11 6.

.

11 12

21 22

dx a x a ydtdy a x a ydt

= + = + (1)

0x y= = . :

( ) ( )11 12

21 22

2 211 22 11 22 12 21

0

Tr det 0

a aa a

a a a a a a A A

= + + + =

.

1) . (1)

2

11 2

1 2 21 1 1

1, , lnt t x xx C e y C e t y CC C

= = = =

.

) 1 2 0 < < , 2 2

1 11 2

0, 1

1 2 21

0, 0,t tt t

xx C e y C e y CC

> >

+ + = = =

.

0x y= = .

) 1 20 < < , ( )2

11 2

0

1 2 21

0, ,t tt t

xx C e y C e y CC

, 0x y= = . ) 0 = , 0x y= = , .

(1) .

3) 1 2 = = . (1)

21 2

1

,t t Cx C e y C e y xC

= = = . ) 0 < , 0x y= = . ) 0 > , 0x y= = . 4) 1 2 = = . (1) :

( ) 11 2 1 21 1 1

1, , ln lnt t Cx x xx C e y C C t e t y CC C C

= = + = = +

) 0 < , 0x y= =

. ) 0 > , 0x y= = .

(1) .

..

5) 1 20, 0 = . (1)

11 2,

tx C e y C= = . 7. .

( )

( )x f x = , (1)

( )f x - 1( , )C a b .

( ) ( ) ( ), ,xc

U x f d c a b = (2) .

(1)

( ) ( )

x p

p f x U x

=

= = . (3)

( )f x , ,

(3). 0x x= (1) 0 , 0x x p= = (3). 0x x= (1) ( )U x . , 0x x= ,

0( ) 0f x = , (2), 0 0( ) ( ) 0U x f x = . , 0x ( )U x ,

. . ( )f x - 2 ( , )C a b . ,

1) 0x x= - ( )U x ,

0 , 0x x p= = (3) ; 2) 0x x= - ( )U x 0( ) 0f x > ,

0 , 0x x p= = (3) . . 0 0x = , 0y x x= .

0x = - , ( ) (0)U x U> 0x .

( ) ( ) ( ) 2, 02pV x p U x U= +

- .

( )( ) 0V Vp U xx p

+ = , , , 0 , 0x x p= = .

0x = - ( )U x (0) 0f > , (0) 0U = , (0) (0) 0U f = <

( ) ( )N ( ) ( )( )

( ) ( )( )

2 2

0

1 10 0 0 , 0 12 2

f xF x

U x U U x U x x U x f x x

=

= + + = <

( )= =

0

0

p

f x.

=( ) 0f x (3)

n = =0, 1,2,...,ix x i n . ( )0, 0ix ( ),x p (2). , (3) , .. ( ) =0 0, 1,2,...,x if x i n .

, . , . . (1) . , , (1) (1) ( (2)). 20. .

0ix x= ( )f x ( )N ( ) ( ) ( )

== + + 0 0 0 0

0

( ) i x i i if x f x f x x x x x .

(2)

( )( )0 0, x i ix p p f x x x = = - .

=0ix x x . ( ),x p ( ) ( )= 0, , 0ix p x ( ) ( )=, 0, 0x p

( ) = = 0, x ix p p f x x . .

( ) ( ) ( )

= = = 2 0 0

1,20

10x i x i

x i

f x f xf x

.

( ) >0 0x if x , , ( ) 0p (2), , , (1) ,

= ( )dp f xdx p

(4)

(4) = ( )pdp f x dx , = +2 ( )2p f x dx C = +2 ( )p f x dx C .

, < 0p , = +2 ( )p f x dx C .

, C , , x . (4), , . , , ( )0, 0x , , t + t .

, ( )0,0x ,

= 0

2 ( )x

x

p f s ds . (5)

, (1) , ( )0 0,x p , , (1) . , , , . t + t . , t , . 40. . 1. .

( ) ( )= 2

2

d x x a x f xdt

, 0a > . 1- :

( ),x p p x a x = = (6) 1 20,x x a= = ( ) 0f x = (6) ( ) ( )=, 0, 0x p ( ) ( )=, , 0x p a . , ( ) = >0 0xf a ; ( ) = < 0xf a a , ( )0,0 - - , ( ), 0a - - .

(6). . 3,

= + = +2 ( ) ( )2p x x a dx C U x C , =

3 2

( )3 2x xU x a .

= + = +2 ( ) 2 ( )p x x a dx C U x C . (7)

( )f x C ,

(6) 1- 3.

1 ( )f x .

2 ( )f x :

= + = +2 ( ) ( )2p f x dx C U x C . , .

( )f x , 1. ( )f x

( ) 0f x > ( ) 0f x < . x 2

2p

, ( ) 0f x > . x a= ( )f x , ..

2

2p .

32a

32a

32a

S2

p

x

x

a

f(x)

x

( )= +22p f x d x C

a

a

. 2

. 3

S1

. 1

1S 2S , ( ) = 2

2pU x

. = x x =

0

( ) ( )a x

a

x a x dx x a x dx , ( ) =

0

0x

x a x dx . = 32ax .

x , 2

2p

.

x = 20

( )2

xp f x ds

( ) < 0f x , , = 2

( )2pU x

x . 3 (0, 0) .

(.3) , , . t .

, : = > 0dxpdt

, x

, .. . C . C ( )

2 ( ). (7) , , x .

C 2 ( ). (7) , C ( ), 0a . . . 3 (7) . . 0x

( ) ( ) ( )= = =2

02 , 0 , 0

d x x a x x x xdt

.

. . 30, , ( )0,0 , t + , t .

0x , , , ( )0,0 .

( ) ( ) =0 0x z , ( )( )0, 0x p . , ( )( )0, 0x p t + , ( )0,0 t + .

) - .

( )= 2 2 22 ( )d x f x x x adt . : ( )= = = = 2 2( ) 0 0 0,f x x x a x x a .

.4 ( )= 2 2( )f x x x a .

2

1

a -a

p

x

S1

S1 S1

-a a

2

2p

x

x

f(x)

a -a

.4

.5

.6

S1

xf 0,x x a= = : ( ) 0xf a . , 5, ( )0,0 - , ( ), 0a ( ), 0a - . = = +2 4 22( )2 4 2

x

a

p x xf x dx a C (. 5). ,

( )f x - , - , .

( ) ( )

= x xa a

f s ds f s ds , ( ), 0a ( ), 0a

: = 2 ( )xa

p f x ds .

6 . 1 2 . , 1 2 .

C . C OY . (.6, ).

C ,

OY , ( ) = 2

2xU x

(.5, ). < x a - (.6, ).

C

= 2

( )2pU x x .

C , 6 . ) .

( ) ( )= + < >1 20 0, 0, 0x x xf f a f a . , ( )0,0 - ,

( )1, 0a ( )2, 0a - . 8 = +2 ( )2p f x dx C .

0C > OY ( ). C , .

= 2

2

( )2

x

a

p f s ds . OY

( )2, 0a . (.9) ( )2, 0a . 1( ,0)a ( )

=

1

2x

a

p f s ds

(.9) .

.

2 -1

S2

.9

.8 2

2p

p

x

x

x S1

S1

S2 -1 2

f(x) .7

3. . :

= 2

2 ( ) sind x f x xdt

.

1-

, sinx p p x = = .

x m= ( ) 0f x = ( ) ( )=, , 0mx p m . , ( ) = = = +

1, 2,

1, 2 1xm k

f m k Zm k

, ,

( ) ( )( )+ = +2 1, 2 1 ,0kx p k - , ( ) ( )=2 , 2 , 0kx p k - . :

= + = +2 sin cos2p xdx C x C . = ( ) cosU x x , ,

= + 2 cos , 2p x C C . 10.

. 10

2C = , . 2 2C < < , ( , ). 2C > , ( ), .

. (0,0) 1) ; 2) , = +2 2( , ) 4 sin

2xV x p p .