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. . III­IV , 2003­2012 .
. , , . . 2005, 2007 , ! , , ( , , - , , , , , - ). . , , . , : DMVN, . . . : DMVN, . . , . , . . . : . . , . , Baron, . . , . . , . , . : 6 2012 . http://dmvn.mexmat.net, http://dmvn.mexmat.ru. dmvn@mccme.ru.

1. 02.06.2004 ( )

1 15 . x , . y , . 0 x 10 0 y 5. z = x + y . x = 0, z z z 3 6 9 3 4 5

20 . z := y , . , .

1.1. 1
1 (3 ). , y1 = 9 sin t y2 = 3t y + p(t)y + q (t)y = f (t),
2

(1)

p, q , f C1 (R).

1 y3 (1), y3 ( ) = - 2 . ?

2 (2 ). (1) y4 = t2 - 1? 3 (2 ). xu - (2y - 3x)u = 0 x y (x, y ) (0, 0). 4 (3 ). a R C1 (R) (2, 1) (2) , u(x + 2, ax + 1) = (x)? 1 (2)


1.2. 2
1 (3 ). , y1 = 4 cos t y2 = 2t y + p(t)y + q (t)y = f (t),
2

(3)
3

p, q , f C1 (R). -
2

y3 (3), y ? 3 (2 ). 4xu + (y - 3x)u = 0 x y (x, y ) (0, 0).

= 1.

2 (2 ). (3) y4 = t2 - 1? (4)

4 (3 ). a R C1 (R) (1, 2) (4) , u(x + 1, ax + 2) = (x)?

1.3. 3
1 (3 ). , y1 = t2 - 3 y2 = 6 cos t y + p(t)y + q (t)y = f (t), (5) - 3 = 1. p, q , f C1 (R).
3

y3 (5), y ?

. v := y2 - y1 y + p(t)y + q (t)y = 0. R. , y3 = y1 + v , : y1 - 3 + · v - 3 3 - 3 + · (6 cos 3 - 3 + 3 (6) , v (6) . = 1, ) = 1,

= 6 co1 3 . , y3 = t2 - 3 + 6 co1 3 6 cos t - t2 + 3 . s s , . , - 3, . , , , () . 2 (2 ). (5) y4 = 3t2 ? . , y4 . u := y4 - y1 v := y1 - y (6). u(t) = 2t2 + 3, v (t) = t2 - 6 cos t - 3.
2

(7)

, u v . , u(t) + µv (t), , . . t2 (2 + µ) + 3( - µ) - 6µ cos t 0. , t. , : 2 + µ = 0. µ, : 9 + 12 cos t 0. , = 0, µ = -2 = 0, . , . , u v , . . . , Wu,v (t) R. : Wu,v (t) = 2 t2 + 3 4t t2 - 6 c o s t - 3 = 6(3t + 2t2 sin t + 3 sin t + 4t cos t). 2t + 6 sin t 2 (8)


t = 0 . . . . , t, Wu,v (t) = 0. , , . . , . , . 3 (2 ). 3xu - (x - 2y )u = 0 x y (x, y ) (0, 0). . : x = 3x; y = 2y - x. A = (10) (9)

30 . : det(A-E ) = (-2)(-3). -1 2 , , . : = 2 10 0 0 0 A - 2E = , h1 = . = 3 A - 3E = , -1 0 1 -1 -1 1 h2 = . {h1 , h2 } diag(2, 3). , -1 , , y

h

1

x h
2

. 4 (3 ). a R C1 (R) (1, 2) (9) , u(x + 1, ax + 2) = (x)?

1.4. 4
1 (3 ). , y1 = 8 sin t y2 = 2t y + p(t)y + q (t)y = f (t),
2

(11)
1 2

p, q , f C1 (R).

y3 (11), y3 ( ) = - . ? 2 (2 ). (11) y4 = t2 - 1? 3 (2 ). 2xu - (y - 3x)u = 0 x y (x, y ) (0, 0). 4 (3 ). a R C1 (R) (2, 1) (12) , u(x + 2, ax + 1) = (x)? 3 (12)


2. 28.06.2004 ( 1)
2.1. 1
1 (2 ). µ R y = (x2 + y 5 ) cos ln y ,


(1) (2)



y (2) = e 2 ,

y (2) = µ.

2 (2). µ = 0 y1 , y1 . 3 (3 ). y1 . 4 (2 ). R y2 , y2 (2) = 30?

2.2. 2
1 (2 ). µ R y = (x3 + y 4 ) sin ey , (3) (4) y (3) = ln , y (3) = µ. 2 (2). µ = 0 y1 , y1 . 3 (3 ). y1 . 4 (2 ). R y2 , y2 (3) = 5?

3. 30.08.2004 ( 2)
3.1. 1
y


+ a(x)y + b(x)y + c(x)y = 0,

. , , c1 c2 c1 (x + 3) + c2 (tg 2x - 1). -x - 3 tg 2x. 2 (2 ). ?
. y3 (0) = 0, y3 (0) = -7, y3 (0) = c c, c , : .

, x + 3 tg 2x - 1 . 1 (2 ). - y3 , y3 (0) = 0, y3 (0) = -7.

a, b, c C(R).

(1)

3 (4 ). 0 ? . a, b, c C(I ) , I . - tg 2x - 1 ± , 4 - - , . . 44 . : , , , - ; . 44 , a, b c, y1 = x + 3 y2 = tg 2x - 1 . , y1 y2 . , 2 s n2 b(x) = c(x) = 0, a(x) = -4 1+sini4x 2x (!). P.S. , : a, b c 0, . 4


3.2. 2
y


+ a(x)y + b(x)y + c(x)y = 0,

a, b, c C U (0) .

(2)

, x + 2 1 - tg 3x . 1 (2 ). - y3 , y3 (0) = 7, y3 (0) = 0. 2 (2 ). ? 3 (4 ). 0 ? . 5 , 9 , , 5 .

4. 2005
4.1. 1
1 (4 ). y1 (t) = cos 8t - 1, y + p(t)y + q (t)y = 0, Ё y (t) = cos 4t? . , . , y (t) yi (t) (i = 1, 2, 3) . , . , , y2 (t). , , ( p, q C 1 (R)), y2 (t) . . : . 2 (2 ). a [0; 3) x = 3x, (1) y = ax + 3 y . . A = (
30 a3

y2 (t) = tg 4t,

y3 (t) = cos(8t - 5)

p, g C 1 (R),

):

|A - E | = ( - 3)2 = 0 1,2 = = 3 > 0. , a = 0 rk(A - E ) = 0 : a = 0 a = 0 . a = 0 rk(A - E ) = 1 ; . ; .

3 (1 ). (1, 0) G R2 , (x, y ) = 1 + 2y x = 1.

4 (3 ). G = {(x, y ) | x > 0, y > 0} , (x, y ) = 1 + 2y x = 1.

5


5. 2007
5.1. 1
1 (1 ). e xµ (0) =
µ-1 1 At

=

e-2t cos 5t -e-2t sin 5t e-2t sin 5t e-2t cos 5t

xµ (t)

x = Ax +

µt2 sh µ

-

. x0 (t).

. µ = 0 x = Ax x(0) = -1 . 1 , x(t) = eAt C . C . t = 0 : x(0) = C = -1 . 1 c t- i : x0 (t) = e-2t - oos 5-sisnn 5t . c s 5t 5t 2 (2 ). x0 (t) 1 . . , . A (. 4): |A - E | = -2 - 5 -5 = ( + 2)2 + 25 = 0 1,2 = -2 ± 5i. -2 -

Re 1,2 = -2 < 0, , , x0 (t) ( ) . : x0 (t) . 3 (3 ). 1, . . . : const.

4 (2 ). A 1. . , : A = -2 -5 . 5 -2 lim sup |xµ (t) - x0 (t)|,
t d At dt e

= AeAt , A =

d At dt t=0 e

.

5 (3 ). ,
µ- 0

|x| = max{|x1 |, |x2 |}.

. . : 0.

6. 04.06.2011 ( )
6.1. 5
1 (3 ). y y , y , . . . , Ё y 1 = c o s 2 t, y2 = cos2 t, y3 = t2 sin t, y4 = sin t2 . (1)

, 2 (2+5 ).

?

x1 = sin ax1 - a tg x2 , x2 = 2 1 + bx2 - 1

(2)

) a = b = 4 X (t) = (x1 (t), x2 (t)) (0, 0). ) (a, b) R2 , > 0 > 0, X (0) < , X (t) < t 0. 6


7. 28.06.2011 ()
7.1. 3
y = y2 -
8

|y |.

(1)

1 (1+2 ). (x0 , y0 ) R2 : ) , ) ? t 2 (2 ). ( ) y (t) = 1, 0.

3 (1+1+2 ). ) , y (7) = 0? ) ? ) R?

8. 02.06.2012 ( )
8.1. 1
y1 (t) = 6t5 · ch 3t · sin 4t y
(n)

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

1 (2+2 ). ) n N ? ) y2 (t) = (t - 2)5 · ch(3t - 6) · sin(4t - 8)? 2 (2+1+3 ). x = 2 x - 4 y - e -t , 4 y = 3 x - 6 y + 5 t- 2 , ) ? ) ? ) : , ? x(1) = 3, x(1) = -2.

8.2. 2
y1 (t) = 4t · sh 2t · cos 5t y
(n) 7

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

3 (2+2 ). ) n N ? ) y2 (t) = (t + 3)6 · sh(2t + 6) · cos(5t + 15)? 4 (2+1+3 ). 7 x = 3x - 9y - 3 t y = 2 x - 6 y + t- 4 ,
-1

,

x(1) = -1, x(1) = 4.

) ? ) ? ) : , ? 7


8.3. 3
y1 (t) = 5t · ch 2t · sin 3t y
(n) 6

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

5 (2+2 ). ) n N ? ) y2 (t) = (t - 4)7 · ch(2t - 8) · sin(3t - 12)? 6 (2+1+3 ). x = 3 x - 6 y - e -t , y = 4 x - 8 y + 2 / 3 t, ) ? ) ? ) : , ? x(1) = 2, x(1) = -5.

8.4. 4
y1 (t) = 7t5 · sh 4t · cos 3t y
(n)

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

7 (2+2 ). ) n N ? ) y2 (t) = (t + 2)7 · sh(4t + 8) · cos(3t + 6)? 8 (2+1+3 ). x(1) = -2, x = 2 x - 3 y - t- 4 , -5 y = 4x - 6y + 3t , x(1) = 3. ) ? ) ? ) : , ?

8.5. 5
y1 (t) = 3t7 · ch 5t · sin 2t y
(n)

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

9 (2+2 ). ) n N ? ) y2 (t) = (t - 3)6 · ch(5t - 15) · sin(2t - 6)? 10 (2+1+3 ). x = 3 x - 2 y - 5 t- 1 , y = 9x - 6y + e-3t , ) ? ) ? ) : , ? 8 x(1) = 4, x(1) = -1.


8.6. 6
y1 (t) = 8t · sh 3t · cos 2t y
(n) 6

+ a1 y
(n-1)

+ . . . + an y = 0

(a1 , . . . , an R).

11 (2+2 ). ) n N ? ) y2 (t) = (t + 4)7 · sh(3t + 12) · cos(2t + 8)? 12 (2+1+3 ). x = 3 x - 4 y - 3 / 5 t, y = 6 x - 8 y + t- 6 , ) ? ) ? ) : , ?
: 6 2012 . http://dmvn.mexmat.net, http://dmvn.mexmat.ru. dmvn@mccme.ru.

x(1) = -5, x(1) = 2.

9