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Сборник задач по высшей математике. Тестовые методы контроля знаний. В 3 томах. Т.3

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 4. " /0 !
(
)
=′′+′′+′′+′
),(yxuu
=
22
11212
22
11212
0>D
(
)
=′′Φ+′′
η
ξ
(
)
=
Φ
+
′−′
β
α
0=D
(
)
=′′Φ+′′
η
ξ
(
)
=
Φ
+
η
ξ
0<D
(
)
=′′Φ+′′+′
η
ξ
(
)
=
ΦΦΦΦΦ
    :   !!      ,     , & !!­    .
§ 11. ! !! "0  / .
  /" "
" !!    
 
221211
$
)
 , ,   .
$
4
1.
2.
 !.
aaaD =
. )     
#112%3* 4.  1$5 )&(&,$5
  3 
aaaD =


uuuyxFuauaua
yxyyxyxx
1
3
4
0,,,,2
. (4.1)
# 4.1
uuuu
2
uuuu
0,,,,
ηξξη
uuuuu
βαββαα
0,,,,
ηξηη
uuuu
ηξξξ

0,,,,

0,,,,
3.
 .   !!  (4.1)
    (
 !    ,   !
,,,,
54321

  .
111
5
uuuyxF
0,,,,
yx
),  -
uuuuu
ηξηηξξ
0,,,,
!" 
22
12211
Cyx
=
ϕ
),(Cyx
=
ψ
),(
(
)
=′′+′′+′′+′
0>D
Cyx
=
ϕ
),(Cyx
=
ψ
),(
(
)
ϕ=ξ
(
)
ψ=η
(
)
=′′ηξΦ+′′
2
η+ξ
2
η−ξ
(
)
=
βαΦ
+
′−′
0=D
Cyx
=
ϕ
),(
(
)
ϕ=ξ
η
ξ′ξ
(
)
=
′′ηξΦ+′
0<D
(
)
=ϕ±
ϕ
(
)
ϕ=ξ
(
)
ϕ=η
2
02
=+ dxadxdyadya
. (4.2)
)  (&)   !!-
 
. $     ­ !!    
221211
  ,  .
 )$5

    :
  
  
#& 
( ),  -
yx,
,
1
=α ,
uuuu
ηξξη
=β   -
 -
uuuyxFuauaua
yx,
  -
0,,,,
.
0,,,,2
yxyyxyxx
. -
 

   :
,
 
 
    . )1  
  
yx
,Re
+
′′
′′
( ),  
yx
,
,   
η
η
3
( ),  
,
ηξΦ+
5
yx
,Im
,    
uuuuu .
ηξηηξξ
0,,,,
=
2
  !, -
yx
0
. "  -
yx
uuuu
0,,,,
ηξηη
.
uuuuu
),(Im,Re    
0,,,,
βαββαα
.
.  ,
Cyxiyx
112
#$ " $   !
η′⋅′+ξ′⋅′=′
η′⋅′+
ξ′⋅′=
()(
)
()(
)
η
′⋅′
+
ξ
′⋅′
+
η′⋅
ξ′+
η′ξ
′′′
+
η′η′⋅
+
ξ′ξ′⋅
′=′
+∆=′′
+∆=
0=∆
u
),(txuu
=
Rt∈
Rx∈
),(txff
=
=′+′′−′
=−′+′
=′+′′+′
=+′−′
=′−
+
′+′
uuu
ηξ
uuu
′′
=
′′
=
′′
+ξ
′′
+ξ
ηξ
2
2
;
xxx
;
yyy
′′
′′
22
+η′⋅ξ′⋅
+η′⋅ξ′⋅
′′
′′
+η
22
+η
)(
+ξ′′⋅
+ξ′′⋅
uuuuuu η′′⋅
ηξηηξηξξ
uuuuuu η′′⋅
ηξηηξηξξ
; (4.3)
xxxxxxxxxx
;
yyyyyyyyyy
uuuuuu
xyxyxyyxyxyxxy
ηξξηηηξξ
%&   , ­  ! , :
  (#"% )
2
tt
fuau
;
  (#"% )
2
t
fuau
; (4.4)
.
 & ('"% )
(   ).
$
t,
 5,
 !,
,
n
,
= ...
+
21
++
,
xxx
n
! & .
 1. ),    !!-     
,)
0)
)
uuu
uuu
$$%
uuu
0
$$$%%
%%%$%
;
0
;
0
;
/)
#)
0
uuu
%$$
uuuu
;
$$$%%
0
.
113
.      4.1  -
=′+′′−′
=−′+′
=′+′′+′
=
+′−
=′−
+
′+′
=′−′′+′′−′′
=′+′′+′′+′′
=′+
′+′
′+′
(
)
=
′+′
′+′
′+′
22
11212
22
11212
22
11212
22
11212
 :
,)
uuu
0
$$$%%
–    -
 ;
0)
$$%
0
uuu
–    -
 ;
)
/)
uuu
uuu
%$$
0
%%%$%
 ;
0
   
;
#)
uuuu
0
$$$%%
–    -
 .
 2. )  !! :
,)
uuuu
021415
yyyxyxx
;
0)
)
uuuu
uuuu
053612
yyyxyxx
xyyxyxx
;
03102
.
.  !!       (4.1)
221211
uuuyxFuauaua
0,,,,2
yxyyxyxx
   
(. . 4.1).
aaaD =
  ,) 
2
,
025,42145,7
>=== aaaD
,   .
  0)
2
,
0366
=== aaaD
    .
  )
   .
114
,
09101
<=== aaaD
 3. '   :
=′−
′+′
′−′
=′+′′+′′+′′
=′+′′+′′+′′
=′−′′+′′−′′
22
12211
dx
y
y
1
=
y
14
=
y
Cxy
+−=
Cxy
+−=
14
Cyx
=
+
Cyx
=
+
14
=′+
′+′
′+′
22
12211
,)
0)
)
. ,)
uuuu
uuuu
uuuu
xyyxyxx
021415
yyyxyxx
yyyxyxx
;
053612
;
03102
.
uuuu
021415
yyyxyxx
–  -
  (.  2). +     (4.2):
2
02
=+ dxadxdyadya
.
*
22
01415
=++ dxdxdydy
.
'  – &   !!­ .
"    
2
,   -

"&  

2
dy
dx
dy dx
=++
01415
, 
2
+′yy
, 
,
.
01415
=+
.
( !!     ­ . * & ()  
,
.
*,  !!    ­    
0)
uuuu
053612
yyyxyxx
. (  
.
 (.  2). +    ­ (4.2)
*
2
02
=+ dxadxdyadya
.
22
03612
=+ dxdxdydy
, 
115
2
−′yy
03612
=+
.
$  
y
6
=
y
Cxy
+
=
6
Cxy
=
6
=′+′′+′′+′′
y
(
)
+±=
(
)
=−±
(
)
=−+
(
)
=−−
=′+′′+′′+′′
yxy
=η−=ξ
;
6
yxy
=η+=ξ
;
6
xxy
=η−=ξ
;
6
xxy
=η+=ξ
;
6
xxy
=η−=ξ
;
6
(
)
=
′+′
′+′
′+′
=′+′′+′′+′′
Cxy
=
6
xy6
−=ξ
η
ξ′ξ
   &
.
"&  !!     ­ ,    
)
, 
uuuu
03102
xyyxyxx
. (   
.
(.  2).    (4.2)  :
3 
22
0102
=+ dxdxdydy
, 
   
2
iy 3191
±=±=
−′yy
0102
=+
.
,

31
Cxiy
, 
31
Cyxi
.
 ,  !!       
31
Cyxi
31
Cyxi
.
 4. .,      -   !! 
yyyxyxx
; )
.
053612
:
;
,)
/)
uuuu
; 0)
; #)
. $      !!     (4.1)
0,,,,2
yxyyxyxx
uuuu
053612
yyyxyxx
– -
  .  
221211
uuuyxFuauaua
   (.  2),       
(  ,)  )).  
(.  3). %
 
 !,  
yx
0
.
η
η
yx
116
%   ,)
ξ′ξ
ξ
′ξ′
=′−
′+′
′−′
Cyx
=
+
Cyx
=
+
14
y
x
+=ξ
y
x
+=η
14
=
ξ
=
ξ
=
η
=
η
=η′′=η′′=η′′=ξ′′=ξ′′=ξ′
′+′=η′⋅′+ξ′⋅′=′
()(
)
′+′
′+′
=
()(
)
′+′
′+′
=
=
η
′⋅′
+
ξ
′⋅′
+
η′⋅
ξ′+
η′ξ
′′′
+
η′η′⋅
+
ξ′ξ′⋅
′=′
′+′
′+′
=
+
′+′
′+′
′−′
′+′
′+′
=′+′−′′+′′+′′
+
η
yx
=
η
yx
16
06
=
   )
10
16
01
=
,     ,)  )
01
η
yx
=
η
yx
   .
 5. %    !! 
uuuu
021415
yyyxyxx
.
. 6  !!   -  ,      ­ . (    (. ­ 2),       (.  3)
.
     :
.
     :
1
,
x
1
,
y
14
x
,
1
,
y
       
0
yyxyxxyyxyxx
.
+ ! (4.3) 
uuuuu
;
yyy
′′
=
′′
=
′′
+ξ
′′
+ξ
uuu 2
′′
2
uuu 19628
;
ηηξηξξ
′′
2
;
ηηξηξξ
uuu 1514
.
ξηηηξξ
ηξηξ
22
+η′⋅ξ′⋅
+η′⋅ξ′⋅
′′
′′
+η
22
+η
+ξ′′⋅
uuuuuu
ηξηηξηξξ xxxxxxxxxx
+ξ′′⋅
uuuuuu
ηξηηξηξξ yyyyyyyyyy
=η′′⋅
=η′′⋅
uuuuuu )(
ηξξηηηξξ xyxyxyyxyxyxxy
%  !!   
117
)1415(1519628 uuuuuu
ηηξηξξηηξηξξ
uuuuu
0)(2)2(14
ηξηηξηξξ
,
 
=′−′−′′
=′+′+
=′+′′+′′+′′
(
)
=−±
Cyx
=
ϕ
),(
ξ
η
),(yx
ϕ
(
)
ϕ=ξ
(
)
ϕ=η
y
x
−=ξ
x
3
=
η
=ξ′
−=ξ
=η′
=η′
=η′′=η′′=η′′=ξ′′=ξ′′=ξ′
′+′=η′⋅′+ξ′⋅′=′
()(
)
′+′
′+′
=
()(
)
uuu
,022169
ηξξη

uuu
022169
ηξξη
,
    .
 6. %    !!

uuuu
03102
xyyxyxx
.
. 6  !!   -  ,      ­ . %     (.  2),        (.  3)
31
Cyxi
,
1    
      
   
yx,Re
,
*
,
 
1
x
, 1
y
, 3
x

+ ! (4.3) 
uuuuu
3 ;
xxx
′′
=
′′
+ξ
′′
2
ηξηξ
22
′′
+η′⋅ξ′⋅
+η
.
:
.
, 0
yx,Im
.
y
,
0
yyxyxxyyxyxx
+ξ′′⋅
.
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ηξηηξηξξ xxxxxxxxxx
 
=η′′⋅
uuu 96
;
ηηξηξξ
22
′′
=
′′
+ξ
′′
+η′⋅ξ′⋅
′′
118
+η
+ξ′′⋅
yyyyyyyyyy
′′
;2
=η′′⋅
uuuuuuu
ξξηξηηξηξξ
=
η
′⋅′
+
ξ
′⋅′
+
η′⋅
ξ′+
η′ξ
′′′
+
η′η′⋅
+
ξ′ξ′⋅
′=′
uuuuuu )(
′−′
=
=′+′+′′+′′−′′−+′′+′′+′′
=′+′+
′+′
=′+′+
′+′
=′′+′′+′′
=′′+′′+′′
=+′′−′′
=′+′+′′
=+′′+′′
=+′′−′′
=′−′+
′+′
ηξξηηηξξ xyxyxyyxyxyxxy
uu 3
.
ξηξξ
%     :
uuuuuuuu
ηξξξξηξξηηξηξξ
,0)3(310)3(296

uuuu
ηξηηξξ
,0)3(399

uuuu
0333
ηξηηξξ
,
    .
 7. ),     -  ,  –  ,   –  5:
,)
2
u
uuu
0
zzyyxx
; 0)
t
=
2
a
u
2
x
+
2
u
; )
2
y
′′
2
uau′′=
.
xxtt
.    ! (4.4)  -  .
,) .
uuu
0
zzyyxx
  5 – -
    .
0)
2
u
t
=
2
a
u
2
x
+
2
u
  (
2
y
   ).
) .
′′
2
uau′′=
   
xxtt
   –  .
  598 – 599   !! ,    .
598.
,)
ξξξη
0
uuu
; 0)
uuu
0
ηξξη
;
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0
uuu
ηηξη
; /)
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0
uuu
ηηξξ
0
ξηξξξη
.
119
;
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0, /
599.
=′+
+
=′−
′+′
=′+
+
′−′
=′−′+
′+′
=′−
′+′
=′′−′′+′′
=′+′′−′′+′′
=′+
′+′
′+′
=′′+′′−′′
=
′−′
′+′
=′+′′+′′+′′
=′+′′+′′+′′
=′′+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+
′+′
′+′
=
′+′
′−′
=′′−′′+′′
=′+′′+′′+′′
=′+
′+′
′+′
=′′+′′−′′
=
′′−′′+′
=′+′′+′′+′′
=′+′′+′′+′′
=
′−′
′−′
,)
uuu
0
ξηη
; 0)
0
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ξξηηη
;
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0
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;
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0
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ξξξηη
.
:
,, , #
  600 – 607     .
600.
601.
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;
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;
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;
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602.
603.
604.
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yyxyxx
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.
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;
uuuu
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;
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;
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;
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;
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,, 0
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uuuu
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yyxyxx
06416
120
;
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,, /
yyyxyxx
.
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