (2.20) y (t0 ) = y0 (2.30). (2.31) = {(t, y ) : |t - t0 | a0 , |y - y0 | b 0 },

a0 , b0 , 0 . , f (t, y ) 0 , . y f L = max (t, y ) (t,y ) y f (t, y ), y (2.29). , 2.1.2 , . , h > 0 , [t0 - h, t0 + h] (2.31), (2.20).

2.2.1. 2.2.2 2.2.1 (2.21), (2.22) (2.23), (2.24).


2.2. , y 41 2.2.3. (2.14), . , . y = f (t, y ), y , y = f (t, p), dy = pdt.

dy , : dy = f (t, p) f (t, p) dt + dp = pdt. t p

t, p. t = ( , c), p = ( , c), t = ( , c), y = f (( , c), ( , c).

t = f (y , y ), t, 2- t = f (y , p), dy = pdt.

dt, : dt = f (y , p) f (y , p) dy dy + dp = . y p p

y , p. y = ( , c), p = ( , c), y = ( , c), t = f (( , c), ( , c)).


42

2. F (t, y , y ) = 0 2- F (t, y , p) = 0, dy = pdt.

, R3 , T (u, v ), Y (u, v ), P (u, v ): t = T (u, v ), y = Y (u, v ), p = P (u, v ).

, dy , dt (u, v ), : Y (u, v ) Y (u, v ) du + dv = u v T (u, v ) T (u, v ) du + dv P (u, v ). u v

u, v . u = ( , c), v = ( , c), t = T (( , c), ( , c)), y = Y (( , c), ( , c)).

2.2.4. 2.2.2. y = (t) F (t, y (t), y (t)) = 0 [t1 , t2 ], y = (t) 2.2.1, = {(t, y ) : y = (t), t [t1 , t2 ]}

, .


2.2. , y 43 , F (t, y (t), y (t)) = 0, y (t0 ) = y0 , y (t0 ) = y0 , (t0 , y0 ) .

, 2.2.1 . , . . F (t, y , p), . 2.2.3. y=
3

y

2

(2.32)

(t - C )3 . 27 y0 (t) (2.32) [t1 , t2 ], t0 [t1 , t2 ] C = t0 , (t0 , 0) y0 (t) (t - t0 )3 y (t, t0 ) = 27 (. . 1.3). F (t, y , p) = p - 3 y 2 , F 2 =- y 33y y0 (t) 0 y (t, C ) = y = 0, (2.26). , F , . y (2.26) F (t, y , p). (t), F (t, (t), (t)) = 0, F (t, (t), (t)) = 0. p


44

2.

, (t, (t), (t)) t F (t, y , p) = 0, F (2.33) (t, y , p) = 0. p (2.33) p (t, y ) = 0. , . : 1. (t, y ) = 0 ; 2. (t, y ) = 0 (2.14), ; 3. (t, y ) = 0 , (2.14). . 2.2.4. (2.32) 2.2.3 (y )3 - y 2 = 0. (2.33) p3 - y 2 = 0, 3p2 = 0 y (t) = 0, . 2.2.5. (y )2 - y 2 = 0. (2.33) p2 - y 2 = 0, 2p = 0 y (t) = 0, . , , y1 (t) = c1 exp{t}, y2 (t) = c2 exp{-t}.


2.3.

45

y (t) = 0 . , y (t) = 0 . 2.2.6. (2.17). (2.33) p2 - (t + y )p + ty = 0, 2p - t - y = 0 y (t) = t, (2.17). , .

2.3. n-
n- . 2.3.1. fi (t, y1 , y2 , . . . , yn ), i = 1, 2, . . . , n t [a, b], (y1 , y2 , . . . , yn ) Rn y1 (t), y2 (t), . . . , yn [a, b] y1 (t) = f1 (t, y1 (t), y2 (t), . . . , y2 (t) = f2 (t, y1 (t), y2 (t), . . . , ... yn (t) = fn (t, y1 (t), y2 (t), . . . , y1 (t0 ) = y01 , y2 (t0 ) = y02 , ..., (t), yn (t)), yn (t)), yn (t)),

(2.34)

yn (t0 ) = y0n ,

(2.35)


46

2.

t0 í [a, b], y01 , y02 , . . . y0n í . (2.34). 2.3.1. y1 (t), y2 (t), . . . , yn (t) (2.34), (2.35) [a, b], : 1. yi (t) [a, b], i 1, 2, . . . , n; 2. yi (t) = fi (t, y1 (t), y2 (t), . . . , yn (t)), t [a, b], i = 1, 2, . . . , n; 3. yi (t0 ) = y0i , i = 1, 2, . . . , n. 2.3.2. f (t, y1 , y2 , . . . , yn ) y1 , y2 , . . . , yn , L > 0, |f (t, y1 , y2 , . . . , yn ) - f (t, y1 , y2 , . . . , yn )| L |y1 - y1 | + |y2 - y2 | + § § § + |yn - yn | , t [a, b], (y1 , y2 , . . . , yn ), (y1 , y2 , . . . , yn ) Rn . (2.36) 2.3.2. (2.34), (2.35) . 2.3.1. fk (t, y1 , y2 , . . . , yn ), k = 1, 2, . . . , n, t [a, b], (y1 , y2 , . . . , yn ) Rn (2.36) L. , y1 (t), y2 (t), . . . , yn (t) y1 (t), y2 (t), . . . , yn (t) (2.34), (2.35) [a, b], yi (t) = yi (t) t [a, b], i = 1, 2, . . . , n. . y1 (t), y2 (t), . . . , yn (t) í (2.34), (2.35), yi (t) = fi (t, y1 (t), y2 (t), . . . , yn (t)) t [a, b], yi (t0 ) = y0i , i = 1, 2, . . . , n. =


2.3.

47

t0 t (2.35), i = 1, 2, . . . , n
t

yi (t) = y0i +
t0

fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d ,

t [a, b].

(2.37)

yi (t), i = 1, 2, . . . , n
t

yi (t) = y0i +
t0

fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d ,

t [a, b].

(2.38)

(2.38) (2.37) (2.36), i = 1, 2, . . . , n t [a, b] |yi (t) - yi (t)| =
t

=
t0

(fi ( , y1 ( ), y2 ( ), . . . , yn ( )) - fi ( , y1 ( ), y2 ( ), . . . , yn ( ))) d
t

L
t0

|y1 ( ) - y1 ( )| + |y2 ( ) - y2 ( )| + § § § + |yn ( ) - yn ( )| d .

z (t) = |y1 (t) - y1 (t)| + |y2 (t) - y2 (t)| + § § § + |yn (t) - yn (t)|. :
t

|yi (t) - yi (t)|

L
t0

z ( )d ,

i = 1, 2, . . . , n,

t [a, b].

,
t

z (t)

nL
t0

z ( )d ,

t [a, b].


48

2.

- 2.1.2 , z (t) = 0, t [a, b]. , yi (t) = yi (t) i = 1, 2, . . . , n, 2.3.1 . 2.3.3. (2.34), (2.35). 2.1.5. , , f (t, y ) . , , . fk (t, y1 , . . . , yn ), . (2.34), (2.35) , 2.1.5, 2.4 2.3.2. fk (t, y1 , y2 , . . . , yn ), k = 1, 2, . . . , n, t [a, b], (y1 , y2 , . . . , yn ) Rn (2.36) L. y1 (t), y2 (t), . . . , yn (t), (2.34), (2.35) [a, b]. . [a, b] yi (t)
t

t [a, b].

yi (t) = y0i +
t0

fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d ,

i = 1, 2, . . . , n.

(2.39)

, y1 (t), . . . , yn (t) ï ï [a, b] (2.39), (2.34), (2.35) [a, b].


2.3.

49

, (2.39) t = t0 , , yi (t) ï (2.35). (2.39) t, , (2.34). , , yi (t) [a, b], ï (2.39). yi (t), ï . k k k y1 (t), y2 (t), . . . , yn (t), k = 0, 1, 2, . . . ,
t k+1 i k k k fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d , t0 k i = 1, 2, . . . , n, t [a, b]. , yi (t) [a, b]. 0 m yi (t) . , yi (t) m+1 , yi (t). fi (t, y1 , y2 , . . . , yn ) t [a, b], (y1 , y2 , . . . , yn ) Rn , (2.40) , m yi +1 (t) [a, b]. B t 0 yi (t) = y0i ,

y

(t) = y0i +

(2.40)

B=

i=1,2,...,n t[a,b] t 0

max

max

fi ( , y01 , y02 , . . . , y0n )d .

, i = 1, 2, . . . , n k = 0, 1, . . . [a, b] |y
k+1 i k (t) - yi (t)|

B (nL)k

|t - t0 |k . k!

(2.41)

k = 0 ,
t

|y (t) - y (t)| =
t0

1 i

0 i

fi ( , y01 , y02 , . . . , y0n )d

B.

(2.41) k = m - 1. ,


50 k = m. (2.40) |y
t m+1 i m (t) - yi (t)|

2.

m m m |fi ( , y1 ( ), y2 ( ), . . . , yn ( ))- t0

-fi ( , y

m-1 1

( ), y

m-1 2

( ), . . . , y

m-1 n

( ))|d

t m L |y1 ( ) - y t0 m § § § + |yn ( ) - y m-1 n m-1 1 m ( )| + |y2 ( ) - y m-1 2

( )| + . . .

( )| d .

,
t

|y

m+1 i

(t) - y (t)|
t0

m i

B (nL)m

| - t0 |m-1 d (m - 1)!

B (nL)m

|t - t0 |m . m!

, (2.41) . [a, b]


y (t) +
m=0

0 i

(y

m+1 i

m (t) - yi (t)),

i = 1, 2, . . . , n.

(2.41), [a, b] |y
m+1 i m (t) - yi (t)|

B (nL)m

(b - a)m , m!

m = 0, 1, . . . .

, , [a, b]. , [a, b]
k-1 k 0 yi (t) = yi (t) + m=0

(y

m+1 i

m (t) - yi (t)),

i = 1, 2, . . . , n


2.3.

51

[a, b] yi (t). ï k + (2.40), , yi (t) ï (2.39), (2.34), (2.35). 2.3.2 . 2.3.1. (2.36) , fk (t, y1 , y2 , . . . , yn ) fk (t, y1 , y2 , . . . , yn ) yj D, t [a, b], (y1 , y2 , . . . , yn ) Rn ,

k , j = 1, 2, . . . , n , D í . , |fk (t, y1 , y2 , . . . , yn ) - fk (t, y1 , y2 , . . . , yn )| |fk (t, y1 , y2 , . . . , yn ) - fk (t, y1 , y2 , . . . , yn )|+ +|fk (t, y1 , y2 , . . . , yn ) - fk (t, y1 , y2 , . . . , yn )| + . . . § § § + |fk (t, y1 , y2 , . . . , yn ) - fk (t, y1 , y2 , . . . , yn )|. , |fk (t, y1 , y2 , . . . , yn ) - fk (t, y1 , y2 , . . . , yn )| D |y1 - y1 | + |y2 - y2 | + § § § + |yn - yn | . , fk (t, y1 , y2 , . . . , yn ) (2.36) L = D. , , 2.3.1 2.3.2. 2.3.1. (y1 (t))3 , 1 + (y1 (t))2 y2 (t) = t2 y2 (t) + cos(y1 (t) + y2 (t)) y1 (t) = t sin(y1 (t) + y2 (t)) + 2.3.1 2.3.2, [a, b].


52

2.

2.3.4. n- n- , , y
( n)

(t) = F (t, y (t), y (t), y (t), . . . , y

(n-1)

(t)),

t [a, b],

(2.42)

F (t, y1 , y2 , . . . , yn ) , y (t) í . y (t) y (t0 ) = y00 , y (t0 ) = y01 , y
(2)

(t0 ) = y02 , . . . , y

(n-1)

(t0 ) = y

0n-1

,

(2.43)

t0 [a, b], y00 , . . . , y0n-1 í . n- , , y (t), (2.42) (2.43). 2.3.3. y (t) (2.42), (2.43) [a, b], y (t) n [a, b] , y (t) (2.42) (2.43). (2.42), (2.43). 2.3.3. F (t, y1 , y2 , . . . , yn ) t [a, b], (y1 , y2 , . . . , yn ) Rn L1 > 0,
n

|F (t, y1 , y2 , . . . , yn ) - F (t, y1 , y2 , . . . , yn )| t [a, b],

L1
i=1

|yi - yi |,

(2.44)

(y1 , y2 , . . . , yn ), (y1 , y2 , . . . , yn ) Rn .

y (t), (2.42), (2.43) [a, b]. . . y (t) (2.42), (2.43)


2.3. [a, b]. y1 (t) = y (t), y2 (t) = y (t), y3 (t) = y (t), ... yn (t) = y
(n-1)

53

(t).

y (t) (2.42), (2.43) [a, b], yi (t), i = 1, 2, . . . , n = y2 (t), y1 (t) y2 (t) = y3 (t), ... (2.45) yn-1 (t) = yn (t), = F (t, y1 (t), y2 (t), . . . , yn (t)) yn (t) yi (t0 ) = y
0i-1

,

i = 1, 2, . . . , n.

(2.46)

(2.45) (2.34) fi (t, y1 , y2 , . . . , yn ) = yi+1 , i = 1, . . . , n - 1, fn (t, y1 , y2 , . . . , yn ) = F (t, y1 , y2 , . . . , yn ). t [a, b], (y1 , y2 , . . . , yn ) Rn (2.36) L = max{1, L1 }. (2.45), (2.46) 2.3.1 . , (2.45), (2.46) , (2.42), (2.43) . (2.42), (2.43). (2.45), (2.46). 2.3.2 [a, b]. [a, b] yi (t), (2.45), (2.46). y1 (t) y (t), , y (t) n [a, b] , y (i-1) (t) = yi (t), i = 1, 2, . . . , n y (t) (2.42), (2.43). y (t) (2.42), (2.43). 2.3.3 .


54

2.

2.3.5. n- [a, b] n- y1 (t) = a11 (t)y1 (t) + a12 (t)y2 (t) + § § § + a1n (t)yn (t) + f1 (t), y2 (t) = a21 (t)y1 (t) + a22 (t)y2 (t) + § § § + a2n (t)yn (t) + f2 (t), (2.47) ... yn (t) = an1 (t)y1 (t) + an2 (t)y2 (t) + § § § + ann (t)yn (t) + fn (t), aij (t), fi (t), i, j = 1, 2, . . . , n í [a, b] . yi (t0 ) = y0i , i = 1, 2, . . . , n. (2.48)

(2.47), (2.48). 2.3.4. aij (t) , fi (t) [a, b], i, j = 1, 2, . . . , n. y1 (t), y2 (t), . . . , yn (t), (2.47), (2.48) [a, b]. . (2.47) (2.34) fi (t, y1 , y2 , . . . , yn ) = ai1 (t)y1 +ai2 (t)y2 +§ § §+ain (t)yn +fi (t), i = 1, 2, . . . , n. fi (t, y1 , y2 , . . . , yn ) t [a, b], (y1 , y2 , . . . , yn ) Rn (2.36) L = max max |aij (t)|.
1 i,j n t[a,b]

, (2.47), (2.48) 2.3.1 2.3.2, [a, b]. 2.3.4 .


2.4. 2.3.6. n-

55

n a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = f (t), (2.49)

ai (t), i = 0, 1, 2, . . . , n, f (t) í [a, b] , a0 (t) = 0 [a, b]. y (t) t0 [a, b] y (i) (t0 ) = y0i , i = 0, 1, . . . , n - 1. (2.50)

2.3.5. ai (t), f (t) [a, b], i = 1, 2, . . . , n, a0 (t) = 0 [a, b]. y (t), (2.49), (2.50) [a, b]. . (2.49) (2.42) F (t, y1 , y2 , . . . , yn ) = an (t) an-1 (t) a1 (t) f (t) - § y1 - § y2 - § § § - § yn . a0 (t) a0 (t) a0 (t) a0 (t)

F (t, y1 , y2 , . . . , yn ) t [a, b], (y1 , y2 , . . . , yn ) Rn (2.44) ai (t) L1 = max max . 1 i n t[a,b] a0 (t) , (2.49), (2.50) 2.3.3 [a, b]. 2.3.5 .

2.4. ( )
fi (t, y1 , y2 , . . . , yn ), i = 1, 2, . . . , n n + 1- n
+1

= {(t, y1 , y2 , . . . , yn ) :

|t-t0 |

T,

|yi -y0i |

A,

i = 1, 2, . . . , n}.


56 y1 (t) = y2 (t) = ... yn (t) = y1 (t0 ) = y01 , y01 , y02 , . . . , y
0n

2.

f1 (t, y1 (t), y2 (t), . . . , yn (t)), f2 (t, y1 (t), y2 (t), . . . , yn (t)), fn (t, y1 (t), y2 (t), . . . , yn (t))

(2.51)

y2 (t0 ) = y02 ,

...,

yn (t0 ) = y0n ,

(2.52)

í .

2.4.1. y1 (t), y2 (t), . . . , yn (t) (2.51), (2.52) [t0 + h, t0 + h], h T, : 1. yi (t) [t0 - h, t0 + h], i = 1, 2, . . . , n; 2. (t, y1 (t), y2 (t), . . . , yn (t)) n
+1

, t [t0 - h, t0 + h];

3. yi (t) = fi (t, y1 (t), y2 (t), . . . , yn (t)), t [t0 - h, t0 + h], i = 1, 2, . . . , n; 4. yi (t0 ) = y0i , i = 1, 2, . . . , n. , 2.3.1, n+1 , n+1 fi (t, y1 , . . . , yn ). 2.4.2. f (t, y1 , y2 , . . . , yn ) n+1 y1 , y2 , . . . , yn , L > 0 , |f (t, y1 , y2 , . . . , yn ) - f (t, y1 , y2 , . . . , yn )| L |y1 - y1 | + |y2 - y2 | + § § § + |yn - yn | , (t, y1 , y2 , . . . , yn ), (t, y1 , y2 , . . . , yn ) n
+1

. (2.53)

(2.51), (2.52) .


2.4.

57

[t0 - T , t0 + T ], , , . . 2.4.1. fi (t, y1 , y2 , . . . , yn ), i = 1, 2, . . . , n, n+1 , n+1 (2.53) |fk (t, y1 , y2 , . . . , yn )| M, (t, y1 , y2 , . . . , yn ) n
+1

,

k = 1, 2, . . . , n.

y1 (t), y2 (t), . . . , yn (t), (2.51), (2.52) [t0 - h, t0 + h], h = min T , A M .

. 2.3.1. . [t0 - h, t0 + h] yi (t)
t

yi (t) = y0i +
t0

fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d ,

i = 1, 2, . . . , n.

(2.54)

, , y1 (t), y2 (t), . . . , yn (t) ï ï ï [t0 - h, t0 + h], |yi (t) - y0i | ï A, t [t0 - h, t0 + h], i = 1, 2, . . . , n (2.55)

(2.54), (2.51), (2.52) [t0 - h, t0 + h]. , (2.55) , (t, y1 (t), y2 (t), . . . , yn (t)) n ï ï ï
+1

t [t0 - h, t0 + h].

(2.54) t = t0 , , yi (t) ï (2.52). (2.54) t, , (2.51). , , yi (t) [t0 - h, t0 + h], ï


58

2.

(2.55) (2.54). yi (t), ï . k k k y1 (t), y2 (t), . . . , yn (t), k = 0, 1, 2, . . . ,
t k+1 i k k k fi ( , y1 ( ), y2 ( ), . . . , yn ( ))d , t0 0 yi (t) = y0i ,

y

(t) = y0i +

i = 1, 2, . . . , n, (2.56)

i = 1, 2, . . . , n.

k , yi (t) [t0 - h, t0 + h] k |yi (t) - y0i |

A,

t [t0 - h, t0 + h].

(2.57)

0 m yi (t) . , yi (t) m+1 , yi (t). fi (t, y1 , y2 , . . . , yn ) m n+1 , (2.56) , yi +1 (t) [t0 - h, t0 + h]. ,

|y

m+1 i

(t) - y0i |

A,

t [t0 - h, t0 + h].

(2.56). ,
t

|y

m+1 i t

(t) - y0i |
t0

m m m |fi ( , y1 ( ), y2 ( ), . . . , yn ( ))|d

M d
t0

M |t - t0 |

Mh

A,

i = 1, 2, . . . , n,

t [t0 - h, t0 + h],

. , i = 1, 2, . . . , n k = 0, 1, . . . [t0 - h, t0 + h] |y
k+1 i k (t) - yi (t)|

A(nL)k

|t - t0 |k . k!

(2.58)


2.4. k = 0 ,
t 1 i 0 i 0 0 0 |fi ( , y1 ( ), y2 ( ), . . . , yn ( ))|d t0

59

|y (t) - y (t)| =

Mh

A.

(2.58) k = m - 1. , k = m: |y
t m m m |fi ( , y1 ( ), y2 ( ), . . . , yn ( ))- t0 m+1 i m (t) - yi (t)|

-fi ( , y
t

m-1 1

( ), y

m-1 2

( ), . . . , y

m-1 n

( ))|d

m L |y1 ( ) - y t0

m-1 1

m ( )| + |y2 ( ) - y

m-1 2

( )| + . . .

m § § § + |yn ( ) - y

m-1 n

( )| d .

,
t m+1 i m i

|y

(t) - y (t)|
t0

A(nL)m

| - t0 |m-1 d (m - 1)!

A(nL)m

|t - t0 |m . m!

, (2.58) . [t0 - h, t0 + h]


y (t) +
m=0

0 i

(y

m+1 i

m (t) - yi (t)),

i = 1, 2, . . . , n.

(2.58) , [t0 - h, t0 + h] |y
m+1 i m (t) - yi (t)|

A(nL)m

hm , m!

m = 0, 1, . . .


60

2.

, , [t0 - h, t0 + h]. ,
k-1 k 0 yi (t) = yi (t) + m=0

(y

m+1 i

m (t) - yi (t)),

i = 1, 2, . . . , n

[t0 - h, t0 + h] yi (t). k (2.57), , ï yi (t) (2.55). ï (2.56), , yi (t) (2.54), ï (2.51), (2.52). .


3.1.

61

3
3.1. n-

t [a, b] y (t) , y (t) = u(t) + iv (t), u(t) v (t) í . y (t) [a, b], u(t) v (t) [a, b]. y (t) [a, b], u(t) v (t) [a, b], y (t) = u (t) + iv (t). y (t). . 3.1.1. y (t) + 2y (t) + 5y (t) = 0. (3.1) y (t) = et , í . (3.1) et , 2 + 2 + 5 = 0. 1 = -1 + 2i, 2 = -1 - 2i. , z = x + iy , ez = ex cos y + iex sin y . , (3.1) y1 (t) = e-t cos 2t + ie
-t

sin 2t,

y2 (t) = e-t cos 2t - ie-t sin 2t.

(3.2)


62

3.

n- . [a, b] a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = f (t) (3.3)

ak (t) f (t) = g (t) + ih(t), g (t), h(t) í , a0 (t) = 0 [a, b]. 3.1.1. y (t) = u(t) + iv (t) (3.3) [a, b], u(t) v (t) n [a, b] [a, b] a0 (t)u(n) (t) + a1 (t)u(n-1) (t) + § § § + an-1 (t)u (t) + an (t)u(t) = g (t), (3.4) a0 (t)v (n) (t) + a1 (t)v (n-1) (t) + § § § + an-1 (t)v (t) + an (t)v (t) = h(t). (3.5) (3.3). (3.3) , y
(m)

(t0 ) = y

0m

,

m = 0, 1, . . . , n - 1,

(3.6)

y0m í y0m = u0m + iv0m , u0m , v0m R, t0 í [a, b]. (3.3), (3.6). 3.1.1. ak (t), k = 0, 1, . . . , n, g (t) h(t) [a, b], a0 (t) = 0, t [a, b]. y (t), (3.3), (3.6) [a, b]. . (3.4) u
(m)

(t0 ) = u

0m

,

m = 0, 1, . . . , n - 1.

(3.7)

2.3.5 2.3.6 (3.4), (3.7) u(t). (3.5) v
(m)

(t0 ) = v0m ,

m = 0, 1, 2, . . . , n - 1

(3.8)


3.1.

63

v (t). y (t) = u(t) + iv (t) (3.3), (3.6) [a, b]. (3.3), (3.6) (3.4), (3.7) (3.5), (3.8). 3.1.1 . 3.1.1. f (t) (3.3) ( h(t) = 0) (3.6) ( v0m = 0, m = 0, 1, . . . , n - 1), (3.3), (3.6) . n- . [a, b] n- y1 (t) = a11 (t)y1 (t) + a12 (t)y2 (t) + § § § + a1n (t)yn (t) + f1 (t), y2 (t) = a21 (t)y1 (t) + a22 (t)y2 (t) + § § § + a2n (t)yn (t) + f2 (t), (3.9) ... yn (t) = an1 (t)y1 (t) + an2 (t)y2 (t) + § § § + ann (t)yn (t) + fn (t), akj (t) í , fk (t) = gk (t) + ihk (t) í , k , j = 1, 2, . . . , n. 3.1.2. y (t) = (u1 (t) + iv1 (t), u2 (t) + iv2 (t), . . . , un (t) + ivn (t)) (3.9), uk (t), vk (t) [a, b], k = 1, 2, . . . , n, u1 (t) = a11 (t)u1 (t) + a12 (t)u2 (t) + § § § + a1n (t)un (t) + g1 (t), u2 (t) = a21 (t)u1 (t) + a22 (t)u2 (t) + § § § + a2n (t)un (t) + g2 (t), (3.10) ... un (t) = an1 (t)u1 (t) + an2 (t)u2 (t) + § § § + ann (t)un (t) + gn (t), v1 (t) = a11 (t)v1 (t) + a12 (t)v2 (t) + § § § + a1n (t)vn (t) + h1 (t), v2 (t) = a21 (t)v1 (t) + a22 (t)v2 (t) + § § § + a2n (t)vn (t) + h2 (t), (3.11) ... vn (t) = an1 (t)v1 (t) + an2 (t)v2 (t) + § § § + ann (t)vn (t) + hn (t) t [a, b].


64

3. yk (t0 ) = y
0k

=u

0k

+ iv0k ,

(3.12)

u0k , v0k í , k = 1, 2, . . . , n. (3.9), (3.12). 3.1.2. akj (t), gk (t), hk (t) [a, b], k , j = 1, 2, . . . , n. y (t), (3.9), (3.12) ï [a, b]. . (3.10) uk (t0 ) = u0k , k = 1, 2, . . . , n. (3.13)

2.3.4 2.3.5 (3.10), (3.13) (u1 (t), u2 (t), . . . , un (t)). (3.11) vk (t0 ) = v0k , k = 1, 2, . . . , n. (3.14)

(v1 (t), v2 (t), . . . , vn (t)). y (t) = (u1 (t) + iv1 (t), u2 (t) + iv2 (t), . . . , un (t) + ivn (t)) ï (3.9), (3.12) [a, b]. (3.9), (3.12) (3.10), (3.13) (3.11), (3.14). 3.1.2 . 3.1.2. fk (t) (3.9) ( hk (t) = 0) (3.12) ( v0k = 0, k = 1, 2, . . . , n), (3.9), (3.12) .


3.2. n-

65

3.2. n-
n- a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = f (t) (3.15)

[a, b] ak (t), k = 0, 1, . . . , n, a0 (t) = 0, t [a, b] [a, b] f (t). n- . 3.2.1. n Ly = a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t). (3.16)

L n [a, b] y (t), Ly (t) C [a, b]. , (3.15) Ly = f (t), t [a, b].

f (t) [a, b], (3.15) , f (t) [a, b], (3.15) . 3.2.1. yk (t), k = 1, 2, . . . , m m

Lyk = fk (t), y (t) =
k=1 m

ck yk (t) , ck

í , Ly = f (t), f (t) =
k=1

ck fk (t).

. L, :
m m m

Ly = L
k=1

ck yk (t) =
k=1

ck Lyk =
k=1

ck fk (t) = f (t),

t [a, b].


66

3.

3.2.1. . . 3.2.2. Ly = f (t), y (t0 ) = y00 , y (t0 ) = y01 , ..., y
(n-1)

(t0 ) = y

0n-1

y (t) = v (t) + w(t), v (t) Lv = f (t), v (t0 ) = 0, v (t0 ) = 0, ..., v
(n-1)

(t0 ) = 0,

w(t) Lw = 0, w(t0 ) = y00 , w (t0 ) = y01 , ..., w
(n-1)

(t0 ) = y

0n-1

.

. y (t) = v (t) + w(t) 3.2.1. y
(k )

(t0 ) = v

(k )

(t0 ) + w

(k )

(t0 ) = 0 + y

0k

= y0 k ,

k = 0, 1, . . . , n - 1.

3.2.3. Ly = 0, y (t0 ) = y00 , y (t0 ) = y01 , ..., y
(n-1)

(t0 ) = y

0n-1


n-1

y (t) =
m=0

ym (t)y

0m

,

ym (t) : Ly
m

= 0,

y

(m) m

(t0 ) = 1,

y

(k ) m

(t0 ) = 0,

k {0, 1, . . . , n - 1}\{m}.


3.3.

67

. y (t) ym (t) 3.2.1. :
n-1

y

(k )

(t0 ) =
m=0

y

(k ) m

(t0 )y

(m) 0

=y

(k ) k

(t0 )y

0k

= y0k ,

k = 0, 1, . . . , n - 1.

3.3.
3.3.1. 1 (t), 2 (t), . . . , m (t), [a, b] . . 3.3.1. 1 (t), 2 (t), . . . , m (t) [a, b],
m

ck C, k = 1, . . . , m,
k=1

|ck | > 0, -

c1 1 (t) + c2 2 (t) + § § § + cm m (t) = 0, t [a, b]. (3.17)

(3.17) ck = 0, k = 1, 2, . . . , n, 1 (t), 2 (t), . . . , m (t) [a, b]. 3.3.1. , , k (t) , ck , k = 1, 2, . . . , m.


68

3. 3.3.1. [a, b] 1 (t) = t3 , 2 (t) = t2 |t|. 2 (t) t = -d d3 = 0, 1 (t) =

0 < a < b, 1 (t) = [a, b]. a < 0 < b, , t = d = min{|a|, b} c1 1 (t) + c2 2 (t) = 0, c1 d3 + c2 c1 d3 - c2 d3 = 0, , c1 = c2 = 0, t3 2 (t) = t2 |t| [a, b].

3.3.2. 3.3.1 , , . 3.3.2. 1 (t), 2 (t), . . . , m (t), (m - 1) [a, b] , t [a, b] W [1 , . . . , m ](t) = det 1 (t) 1 (t) . . .
(m-1) 1

2 (t) 1 (t) . . .
(m-1) 2

... ... .. . ...

m (t) m (t) . . . m
(m-1)

.

(t)

(t)

(t)

. 3.3.1. (m-1) [a, b] 1 (t), 2 (t), . . . , m (t) [a, b], : W [1 , . . . , m ](t) = 0, t [a, b].

. k (t) [a, b], c1 , c2 , . . . , cn , [a, b] (3.17). m - 1 : c1
(k ) 1

(t) + § § § + cm

(k ) m

(t) = 0,

k = 0, 1, . . . , m - 1,

t [a, b]. (3.18)


3.3.

69

(3.18) , - t [a, b]. , t [a, b]. 3.3.1. (m - 1) [a, b] 1 (t), 2 (t), . . . , m (t) t0 [a, b], W [1 , . . . , m ](t0 ) = 0, [a, b]. , , , . . 3.3.2. m = 2 [-1, 1] , : 1 (t) = t3 , 2 (t) = t2 |t|, W [1 , 2 ](t) = det t3 3t2 t2 |t| 3t|t| 0.

, 3.3.1, [-1, 1]. 3.3.2. n c [a, b] aj (t), j = 0, . . . , n, a0 (t) = 0 [a, b]: a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = 0. (3.19)

y1 (t), y2 (t), . . . , yn (t), (3.19) n. , . .


70

3.

(3.19) . , . 3.3.2. y1 (t), y2 (t), . . . , yn (t) (3.19) [a, b] : W [y1 , . . . , yn ](t) 0 [a, b] y1 (t), y2 (t), . . . , yn (t) ; W [y1 , . . . , yn ](t) = 0 t [a, b] y1 (t), y2 (t), . . . , yn (t) [a, b]. . - t0 , yk (t), , W [y1 , . . . , yn ](t0 ) = 0. c1 , c2 , . . . , cn : c1 y1 (t0 ) + c2 y2 (t0 ) + § § § + cn yn (t0 ) = 0, c1 y1 (t0 ) + c2 y2 (t0 ) + § § § + cn yn (t0 ) = 0, (3.20) ... (n-1) (n-1) (n-1) (t0 ) = 0. c1 y1 (t0 ) + c2 y2 (t0 ) + § § § + cn yn (W [y1 , . . . , yn ](t0 ) = 0),
n

c1 , c2 , . . . , cn ,
k=1

|ck | > 0.


n

y (t) =
k=1

ck yk (t).

3.2.1 , (3.19), (3.20) , y
(m)

(t0 ) = 0,

m = 0, 1, . . . , n - 1.

, y (t) (3.19)


3.4.

71

t0 . [a, b]. ,
n

y (t) =
k=1

ck yk (t) = 0,

t [a, b],

yk (t), k = 1, 2, . . . , n . 3.3.1 , , , [a, b]. t [a, b] , W [y1 , . . . , yn ]( t ) = 0. , [a, b], yk (t), k = 1, 2, . . . , n . 3.3.3. 3.3.2 [-1, 1] 1 (t) = t3 , 2 (t) = t2 |t|

a0 (t)y (t) + a1 (t)y (t) + a2 (t)y (t) = 0, t [-1, 1]

a0 (t), a1 (t), a2 (t) a0 (t) = 0, W [1 , 2 ](t) 0 [-1, 1].

3.4.
3.4.1. 3.4.1. n- (3.19) [a, b] n .


72

3.

3.4.1. (3.19) [a, b]. . B bij , i, j = 1, 2, . . . , n , det B = 0. yj (t) (3.19) yj (t0 ) = b1j , yj (t0 ) = b2j , . . . , y
(n-1) j

(t0 ) = bnj ,

j = 1, 2, . . . , n.

(3.21)

2.3.5 n- yj (t) . W [y1 , . . . , yn ](t), (3.21), , W [y1 , . . . , yn ](t0 ) = det B = 0. , 3.3.2 [a, b], yj (t) [a, b]. , (3.19) . 3.4.1. 3.4.1 , (3.19) . , B , det B = 0, (3.19). 3.4.2. aj (t) , (3.19) . 3.4.2. 3.4.2. n- (3.19) n , (3.19) . 3.4.2. y1 (t), y2 (t), . . . , yn (t) í (3.19) [a, b]. y
OO

(t) = c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t),

cj C.

(3.22)


3.4.

73

. (3.19) , ck yOO (t), (3.22), (3.19). , (3.19) (3.22) ck . y (t) í (3.19). ck c1 y1 (t0 ) + c2 y2 (t0 ) + § § § + cn yn (t0 ) c1 y1 (t0 ) + c2 y2 (t0 ) + § § § + cn yn (t0 ) c1 y
(n-1) 1

(t0 ) + c2 y

(n-1) 2

(t0 ) + § § § + cn y

(n-1) n

= y (t0 ), = y (t0 ), ... (t0 ) = y (n-1) (t0 ),

(3.23)

t0 í [a, b]. t0 , y1 (t), y2 (t), . . . , yn (t) . , (3.23) c1 , c2 , . . . , cn .
n

y (t) =
k=1

ck yk (t).

(3.19). c1 , c2 , . . . , cn (3.23), y (t) , y
(k )

(t0 ) = y

(k )

(t0 ),

k = 0, 1, . . . , n - 1.

, y (t) y (t) (3.19) t0 . :
n

y (t) = y (t) =
k=1

ck yk (t).

3.4.2 .


74

3.

3.4.1. 3.4.2 , (3.19) n . , , , a0 (t) [a, b]. 3.4.1. [-1, 3] y1 (t) = t, y2 (t) = t3 , y3 (t) = |t|3 .

t2 y - 3ty + 3y = 0, t [-1, 3],

a0 (t) = t2 , t = 0 [-1, 3] , a0 (t) = 0 t [a, b] 3.4.2 . 3.4.3. (3.19) , . (. 3.4.1 ) (3.22) cj R . 3.4.3. [a, b] aj (t), j = 0, . . . , n, a0 (t) = 0, t [a, b]

[a, b] f (t): a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = f (t). (3.24)

n- (3.24). .


3.4.

75

3.4.3. n- (3.24) n , (3.24) . 3.4.3. y1 (t), y2 (t), . . . , yn (t) í (3.19) [a, b], yH (t) í () (3.24). (3.24) y
OH

(t) = yH (t) + y

OO

(t) = = yH (t) + c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t), (3.25)

c1 , c2 , . . . , cn í . . cj C (3.25) (3.24) . , (3.25) (3.24), y (t) (3.24) c1 , c2 , . . . , cn , [a, b] y (t) = yH (t) + c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t). (3.26)

y (t) í (3.24). y (t) = y (t) - yH (t) (3.24) (3.19). 3.4.2 cj , y (t) = c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t), (3.26). 3.4.4. 3.4.3 , (3.24) (3.19) -


76

3.

(3.24). yH (t) (3.24) , (3.19). , (3.22) , c1 , c2 , . . . , cn [a, b] c1 (t), c2 (t), . . . , cn (t), : yH (t) = c1 (t)y1 (t) + c2 (t)y2 (t) + § § § + cn (t)yn (t). (3.27)

ck (t) ck (t) (3.27) t [a, b] c1 (t)y1 (t) + c2 (t)y2 (t) + § § § + cn (t)yn (t) (1) (1) (1) c1 (t)y1 (t) + c2 (t)y2 (t) + § § § + cn (t)yn (t) c1 (t)y c1 (t)y
(n-2) 1 (n-1) 1

(t) + c2 (t)y (t) + c2 (t)y

(n-2) 2 (n-1) 2

(t) + § § § + cn (t)y (t) + § § § + cn (t)y

(n-2) n (n-1) n

= = ... (t) = (t) =

0, 0, 0, f (t) . a0 (t)

yk (t) , ck (t) , ck (t) = gk (t), k = 1, 2, . . . , n.
t

, ck (t) =
t0

gk ( )d .

(3.27) yH (t) yH (t) y
(n-1) H

= c1 (t)y1 (t) + c2 (t)y2 (t) + cn (t)yn (t), = c1 (t)y1 (t) + c2 (t)y2 (t) + cn (t)yn (t), ... (n-1) (n-1) (t) = c1 (t)y1 (t) + c2 (t)y2 (t) + cn (t)y

(n-1) n

(t),


3.4.
n

77

y

( n) H

(t) = c1 (t)y

( n) 1

(t) + c2 (t)y = c1 (t)y

( n) 2

(t) + cn (t)y

( n) n

(t) +
k=1

ck (t)y
( n) n

(n-1) k

(t) =

( n) 1

(t) + c2 (t)y

( n) 2

(t) + cn (t)y

(t) +

f (t) . a0 (t)

, (3.27) (n - 1) , cj (t) t . yH (t) (3.24), LyH (t) = a0 (t) § f (t) + a0 (t) a0 (t)
n n

ck (t)y
k=1 n -1

( n) k

(t) + a1 (t)
k=1

ck (t)y
n

(n-1) k

(t) + . . .

§ § § + an

(t)
k=1

ck (t)yk (t) + an (t)
k=1

ck (t)yk (t).

(3.16) L,
n

LyH (t) = f (t) +
k=1

ck (t)Lyk (t) = f (t) + 0 = f (t),

t [a, b],

yk (t), k = 1, 2, . . . , n (3.19), Lyk (t) = 0. , ,
n t

yH (t) = c1 (t)y1 (t) + c2 (t)y2 (t) + § § § + cn (t)yn (t) =
k=1

yk (t)
t0

gk ( )d

(3.24). 3.4.5. n- c aj R, j = 0, . . . , n, a0 = 0: a0 y
( n)

(t) + a1 y

(n-1)

(t) + § § § + an

-1

y (t) + an y (t) = 0.

(3.28)


78

3.

Ly = 0, L Ly = a0 y
( n)

(t) + a1 y

(n-1)

(t) + § § § + an

-1

y (t) + an y (t).

L M () = a0 n + a1
n-1

+ § § § + an

-1

+ an .

(3.29)

M () , M () = 0 (3.30) . , exp{0 t} (3.28) , 0 (3.30). 1 , . . . , , M (j ) = 0, k1 , . . . , k , k1 + § § § + k = n. , M () = a0 ( - 1 )k1 ( - 2 )k2 . . . ( - )k . (3.31)

3.4.1. n g (t) C
n

L exp{t}g (t) = exp{t}
m=0

M

(m)

()g m!

(m)

(t)

.

. dp exp{t}g (t) = dtp
p p p Cn m=0

dp-m exp{t} dtp-m

dm g (t) = dtm
p-m (m)

= exp{t}
m=0

p(p - 1) . . . (p - (m - 1)) m! = exp{t}

g

(t) =
p

p

m=0

dm dm

g

(m)

(t) . m!


3.4. ,
n

79

L exp{t}g (t) =
p=0

an

-p n

dp exp{t}g (t) = dtp
p

= exp{t}
p=0

an

-p m=0

dm dm
n

p

g

(m)

(t) m! dm dm
p

n

= exp{t}
p=0

an

g

(m)

-p m=0

(t) , m!

dm p /d

m

= 0, m = p + 1, . . . , n. ,

n

L exp{t}g (t) = exp{t}
m=0

g

(m)

(t) dm m! dm

n

an
p=0

-p



p

=
(m)

n

= exp{t}
m=0

g

(m)

(t) M m!

().

3.4.2. j (3.30) kj exp{j t}, t exp{j t}, ..., t
kj -1

exp{j t}

(3.28). . j í (3.30) kj , (3.31) M () = ( - j )kj R(), R() í n - kj . , M
(m)

(j ) =

dm M () dm

= 0,
=j

m = 0, 1, . . . , kj - 1.


80

3.

3.4.1 g (t) = tp , p = 0, 1, . . . , kj - 1
n p

L exp{j t}t

= exp{j t}
m=0 n

t

p (m)

m! M

M
(m)

(m)

(j ) =

= exp{j t}
m=k
j

t

p (m)

m!

(j ) = 0 ( p < kj ).

, , exp{j t}, t exp{j t}, ..., tk
j

-1

exp{j t},

j = 1, . . . , .

(3.32)

(3.28). n (3.28). 3.4.4. (3.32) (3.28) [a, b].

-

. , (3.32) [a, b]. , (3.32) :
k1 -1 k2 -1 k -1

C1,k tk exp{1 t} +
k=0 k=0

C2,k tk exp{2 t} + § § § +
k=0

C

,k

tk exp{ t} 0,

P1 (t) exp{1 t} + P2 (t) exp{2 t} + § § § + P (t) exp{ t} 0, (3.33)

sj = deg Pj (t) kj - 1, j = 1, . . . , . , P (t) , P (t) = p ts + . . . , p = 0. (3.33) exp{-1 t} P1 (t) + P2 (t) exp{(2 - 1 )t} + § § § + P (t) exp{( - 1 )t} 0.


3.4.

81

s1 + 1 . ds1 +1 P1 (t) deg P1 (t) = s1 , 0. dts1 +1 , (Pj (t) exp{²t}) = (²Pj (t) + Pj (t) ) exp{²t}, ² = j - 1 = 0,

. ds1 +1 (Pj (t) exp{(j - 1 )t}) = Qj (t) exp{(j - 1 )t}, dts1 +1 deg Qj (t) = sj , Qj (t) = (j - 1 )s Q2 (t) exp{(2 - 1 )t} + § § § + Q (t) exp{( - 1 )t} 0. exp{(1 - 2 )t} s2 + 1 R3 (t) exp{(3 - 2 )t} + § § § + R (t) exp{( - 2 )t} 0, Rj (t) = (j - 2 )
s2 +1

1

+1

pj t

s

j

+ ....

deg Rj (t) = sj , j = 3, . . . , .

(j - 1 )

s1 +1

pj t

s

j

+ ...,

, S (t) exp{( -
-1

)t} 0, )
s
-1

deg S (t) = s , . . . ( - 2 )s
2

S (t) = ( -

-1

+1

+1

( - 1 )s

1

+1

p ts + . . . .

P (t) p = 0. (3.32). 3.4.6. (3.28) ,


82

3.

. (3.29) . , : = + i , = - i , , R. (3.32) , Mn (), , : y (t) = ts exp{t}(cos t + i sin t), y (t) = ts exp{t}(cos t - i sin t).

: yR (t) = Re y (t) = ts exp{t} cos t, yI (t) = Im y (t) = ts exp{t} sin t. (3.34)

yR (t), yI (t) (3.28) . n (3.28) . [a, b]. , rj R [a, b]. , § § § + r1 yR (t) + r2 yI (t) + § § § = 0,
2 2 r1 + r2 > 0.

(3.34) , § § § + 0.5(r1 - ir2 )y (t) + 0.5(r1 + ir2 )y (t) + § § § = 0,
2 2 r1 + r2 > 0.

, c (3.32) , .


3.5.

83

3.4.2. , y1 (t) = 1, y2 (t) = sin(2t).

y1 (t) = exp{0 § t}, y2 (t) = Im exp{2it}.

, y3 (t) = Re exp{2it} . exp{0 § t}, exp{2it}, exp{-2it},

3, 1 = 0, 2 = 2i, 3 = -2i. M () = ( - 2i)( + 2i) = 3 + 4 y + 4y = 0.

3.5. n-
3.5.1. y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = 0,

(3.35)

. , : , , . .


84

3.

3.5.1. am (t) [a, b], m = 1, 2, . . . , n. (3.35) . . y1 (t), y2 (t), . . . , yn (t) í (3.35). , n- [a, b] bm (t), m = 1, 2, . . . , n, y1 (t), y2 (t), . . . yn (t) . , am (t) = bm (t), t [a, b], m = 1, 2, . . . , n. , yk (t) , y
( n) k (t) ( n) yk (t)

+ a1 (t)y + b1 (t

(n-1) k (n-1) )yk

(t) + § § § + an (t) + § § § + b

-1

(t)yk (t) + an (t)yk (t) = 0, (t)yk (t) + bn (t)yk (t) = 0,

t [a, b], t [a, b],

n-1

k = 1, 2, . . . , n. k , (a1 (t)-b1 (t))y
(n-1) k

(t)+§ § §+(an

-1

(t)-b

n-1

(t))yk (t)+(an (t)-bn (t))yk (t) = 0,

t [a, b] k = 1, 2, . . . , n. , t0 (a, b) , a1 (t0 ) = b1 (t0 ). a1 (t), b1 (t) > 0, a1 (t) = b1 (t), t [t0 - , t0 + ] [a, b]. am (t) - bm (t) , a1 (t) - b1 (t)

a1 (t) - b1 (t) pm (t) = y
(n-1) k

(t)+p2 (t)y

(n-2) k

(t)+§ § §+p

n-1

(t)yk (t)+pn (t)yk (t) = 0, t [t0 -, t0 +],

k = 1, 2, . . . , n. , , n y1 (t), y2 (t), . . . , yn (t) (n - 1)- pm (t). , (n - 1)- n - 1 . , a1 (t) = b1 (t), t [a, b]. . 3.5.1 .


3.5.

85

, . 3.5.2. n [a, b] y1 (t), y2 (t), . . . , yn (t) , W [y1 , y2 , . . . , yn ](t) [a, b]. n- , y1 (t), y2 (t), . . . , yn (t) . . y (t) y1 (t) y2 (t) y1 (t) y2 (t) y1 (t) y2 (t) . . det . . . . (n-1) (n-1) y1 (t) y2 (t) ( n) ( n) y1 (t) y2 (t) [a, b] n- . . . . . . . . . . . yn (t) yn (t) yn (t) . . . y (t) y (t) y (t) . . .

= 0. . (n-1) (t) y (n-1) (t) . . . yn ( n) y (n) (t) ... yn (t)

(3.36)

, , (3.36) n- , . y (n) (t) , y1 (t), y2 (t), . . . yn (t), [a, b]. , (3.35) [a, b] . y1 (t), y2 (t), . . . yn (t) , y (t) = yk (t) (3.36) . 3.5.2 . 3.5.1. , y1 (t) = t, y2 (t) = exp{t2 }, y3 (t) = t2 , y4 (t) = 3t - 2t2 .


86

3.

, y4 (t) = 3y1 (t) - 2y3 (t), y1 (t), y2 (t) y3 (t) t exp{t2 } t2 2t exp{t2 } 2t = W [y1 , y2 , y3 ](t) = det 1 2 2 2 0 2 exp{t } + 4t exp{t } 2 = -2 exp{t2 }(2t4 - t2 + 1) = 0, t R. 3.5.2, t exp{t2 } t2 y 1 1 2t exp{t2 } 2t y = 0. det 0 (2 + 4t2 ) exp{t2 } 2 y W [y1 , y2 , y3 ](t) 3 2 0 (12t + 8t ) exp{t } 0 y 3.5.2. [1, 2] , y1 (t) = 1, y2 (t) = cos(t), y3 (t) = sin2 (t/2).

, y3 (t) = 0.5(y1 (t) - y2 (t)), y1 (t) y2 (t) W [y1 , y2 ](t) = det 1 cos t 0 - sin t = - sin t = 0, t [1, 2].

3.5.2, 1 cos t y 1 = 0, y - ctg(t) y = 0. det 0 - sin t y W [y1 , y2 ](t) 0 - cos t y


3.5. 3.5.2. -

87

(3.36), . . D(t) í n- , , [a, b]. D (t) D(t) n , D(t) . (t) = W [y1 , y2 , . . . , yn ](t), n [a, b] y1 (t), y2 (t), . . . , yn (t), (t) = det y y1 (t) y1 (t) . . . y2 (t) y2 (t) . . . ... ... .. . y y
n-1 n-1

. (n-2) (n-2) (n-2) (n-2) (t) (t) y2 (t) . . . yn-1 (t) yn 1 ( n) ( n) ( n) ( n) y1 (t) y2 (t) . . . yn-1 (t) yn (t) . . .

(t) (t)

yn (t) yn (t) . . .



, (t). , , , , , . , , , (t). y1 (t), y2 (t), . . . , yn (t) í (3.35). 3.5.1 , . , (3.36) (t), (3.35). (3.36) , a1 (t) = - (t) . (t)


88

3.

t0 t, -
t

(t) = (t0 ) exp -
t0

a1 ( )d ,

t [a, b].

3.5.1. a1 (t) = 0, t [a, b], W [y1 , y2 , . . . , yn ](t) [a, b].


4.1.

89

4
4.1.
[a, b] [a, b] ai,j (t) fk (t): dy (t) = A(t)y (t) + f (t), dt a11 (t) § § § a1n (t) . . , .. . A(t) = . . . . an1 (t) § § § ann (t) f1 (t) . f (t) = . . . fn (t) t [a, b], (4.1)

, y (t) = (y1 (t), . . . , yn (t)) (4.1) , , - y (t) = u(t) + iv (t), u(t) = (u1 (t), . . . , un (t)) , v (t) = (v1 (t), . . . , vn (t)) ,

uj (t), vj (t) , j = 1, . . . , n, , , . 4.1.1. 4.1.1. (4.1) , f (t) [a, b]. (4.1) .


90

4. - -

= (0, . . . , 0) .

4.1.1. y (t) í , y (t) C. y 1 (t) y 2 (t) í , y (t) = y 1 (t) + y 2 (t) . . dy (t)/dt = A(t)y (t), d{y (t)} dy (t) = = A(t)y (t) = A(t){y (t)}. dt dt dy (t)/dt = A(t)y (t), = 1, 2,

dy (t) d{y 1 (t) + y 1 (t)} dy (t) dy 1 (t) = =1 + = dt dt dt dt = A(t)y 1 (t) + A(t)y 2 (t) = A(t)y (t).

4.1.1. y (t) í m



= 1, . . . , m, y (t) =
=1

y (t) -

C. 4.1.2. [a, b] ai,j (t), i, j = 1, 2, . . . , n dy (t) = A(t)y (t), dt a11 (t) § § § a1n (t) . . , .. . A(t) = . . . . an1 (t) § § § ann (t) y1 (t) . y (t) = . . . yn (t) t [a, b], (4.2)


4.1. n - y j (t) = (y1j (t), . . . , ynj (t)) , j = 1, . . . , n.

91

Y (t), : y11 (t) § § § y1n (t) . . . .. . Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)) = . . . . yn1 (t) § § § ynn (t)

(4.3)

(4.2) d Y (t) = A(t)Y (t), (4.4) dt , , d Y (t)/dt = dyij (t)/dt . , (4.4) [a, b] (4.3), (4.4) . (4.4) (4.2) , : ""A(t) Y (t) í . (4.2) (4.4) . 4.1.1. - y 1 (t), y 2 (t), . . . , y n (t) (4.2) [a, b] , Y (t) (4.3) (4.4). . y 1 (t), y 2 (t), . . . , y n (t) (4.2) Y (t) (4.3). dy j (t) = A(t)y j (t), dt j = 1, . . . , n,


92

4.

, , d Y (t) = dt dy 1 (t) dy 2 (t) dy (t) , ,..., n dt dt dt =

= (Ay 1 (t), Ay 2 (t), . . . , Ay n (t)) = A(t)Y (t). (4.4). , (4.4) , .

4.1.2. Y (t) (4.4). : 1. c = (c1 , c2 , . . . , cn ) , cj C, y (t) = Y (t)c (4.2); 2. B = bi,j , bi,j C, i, j = 1, . . . , n, X (t) = Y (t)B (4.4). . 1. Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)) (4.4), 4.1.1 - y j (t) (4.2),
n

y (t) = Y (t)c =
j =1

cj y j (t).

2. , : dX (t) d d Y (t) = {Y (t)B } = §B = dt dt dt = {A(t)Y (t)} B = A(t) {Y (t)B } = A(t)X (t).


4.2.

93

4.2. -
4.2.1. - - y 1 (t), y 2 (t), . . . , y m (t), [a, b], y j (t) = (yj 1 (t), . . . , yj m (t)) , j = 1, . . . , m, m N. . 4.2.1. - y 1 (t), y 2 (t), . . . , y m (t) [a, b], m

c1 , c2 , . . . , cm ,
j =1

|cj | > 0 , t [a, b]. (4.5)

c1 y 1 (t) + c2 y 2 (t) + § § § + cm y m (t) = ,

(4.5) , c = (0, . . . , 0) , - y 1 (t), y 2 (t), . . . , y m (t) [a, b]. (4.5) , Y (t) m ½ m Y (t)c = , t [a, b] (4.6)

c = (c1 , . . . , cm ) . 4.2.1. - , cj , j = 1, . . . , m. 4.2.2. [a, b] y 1 (t), y 2 (t), . . . , y m (t) t [a, b] Y (t) = (y 1 (t), y 2 (t), . . . , y m (t)): (t) = det Y (t).


94

4.

- . 4.2.1. y 1 (t), y 2 (t), . . . y m (t) [a, b], : (t) = 0, t [a, b].

. (4.6) c = (c1 , . . . , cm ) , t0 [a, b] Y (t0 )c = . (4.7)

(4.7) , Y (t0 ) c. , det Y (t0 ) = 0. , (4.5), Y (t) t [a, b]. - , , . - . 4.2.1. m = 2 [-1, 1] -, : y 1 (t) = t t
3 2

, y 2 (t) =

t2 |t| , Y (t) = t|t|

t t

3 2

t2 |t| , (t) = det Y (t) 0. t|t|

- . , c = (c1 , c2 ) Y (t)c = [-1, 1], t = 1 c1 + c2 = 0, t = -1 í c1 - c2 = 0, c1 = c2 = 0.


4.2.

95

4.2.2. n- - y 1 (t), y 2 (t), . . . y n (t), (4.2), Y (t) í (4.3). , - . . 4.2.2. y 1 (t), y 2 (t), . . . , y n (t) í - (4.2) [a, b]. t0 [a, b], det Y (t0 ) = 0, - y 1 (t), y 2 (t), . . . , y n (t) [a, b] det Y (t) = 0, t [a, b]. . c = (c1 , . . . , cn ) Y (t0 )c =
0

(4.8)

c = (c0 , . . . , c0 ) n 1 Y (t0 ), . y (t) = Y (t)c0 . , y (t) í (4.2) 4.1.2 , , y (t0 ) = (4.8). , t = t0 : dy (t) = A(t)y (t), y (t0 ) = . dt 2.1.2 í . = y (t) = Y (t)c0 = c0 y 1 (t) + c0 y 2 (t) + § § § + c0 y n (t), 1 2 n t [a, b],

- [a, b]. 4.2.1 det Y (t) = 0, t [a, b].


96

4.

4.2.1 4.2.2 - . 4.2.3. - y 1 (t), y 2 (t), . . . , y n (t), (4.2) [a, b], , det Y (t) 0 ( - ), , det Y (t) = 0, t [a, b] ( - ). 4.2.2. 4.2.3, y 1 (t) = t t
3 2

,

y 2 (t) =

t2 |t| t|t|

4.2.1 [-1, 1] .

4.3.
4.3.1. 4.3.1. dy (t) = dt A(t)y (t) n [a, b] n y 1 (t), y 2 (t), . . . , y n (t) . Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)) . (4.2.3) det Y (t) = (4.1.2) (4.4), [a, b] , 0.


4.3.

97

4.3.1. (4.2) [a, b] . . t0 [a, b] d Y (t) = A(t)Y (t), dt Y (t0 ) = E , (4.9)

E í . , , (4.9) n dy j (t) = A(t)y j (t), dt y j (t0 ) = (0, . . . , 0, 1 , 0, . . . , 0) ,
j

j = 1, . . . , n,

. [a, b] y j (t) , Y (t) (4.9), 2.1.2. Y (t) (4.9) 1, det Y (t0 ) = det E = 1, y 1 (t), y 2 (t), . . . , y n (t) 4.2.3 . , y 1 (t), y 2 (t), . . . , y n (t) í , Y (t) í . 4.3.1. . (4.9) Y (t0 ) = B , det B = 0, . 4.3.2. aij (t) , . 4.3.2. 4.3.2. n- n , .


98

4.

4.3.2. Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)) í dy (t) = A(t)y (t) dt [a, b]. y
OO

(t) = c1 y 1 (t) + c2 y 2 (t) + § § § + cn y n (t) = Y (t)c,

(4.10)

c1 , c2 , . . . , cn í , c = (c1 , c2 , . . . , cn ). . 4.1.2 - Y (t)c c Cn . , y (t) c Cn , [a, b] y (t) = Y (t)c. (4.11)

c t0 [a, b] y 0 = y (t0 ). c = (c1 , c2 , . . . , cn ) : Y (t0 )c = y 0 . (4.12)

Y (t0 ) c det Y (t0 ) = 0 c = (c1 , c2 , . . . , cn ) . y (t) = Y (t)c y (t) dy (t) (4.13) = A(t)y (t), y (t0 ) = y 0 , dt , (4.11). , y (t) c Cn (4.11) . 4.3.1. 4.3.2 (4.13) y 0 . , (4.12) c = Y -1 (t0 )y 0 (4.11) y (t) = Z (t, t0 )y 0 , Z (t, t0 ) = Y (t)Y
-1

(t0 ).

(4.14)


4.3.

99

Z (t, t0 ) . t dZ (t, t0 ) = A(t)Z (t, t0 ), dt Z (t0 , t0 ) = Y (t0 )Y
-1

(t0 ) = E .

4.3.3. aij (t) , . ( ) (4.10) c Rn . 4.3.3. , f (t) = (f1 (t), f2 (t), . . . , fn (t)) : dy (t) = A(t)y (t) + f (t), dt t [a, b]. (4.15)

, Y (t) (4.15) dy (t)/dt = A(t)y (t) A(t). 4.3.3. n- (4.15) n , (4.15) . 4.3.3. y OH (t) (4.15) y
OH

(t) = Y (t)c + y H (t),

c = (c1 , c2 , . . . , cn ) Cn ,

(4.16)

y H (t) í () (4.15). . (4.15) - y OH (t) (4.15) c Cn .


100

4.

, , y (t) (4.15) c Cn , [a, b] y (t) = Y (t)c + y H (t). (4.17)

y (t) í (4.15). y (t) = y (t) - y H (t) , dy (t)/dt = A(t)y (t). 4.3.2 c Cn , y (t) = Y (t)c, (4.17). (4.14) Z (t, ). 4.3.4. t0 [a, b]
t

y H (t) =
t0

Z (t, )f (t)d ,

t [a, b],

(4.18)

(4.15), y H (t0 ) = 0. . , , (4.10) , c - c(t) = (c1 (t), c2 (t), . . . , cn (t)) , : y (t) = Y (t)c(t). (4.19) d Y (t)/dt = A(t)Y (t), dy (t) d Y (t) dc(t) dc(t) = c(t) + Y (t) = A(t)Y (t)c(t) + Y (t) . dt dt dt dt (4.20)

(4.19) (4.20) (4.15), - c(t): Y (t) dc(t) = f (t). dt (4.21)


4.4. : t0 t. ,

101

dc(t)/dt = Y -1 (t)f (t) , - ,
t

c(t) =
t0

Y

-1

( )f ( )d .

(4.19)
t t

y (t) = Y (t)c(t) = Y (t)
t0

Y

-1

( )f ( )d =
t0

Z (t, )f ( )d .

4.3.2. y (t) = y (t; y 0 ) dy (t) = A(t)y (t) + f (t), dt t [a, b]

t0 [a, b] y (t0 ) = y y (t; y 0 ) = Z (t, t0 )y 0 +
t0 0 t

Z (t, )f ( )d .

(4.22)

4.4.
A(t) A = (ai,j ), ai,j R, i, j = 1, . . . , n: dy (t) = Ay (t). dt (4.23)


102

4.

y (t) = ay (t), y (t) = h exp{at} h C, (4.23) y (t) = h exp{t}, h = (h1 , h2 , . . . , hn ) Cn , C. (4.24)

- (4.24) (4.23) C, (4.25) (A - E )h = h. , A, h í A. M (): M () = det(A - E ) = 0. (4.26)

4.4.1. , n, n ( ), 1 , . . . , n , j C. , , n A. , n. , Cn . 4.4.1. A n h1 , h2 , ..., hn ,

1 , 2 , ..., n .


4.4. : -

103

y 1 (t) = h1 exp{1 t}, y 2 (t) = h2 exp{2 t}, . . . y n (t) = hn exp{n t} (4.27) (4.23) [a, b]. . [a, b]. j = 1, . . . , n j hj (4.25), y j (t) = hj exp{j t} (4.23) [a, b] . [a, b] . , 4.2.3 , , det Y (t) = 0 t [a, b], Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)). [c, d], [a, b] t = 0: [a, b] [c, d], 0 [c, d].

- (4.27) (4.23) [c, d]. t = 0 det Y (0) = det(h1 , h2 , . . . , hn ) = 0, Y (0) í h1 , h2 , . . . , hn í . 4.2.3 det Y (t) = 0 [c, d], [a, b]. 4.4.2. , , A , n. j kj : 1 , k1 , 2 , k2 , ..., ..., k , , i = j i = j, kj 1, k1 + k2 + § § § + k = n.


104

4.

{1 , . . . , } k . , k -, (4.23). s = dim Ker(A - E ) , , , s = k , (4.27). , s < k , , 1 1 1 , h1 , h2 , . . . , hs , 1 k hj m hj , m = 2, . . . , pj , j = 1, . . . , s, pj 1, p1 + p2 + § § § + ps = k , h1 , . . 2 h1 , . . . . . .. p1 h1 , . . Ah Ah Ah Ah
1 j 2 j 1

hj , 2 hj , . . . . pj . hj ,

. .

1

... ...

hs , 2 hs , . . ... . ps . . . hs ,

1

=

hj ,
2 1

1

m j

pj j

= h j + hj , ... m m-1 = h j + hj , ... pj pj -1 = h j + hj .

(4.28)

k y 1 (t) = hj exp{t}, j y 2 (t) = j . . . hj +
2 1

t h 1!

1 j

exp{t},

(4.29)


4.4. :
m

105

y m (t) = hj + j

t h 1!

m-1 j

+

t2 h 2!

m-2 j

+ §§§ +

tq h q!

m-q j

+ ...

§§§ +

tm-1 h (m - 1)! . . .
pj -1 j

1 j

exp{t},

y j j (t) = h

p

pj j

+

t h 1!

+

t2 h 2!

pj -2 j

+ §§§ +

tq h q!

pj -q j

+ ...

§§§ +

tpj -1 h (pj - 1)!

1 j

exp{t},

j = 1, . . . , s. , (4.23). y m (t), dy m (t)/dt j j , (4.28). dy m (t) j = dt = h
m-1 j

+

t h 1!

m-2 j

+

t2 h 2!

m-3 j

+ hj +

m

t2 m-2 h 2! j tm-1 1 + h exp{t} = (m - 1)! j t t2 m-2 tq m m-1 m-q = Ahj + Ahj + Ahj + § § § + Ahj + ... 1! 2! q! tm-1 1 §§§ + Ah exp{t} = Ay m (t), j (m - 1)! j m = 1, . . . , pj , j = 1, . . . , s. t h 1!
m-1 j

+

tq m-q-1 tm-2 1 hj + §§§ + h+ q! (m - 2)! j tq m-q tm-2 2 + § § § + hj + §§§ + h + q! (m - 2)! j + §§§ +

, y m (t) í (4.23). j , n -, {1 , . . . , } (4.29), [a, b]. [c, d], [a, b] [c, d], 0 [c, d]. - (4.29)


106

4.

(4.23) [c, d]. t = 0 , Y (0) , A, Cn . 4.2.3 , det Y (t) = 0 [c, d], [a, b]. (4.23) [a, b] , , . . 4.4.2. n -, 1 , 2 , . . . (4.29), (4.23) [a, b]. 4.4.3. , . . 4.3.1 4.3.1 . , ? . . , . , ( ) : = p + iq , = p - iq , M () = 0, M ( ) = 0. 4.4.2 -, , , . y (t) = (y1 (t), . . . , yn (t)) ,
y (t) = (y1 (t), . . . , yn (t))


4.4. : , y R (t) = Re y (t), y R (t) = 0.5(y (t) + y (t)), y I (t) = 0.5i(y (t) - y (t)), y I (t) = Im y (t).

107

(4.30)

y R (t), y I (t) í . - n . [a, b]. , rj R [a, b]. , § § § + r1 y R (t) + r2 y I (t) + § § § = 0,
2 2 r 1 + r 2 > 0.

(4.30) -, § § § + 0.5(r1 - ir2 )y (t) + 0.5(r1 + ir2 )y (t) + § § § = 0,
2 2 r1 + r2 > 0.

, c - , .


108

A

A
A.1.
m m + n (u1 , . . . , um , x1 , . . . , xn ) Rm+n : F1 (u1 , . . . , um , x1 , . . . , xn ) = 0, ... (A.1) Fm (u1 , . . . , um , x1 , . . . , xn ) = 0. (A.1) u1 , . . . , um . (A.1) D Rn u1 = 1 (x1 , . . . , xn ), . . . , um = m (x1 , . . . , xn ) (A.2) , (A.1) : Fi (u1 (x1 , . . . , xn ), . . . , um (x1 , . . . , xn ), x1 , . . . , xn ) = 0, (x1 , . . . , xn ) D, i = 1, . . . , m. F1 , . . . , Fm F1 F1 ... u1 u2 F F2 2 D(F1 , . . . , Fm ) ... = det u1 u2 ... D(u1 , . . . , um ) ... ... Fm Fm ... u1 u2 u1 , . . . , u F1 um F2 um ... Fm um ,
m

-

(u1 , . . . , um , x1 , . . . , xn ).


A.1.1. m F1 (u1 , . . . , um , x1 , . . . , xn ), ..., Fm (u1 , . . . , um , x1 , . . . , xn )

109

N0 = N0 (u0 , . . . , u0 , x0 , . . . , x0 ), 1 m 1 n Fi / uj N0 , i, j = 1, . . . , m. , Fi (N0 ) = 0, i = 1, . . . , m, D(F1 , . . . , Fm ) (N0 ) = 0, D(u1 , . . . , um )

1 , . . . , m M0 (x0 , . . . , x0 ), n 1 m (A.2), |ui - u0 | < i , i = 1, . . . , m i (A.1), M0 . [2], . 13, ç2.

A.2.
m n u1 = 1 (x1 , . . . , xn ), ... um = m (x1 , . . . , xn ).

(A.3)

, i (x1 , . . . , xn ), i = 1, . . . , m, n- D. . k {1, . . . , m} í . A.2.1. uk D (A.3), x = (x1 , . . . , xn ) D uk (x) = (u1 (x), . . . , u
k-1

(x), u

k+1

(x), . . . , um (x)),

(A.4)


110

A

í , . u1 , . . . , u
m

D, D . , D (A.4) k {1, . . . , m}, u1 , . . . , um D. A.2.1. m n m (A.3) M0 = M0 (x0 , . . . , x0 ). 1 n , - m M0 , M0 . i (x1 , . . . , xn ), i = 1, . . . , m M0 (x0 , . . . , x0 ), n 1 M0 . (A.3) 1 1 1 ... x1 x2 xn 2 2 2 ... (A.5) x1 x2 xn , ... ... ... ... m m m ... x1 x2 xn m n . A.2.2. (A.5) 1) r- M0 (x0 , . . . , x0 ); n 1 2) (r + 1)- M0 ( r = min(m, n), ).




111

r , r- , M0 , r . [2], . 13, ç3.


112

B

B
3 n- , 4 n- . , 3 4.

B.1. -
3.3, 4.2 -. - . 1 (t), 2 (t), ..., m (t)

(m - 1) [a, b]. j (t) - j (t), j = 1, . . . , m, m - 1 : j (t) = (j (t), j (t), . . . ,
(m-1) j

(t)) ,

j = 1, . . . , m.

(B.1)

B.1.1. 1 (t), 2 (t), . . . , m (t), (m - 1) [a, b] ,


B.1.

113

, (B.1) - 1 (t), 2 (t), . . . , m (t) [a, b]. . (3.17) c1 , c2 , . . . , cm , [a, b] m c1 1 (t) + c2 2 (t) + § § § + cm m (t c1 1 (t) + c2 2 (t) + § § § + cm m (t .. (m-1) (m (t) + c2 2 (t) + § § § + cm m -1) (t ) = 0, ) = 0, . ) = 0,

(B.2)

c1

(m-1) 1

(3.17), (3.17) . (B.1) (B.2) c1 1 (t) + c2 2 (t) + § § § + cm m (t) = 0, (B.3)

(4.5) - 1 (t), 2 (t), . . . , m (t). , (B.1) 1 (t), 2 (t), . . . , m (t) (B.3) (B.2). (B.2) (3.17). - 3.3.1 4.2.1 -. B.1.1. (m - 1) [a, b] 1 (t), 2 (t), . . . , m (t), [a, b], : W [1 , . . . , m ](t) = 0, t [a, b].


114

B

. 1 (t), 2 (t), ..., m (t)

B.1.1 - 1 (t), 2 (t), ..., m (t).

(B.1) : (t) = det(1 (t), 2 (t), . . . , 1 (t) 2 (t) 1 (t) 1 (t) = det . . . . . . (m-1) (m-1) 1 (t) 2 (t)
m

(t)) = ... ... .. . m (t) m (t) . . .
(m-1)

= W [1 , . . . , m ](t).

. . . m

(t)

W [1 , . . . , m ](t) 4.2.1, (t) = 0.

B.2.
n c [a, b] aj (t) R, j = 0, . . . , n, a0 (t) = 0: a0 (t)y
( n)

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = 0.

(B.4)

(B.4) dy (t) = A(t)y (t), dt (B.5)


B.2. A(t) = 0 0 . . . 0 an (t) - a0 (t) 1 0 . . . 0 an-1 (t) - a0 (t) 0 1 . . . 0 an-2 (t) - a0 (t) ... ... 0 0 . . .

115

... , ... 1 a1 (t) ... - a0 (t)

: y (t) í (B.4), y (t) = (y (t), y (t), . . . , y (n-1) (t)) (B.5). , - y (t) = (y1 (t), y2 (t), . . . , yn (t)) (B.5), y1 (t) (B.4). y1 (t), y2 (t), . . . , yn (t), (B.4) n. -, (B.4) (B.5). B.2.1. y1 (t), y2 (t), . . . , yn (t) (B.4) [a, b] : § W [y1 , . . . , yn ](t) 0 [a, b] y1 (t), y2 (t), . . . , yn (t) ; § W [y1 , . . . , yn ](t) = 0 t [a, b] y1 (t), y2 (t), . . . , yn (t) [a, b]. . t0 [a, b] : W [y1 , . . . , yn ](t0 ) = 0. - y j (t) = (yj (t), yj (t), . . . , y
(n-1) j

(t)) ,

j = 1, . . . , n,

(B.5), Y (t) = (y 1 (t), y 2 (t), . . . , y n (t))


116

B

t = t0 : det Y (t0 ) = W [y1 , . . . , yn ](t0 ) = 0. 4.2.3 , det Y (t) 0 [a, b], - y 1 (t), y 2 (t), . . . , y n (t) . B.1.1 y1 (t), y2 (t), . . . , yn (t) W [y1 , . . . , yn ](t) 0 [a, b]. t0 [a, b] , det Y (t0 ) = W [y1 , . . . , yn ](t0 ) = 0, , 4.2.3 , , W [y1 , . . . , yn ](t) = det Y (t) = 0 [a, b], - y 1 (t), y 2 (t), . . . , y n (t) . B.1.1 y1 (t), y2 (t), . . . , yn (t) .

B.3.
, (B.4) n [a, b] n . B.3.1. (B.4) [a, b] aj (t), j = 1, . . . , n, a0 (t) = 0, [a, b]. . (B.4) (B.5). 4.3.1 Y (t), - (B.5). B.1.1 - (B.4) .


B.4.

117

B.3.2. y1 (t), y2 (t), . . . , yn (t) í (B.4) [a, b]. y
OO

(t) = c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t),

cj C,

j = 1, . . . , n. (B.6)

. (B.6) (B.4) . , (B.6) (B.6). , y (t) (B.6) - y (t) = (y (t), y (t), . . . , y - y j (t) = (yj (t), yj (t), . . . , y
(n-1) j (n-1)

(t)) ,

(t)) ,

j = 1, . . . , n,

. (B.5), B.1.1 - y 1 (t), y 2 (t), . . . , y n (t) [a, b] (B.5) . 4.3.2 (B.5), y (t), c1 , c2 , . . . , cn , [a, b] y (t) = c1 y 1 (t) + c2 y 2 (t) + § § § + cn y n (t), y (t) = c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t). , y (t) c1 , c2 , . . . , cn . B.3.2 .

B.4. ,
[a, b] aj (t) R, j = 0, . . . , n,


118 a0 (t) = 0 f (t): a0 (t)y
( n)

B

(t) + a1 (t)y

(n-1)

(t) + § § § + an

-1

(t)y (t) + an (t)y (t) = f (t). (B.7)

y1 (t), y2 (t), . . . , yn (t) í (f (t) 0) [a, b], yH (t) í () (B.7). 3.4.3 (B.7) y
OH

(t) = yH (t) + y

OO

(t) = yH (t) + c1 y1 (t) + c2 y2 (t) + § § § + cn yn (t), cj C, j = 1, . . . , n. (B.8)

yH (t) (B.7) , (B.4). , (B.6) , c1 , c2 , . . . , cn [a, b] c1 (t), c2 (t), . . . , cn (t), : yH (t) = c1 (t)y1 (t) + c2 (t)y2 (t) + § § § + cn (t)yn (t). - y j (t) = (yj (t), yj (t), . . . , y
(n-1) j

(B.9)

(t)) ,

j = 1, . . . , n,

(B.5), Y (t) = (y 1 (t), y 2 (t), . . . , y n (t)) í , y H (t) = (yH (t), yH (t), . . . , y
(n-1) H

(t)) .

- c(t) = (c1 (t), c2 (t), . . . , cn (t)) , y H (t) = Y (t)c(t) : dy (t) = A(t)y (t) + f (t), dt (B.10)


B.5.

119

f (t) = (0, 0, . . . , 0, f (t)/a0 (t)) , A(t) (B.5). 4.3.4 (4.18) ,
t

y H (t) =
t0

Z (t, )f (t)d ,

Z (t, ) = Y (t)Y

-1

( ),

-. - c(t) (4.21) 4.3.4 Y (t)dc(t)/dt = f (t), y y y1 (t) y1 (t) . . .
(n-2) 1 (n-1) 1

y2 (t) y2 (t) . . .
(n-2) 2 (n-1) 2

... ...

(t) y (t) y

... (n (t) . . . yn (n (t) . . . yn

yn (t) yn (t) . . .
-2) -1)



(t) (t)

c1 (t) c2 (t) . . . cn-1 (t cn (t)





= 0 ) f (t) a0 (t)

0 0 . . .

.

det Y (t) = 0, ck (t) = gk (t), t [a, b]. ,
t

ck (t) =
t0

gk ( )d ,

k = 1, 2, . . . , n,

(B.7)
n t

yH (t) =
k=1

yk (t)
t0

gk ( )d .

(B.4) (B.9).


120

B

B.5.
n c aj R, j = 0, . . . , n, a0 = 0: a0 y
( n)

(t) + a1 y

(n-1)

(t) + § § § + an

-1

y (t) + an y (t) = 0.

(B.11)

(B.11) (B.11) dy (t) = Ay (t), dt 0 1 0 0 0 1 . . . . . . . A= . . 0 0 0 a an-1 an-2 n - - - a0 a0 a0

... ...

... ... 1 a1 ... - a0

0 0 . . .



(B.12)

. A (B.12). , . , , - 1 0 ... 0 0 - 1 ... 0 . . . . . . . = 0. . . . ... . det(A - E ) = det . 0 0 0 ... 1 a an-1 an-2 a1 n - - ... - - - a0 a0 a0 a0 ,


(B.11): a0 n + a1
n-1

121

+ § § § + an

-1

+ an = 0.

(B.13)

, j {1 , . . . , } k1 , . . . , k (k1 + § § § + k = n) 1 hj , dim Ker (A - j E ) = 1, kj > 1, kj - 1 hj , hj , . . . , hj ,
2 3 k
j

j = 1, . . . , .

(4.29) 4.4.2: hj exp{j t}, ..., h
k j
j

1

hj + + t h 1!

2

t h 1!

1 j

exp{j t}, . . . t2 h 2!
kj -2 j

kj -1 j

+

+ §§§ +

tkj -1 h (kj - 1)!

1 j

exp{j t}, j = 1, . . . , .

- (B.11): b1 exp{j t}, j ..., b
k j
j

b2 + j + t b 1!

t b 1!

1 j

exp{j t}, . . . t2 b 2!
kj -2 j

kj -1 j

+

+ §§§ +

tkj -1 b (kj - 1)!

1 j

exp{j t}, (B.14)

bm í hj , j = 1, . . . , . j , b1 = 0, (B.14) j . (B.11) exp{j t}, t exp{j t}, ..., tk
j

m

-1

exp{j t},

j = 1, . . . , .

(B.15)

3.4.2 (B.15) [a, b]


122 (B.11).


1. .. . .: - , 2007. 2. .., .., .. . 1. .: - , 1985. 3. .. . .: , 2003. 4. .. . .: , 1983. 5. .., .., .. . .: , 1985. 6. .. . .: , 2004. 7. .. . : - , 2000. 8. .. . .: , 2002.