Специальные функции. Мицик М.Ф. - 5 стр.

UptoLike

ВУЗ: 

ИСОиП ДГТУ | Шахты

Составители: 

Рубрика: 

    
2
u
,   
1
u
   
, :
( ) ( )
0
1221
=
′
′
−
′
′
uxuuxu

( )
[ ]
0
2121
=
′
′
−
′
uuuux
,

( )
Cuuuux =
′
−
′
2121
, (10)

−
C
 ,   ,    

(
)
xu
1

(
)
xu
2
, 
.0
2121
21
21
≠
′
−
′
=
′′
uuuu
uu
uu
 (10) :
.
2
1
2
xu
u
u
=
′
     
[ ]
0
; xx
,  

0
=
x

( )
xu
1
,  (9).
 3. 
(
)
−xu
1
 
 
0
=
x
     
( )
xu
1
, 
  
( )
xu
1
 
(
)
xu
2

0
→
x
.
.

0
ε
, 
[ ]
ε;0

( )
xu
1
   . , 
ε
<
0
x
 
[
]
0
;0 x

( )
xu
1
:
( ) ( )
0,
1
≥=α
α
xzxxu
, (11)

( )
0≠xz

[ ]
0
;0 xx
∈
.
 (11) (9), :
( ) ( )
( )
+=
∫
+
0
212
12
x
x
tzt
dt
CCxzxxu
α
α
.
:
( ) ( )
( )
+=
∫
+
0
122
12
x
x
t
dt
z
C
CCxzxxu
α
α
ξ
.
                                                           u2 ,                   u1
                      ,                 :
                   ′              ′
         u1 (xu2′ ) − u 2 ( xu1′ ) = 0                             [x(u1u2′ − u1′u2 )]′ = 0 ,
                               x (u1u ′2 − u1′u2 ) = C ,                                                       (10)
    C−                                                 ,                           ,
                          u1 (x)       u2 ( x )                ,
                               u1 u2
                                       = u1u2′ − u1′u2 ≠ 0 .
                               u1′ u2′
               (10)                                                :
                                      ′
                                u2 
                                 = 2 .
                                 u1  xu
                                                                                       [x ; x0 ] ,
                           x=0                                     u1 (x) ,                (9).
                           3.            u1 (x ) −
             x=0                                                                                          u1 (x) ,
                                                      u1 (x)                               u2 ( x )           x→0
                                            .
                           .
                 ε >0                                      ,                                    [0; ε ]
u1 (x)                                            .            ,              x0 < ε                         [0; x0 ]
              u1 (x)                                                :
                               u1 (x ) = xα z (x ), α ≥ 0 ,                                                    (11)
         z (x ) ≠ 0              x ∈ [0; x0 ].
                               (11) (9),               :
                                           x0
                                                  dt 
                u2 (x ) = x z (x ) C1 + C ∫ 2α +1 2  .
                                   α

                                            x
                                               t z (t ) 
                                                                                       :
                                                x0
                                                     dt 
                u2 (x ) = x z(x ) C1 + C 2 ∫ 2α +1  .
                                   α       C
                                         z (ξ ) x t    

Страницы