Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Сборник задач по высшей математике. Тестовые методы контроля знаний. В 3 томах. Т.3
.pdf
605.
=
′
′−′
′+′
′
=′+′′+′′+′′
=′+′′+′′+′′
=
′′+′′−′
′
=′′−′′+′′
=′+′′+′′+′′
=′+′′+′′+′′
=
′
′+′
′−′
′
=
′
′−′
′+′
′
=′+
′
′+′
′+′
′
=′+
′
′+′
′+′
′
=+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+
′
′+′
′+′
′
=
′
′+′
′−′
′
=′′−′′+′′
=′+′′+′′+′′
=′+′′+′′−′′
=+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+
′
′+′
′+′
′
=′′−′′−′′
,)
uuu
0423
yyxyxx
;
606.
607.
0)
)
/)
,)
0)
)
/)
,)
0)
)
/)
uuuu
uuuu
uuu
yyxyxx
uuu
yyxyxx
uuu
yyxyxx
uuu
yyxyxx
uuu
02
yyxyxx
043
yyyxyxx
.
;
.
xyyxyxx
;
096
;
:
0916
xyyxyxx
yyyxyxx
;
0196
;
:
,
,, 0
;
0816
;
0255
yyyxyxx
;
:
,, 0
xyyxyxx
044
096
uuuu
uuuu
096
0816
uuuu
uuuu
.
608 – 615 .
608.
609.
610.
,)
0)
)
/)
,)
0)
)
/)
,)
0)
uuu
uuu
uuu
uuu
yyxyxx
uuu
047
yyxyxx
yyxyxx
yyxyxx
yyxyxx
;
.
;
04914
xyyxyxx
yyyxyxx
;
01614
;
:
096
;
096
;
:
0168
;
0
0,
uuuu
uuuu
0204
096
;
uuuu
xyyxyxx
uuuu
yyyxyxx
093
.
0168
uuuu
xyyxyxx
)
/)
uuuu
uuu
yyxyxx
044
yyyxyxx
0168
.
;
121
:
0,
611.
=
′
′−′
′+′
′
=′+′′+′′−′′
=′′−′′−′′
=′++′′+′′
=′′−′′+′′
=′+′′−′′−′′
=′+′′+′′+′′
=
′
′+′
′−′
′
=
′
′−′
′+′
′
=′+
′
′+′
′+′
′
=′+
+
′
′+′
′
=+′′−′′
=′+′′+′′+′′
=
′
′−′
′+′
′
=′′−′′−′′
=′+′′+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+
+
′
′+′
′
=+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+′′+′′−′′
=
−
′
′−′
′
,)
uuu
06416
yyxyxx
;
612.
613.
0)
)
/)
,)
0)
)
/)
,)
0)
)
/)
06416
uuuu
;
xyyxyxx
xyyxyxx
yyyxyxx
;
;
06416
;
06416
.
044
;
044
;
:
:
0612
;
096
;
:
0, /
, /
, /
uuuu
uuu
yyxyxx
uuu
uuu
yyxyxx
uuu
044
yyxyxx
uuuu
xyyxyxx
uuuu
yyyxyxx
044
.
0612
yyxyxx
uuuu
uuuu
yyyxyxx
uuu
096
yyxyxx
.
614.
615.
,)
0)
)
/)
,)
0)
)
/)
uuuu
uuu
uuu
uuu
yyxyxx
uuu
yyxyxx
02510
yyxyxx
yyxyxx
;
02510
uuuu
yyyxyxx
096
;
uuuu
xyyxyxx
uuuu
yyyxyxx
0193
.
;
096
052510
;
;
02510
.
:
,, /
xyyxyxx
0916
;
:
616 – 623 .
616.
,)
0)
uuu
03010
yyxyxx
;
uuuu
03010
xyyxyxx
;
)
/)
uuuu
uuu
03010
yyxyxx
03010
122
;
:
0,
yyyxyxx
.
617.
=
′
′−′
′+′
′
=′+
′
′+′
′+′
′
=′+′′+′′+′′
=′′+′′−′′
=′′−′′+′′
=
′+′′+′′+′
′
=′+′′+′′+′′
=
+
′
′−′
′
=′+′′+′′+′′
=′+′′+′′+′′
=
′
′−′
′+′
′
=
′
′+′
′−′
′
=′′−′′+′′
=′+
′
′+′
′+′
′
=′+′′+′′+′′
=′′+′′−′′
=′+′′+′′+′′
=′′−′′+′′
=′+
′
′+′
′+′
′
=+′′−′′
=′′−′′+′′
=′+′′+′′+′′
=′+
′
′+′
′+′
′
=′′−′′−′′
,)
uuu
0918
yyxyxx
;
618.
619.
0)
)
/)
,)
0)
)
/)
,)
0)
)
uuuu
uuu
yyxyxx
0918
xyyxyxx
uuuu
08118
;
01009
yyyxyxx
.
;
:
uuu
uuu
048
yyxyxx
yyxyxx
;
uuuu
048
xyyxyxx
uuuu
;
0164
yyyxyxx
0254
.
;
:
, /
yyyxyxx
xyyxyxx
;
093
;
036
;
uuuu
uuuu
uuu
036
yyxyxx
620.
621.
622.
/)
,)
0)
)
/)
,)
0)
)
/)
,)
uuu
063
yyxyxx
.
:
,, /
uuu
uuu
yyxyxx
0112
yyxyxx
;
uuuu
uuuu
0211
0211
yyyxyxx
xyyxyxx
.
;
0112
;
:
xyyxyxx
;
yyyxyxx
.
;
0124
;
0122
;
:
,,
uuuu
uuu
0124
yyxyxx
uuuu
uuu
uuu
yyxyxx
0412
yyxyxx
042
0)
)
/)
uuuu
uuuu
uuu
yyxyxx
0145
xyyxyxx
yyyxyxx
0254
;
044
;
.
:
0
123
623.
=
′
′−′
′+′
′
=′+
′
′+′
′+′
′
=′+′′+′′+′′
=′′−′′−′′
),(),,(yxy
x
ψ
ϕ
),(),,(yxy
x
ψ=ηϕ=
ξ
=′+
′
′+′
′+′
′
yyxxyyx
=ψ−=ϕ
),(;2),
(
xyxxyyx
=ψ+=ϕ
),(;2),
(
xyxxyyx
=ψ−=ϕ
),(;2),
(
yyxxyyx
=ψ−=ϕ
),(;2),
(
yxyxxyy
x
+=ψ+=
ϕ
),(;2),
(
=
′−′′+′′−′
′
yyxxyyx
=ψ−=ϕ
),(;2),
(
xyxxyyx
=ψ+=ϕ
),(;2),
(
xyxxyyx
=ψ−=ϕ
),(;2),
(
yyxxyyx
=ψ−=ϕ
),(;2),
(
yxyxxyy
x
+=ψ+=
ϕ
),(;2),
(
=′−
′
′+′
′+′
′
yyxxyyx
=ψ−=ϕ
),(;3),
(
yyxxyyx
=ψ+=ϕ
),(;3),
(
xyxxyyx
=ψ−=ϕ
),(;3),
(
xyxxyyx
=ψ+=ϕ
),(;3),
(
xyxxyyx
=ψ−=ϕ
),(;3),
(
,)
uuu
0416
yyxyxx
;
0)
)
/)
uuuu
uuu
yyxyxx
045
xyyxyxx
uuuu
;
0164
yyyxyxx
0254
.
;
624 – 631 !
.
624.
uuuu
,)
0)
)
0544
xyyxyxx
:
;
;
;
:
,
-
625.
626.
/)
#)
,)
0)
)
/)
#)
,)
0)
)
;
.
uuuu
0344
xyyxyxx
:
:
,,
;
;
;
;
.
uuuu
096
xyyxyxx
:
:
0, #
;
;
;
/)
#)
124
;
.
:
,,
627.
=′+′′+′′−′′
yxyxxyy
x
+=ψ+=
ϕ
),(;3),
(
xyxxyyx
=ψ−=ϕ
),(;3),
(
xyxxyyx
=ψ−=ϕ
),(;3),
(
yyxxyyx
=ψ−=ϕ
),(;3),
(
xyxxyyx
=ψ+=ϕ
),(;3),
(
=′+′′+′′+′′
xyxxyyx
=ψ+=ϕ
),(;4),
(
yyxxyyx
=ψ+=ϕ
),(;4),
(
xyxxyyx
=ψ−=ϕ
),(;4),
(
xyxxyyx
=ψ−=ϕ
),(;4),
(
yyxxyyx
=ψ−=ϕ
),(;4),
(
=′−
′
′+′
′−′
′
yyxxyyx
=ψ−=ϕ
),(;4),
(
xyxxyyx
=ψ+=ϕ
),(;4),
(
xyxxyyx
=ψ−=ϕ
),(;4),
(
yyxxyyx
=ψ−=ϕ
),(;4),
(
yxyxxyy
x
+=ψ+=
ϕ
),(;4),
(
=′+′′+′′+′′
yxyxxyy
x
+=ψ−=
ϕ
),(;5),
(
yyxxyyx
=ψ+=ϕ
),(;5),
(
yyxxyyx
=ψ−=ϕ
),(;5),
(
xyxxyyx
=ψ+=ϕ
),(;5),
(
yyxxyyx
=ψ+=ϕ
),(;5),
(
=′−′′+′′−′′
yyxxyyx
=ψ+=ϕ
),(;5),
(
xyxxyyx
=ψ+=ϕ
),(;5),
(
yxyxxyy
x
+=ψ+=
ϕ
),(;5),
(
uuuu
0296
xyyxyxx
:
628.
629.
,)
0)
)
/)
#)
,)
0)
)
/)
#)
;
;
;
;
.
uuuu
03168
xyyxyxx
:
:
,, #
;
;
;
;
.
uuuu
02168
xyyxyxx
:
:
/, #
630.
631.
,)
0)
)
/)
#)
,)
0)
)
/)
#)
;
;
;
;
.
uuuu
02510
xyyxyxx
:
:
0, #
;
;
;
;
.
uuuu
042510
xyyxyxx
:
:
,,
,)
0)
)
;
;
;
125
/)
xyxxyyx
=ψ−=ϕ
),(;5),
(
yyxxyyx
=ψ−=ϕ
),(;5),
(
=′−
′
′+′
′−′
′
=′+′+
′
′
=′+
′
′+′
′−′
′
=′−′−
′
′
=′−
′
′+′
′+′
′
=′−′−
′
′
=′+′′+′′+′′
=′+′+
′
′
=′−′′+′′−′′
=′+′+
′
′
=′+′′+′′+′′
=′+′+′′
=′−
′
′+′
′+′
′
=′±′+
′
′+′
′
=′+
′
′+′
′−′
′
=′±′+
′
′+′
′
=′−′′+′′+′′
=′±′+′′+′
′
=′+
′
′+′
′−′
′
=′±′+′′+′
′
=′−′′+′′−′′
=′−′′+′
′
=′−
′
′+′
′+′
′
=′±′+′′+′
′
0=t
;
#)
.
:
,,
632 – 643 !! .
632.
633.
634.
635.
636.
637.
uuuu
uuuu
uuuu
uuuu
uuuu
xyyxyxx
uuuu
0334
xyyxyxx
xyyxyxx
xyyxyxx
xyyxyxx
xyyxyxx
.
0356
.
0256
.
0267
.
078
.
0489
.
:
:
:
:
:
:
uuu
ξ
ξ
787
uuu
ξ
uuu
787
uuu
ξ
uuu
787
ξ
uuu
ξ
0394
031516
787
058
021225
787
0736
043249
787
638.
639.
640.
641.
642.
643.
uuuu
uuuu
uuuu
uuuu
xyyxyxx
uuuu
yyyxyxx
§ 12. 6 !!
"0 /
0222
xyyxyxx
xyyxyxx
xyyxyxx
uuuu
.
0422
.
0484
.
084
.
054
.
03106
xyyxyxx
.
:
:
:
:
:
:
uuu
ξηηξξ
uuuu
ξηηξξ
uuuu
ξηηξξ
uuuu
ξηηξξ
ξηηξξ
0
uuuu
ξηηξξ
022
7
044
7
022
7
uuuu
0)(2
7
039
7
#!! &. # !
, (
),
, ( -
126
). , $" ();
0=f
(
)
+∞
∞−∈
()(
)
ϕ
=
()(
)
ψ
=
′
+
−
at
x
[
]
∈
(
)
ϕ
=
()(
)
ψ=′
(
)
(
)
=
k
∞
=
1
l
l
π
π
0=f
[
]
∈
(
)
ϕ
=
(
)
(
)
=
l
l
l
π
– * $". ,
, )%.
% 3& &
(
) ( ),
& :
1
+" ()
′′
2
uau′′=
,
xxtt
,x
) (, #):
1
,
( ) ( ) ( )( ) ( )
2
2
) $" :
′′
txu
):
2
uau′′=
,
xxtt
0,
,
0==x
( )
lx ,0
,
txu
=lx
=
,
,
t
=0
atxatxtxu
2
0
t
=
0,
.
atxu
k
π
l
xtxu
,
1
atx
ξψ++ϕ+−ϕ= d
,
tt
=0
ξ
xtxu
a
)(,
xtxu
,
,
tt
=0
atk
b
+
sincos,
k
π
xtxu
,
π
atk
sin
l
xk
,
l
π
2
l
( )
x
ψ
0
ak
sin
xk
dx
l
l
2
( )
a
k
x
ϕ=
0
sin
xk
b
dx
,
=
k
% & &
(
) (
) :
) $" :
′
2
uau′′=
,
xxt
lx ,0
,
0
t
=
)(,
xtxu
,
txu
0,
,
0==x
txu
=lx
):
2
ak
π
( )
=
∞
k
−
eCtxu
k
=
1
t
l
sin,
π
xk
,
C
k
2
l
x
( )
ϕ=
0
sin
xk
dx
0,
.
127
1. ' & !!
2
)(x
ϕ
(
)
)(x
ϕ
642
(
)
=
(
)
=
′′
yy
2
xyyxu
−+=
3
.
. 6 & , -
. % ,
32
yxydyxyyxu
)(
x
ϕ+−+=−+=
.
$
′
( )
y
3
– !, - , -
2
xy
,
.
% -
yxu ,
!
),(
:
xy
3
yxyyxu ψ+ϕ+−+=
2
)(
ϕ+−+=
dyx
3422
xyyxy
)()(
xxy
,
642
– !.
*, & !!
342
),(
yxu ψ+ϕ+−+=
xyyxy
)()(
xxy
.
2. ), ! -
& !!
u
,)
0)
)
/)
21,1
:
22432
yxyxyx ++−
;
24223
yxyxyx ++−
;
24223
2
yxyxyx −+−
;
22432
yxyxyx ++−
;
xx
′′
122 yxyu
−=
223
,
#)
3 yxyxyx −+−
32432
.
. % , !
u
21,1
:
128
,)
(
)
(
)
(
)
=
(
)
(
)
′
′
=′′+′′−′′
=′−′′+′′−′′
Cxy
=
+
Cxy
=
+
14
x
y
+=ξ
xy14
+=η
=
′
′+′
′−′
′
=
′
′
ξ
η=ξ′′=′
η
ξ
21,1=u
; 0)
21,1=u
; )
u
11,1
; /)
*, ,); 0); /); #).
21,1=u
; #)
21,1=u
.
'
,)
0)
/)
#)
′
x
′
x
′
x
′
x
u
!:
xx
233
+−=
yxxyu
422
+−=
xyyxu
233
+−=
233
+−=
yxxyu
′′
142
,
xx
′′
123
,
xx
xyxxyu
242
,
342
,
xx
42
;
223
;
223
2122
+−=
yxyu
223
.
;
122 yxyu
−=
26 yxyu
−=
′′
xx
′′
122 yxyu
−=
, & !! !
,)
yxyxyx ++−
22432
#)
3 yxyxyx −+−
32432
.
3. ' & !!
uuu
01415
yyxyxx
.
. 2, 3 § 11
uuuu
021415
yyyxyxx
.
%
!! , –
– (. 2)
(. 3):
.
, !!
01415
yyxyxx
)(
,
u
0
ξη
(. § 11). "&
. %
ξηη
uuu
, &
fduu
1
f
– !, - , -
1
)(
,
η. *-
129
η (
ξ
ξ+η=ηη=
η′=
)(),
(
ξ
η
g
f
)()
(
ξ+η
=
gfu
)()14(xygxyf
u
+++
=
1
+
x
)1,1(
u
+
−
at
x
1
+
x
1,1
=
=
t
x
4
17
):
)()()(
gfdfduu
η
1
,
– !.
*,
,
, ,
& !!
.
4. "& 3&
5
′′
=
2
′′
),(),,(,
txuxuau
=+∞−∞∈
.
x
′
txxtt
4
00
==
tt
1,2cos)0,(,
==
axxu
. + ! #
1
,
( ) ( ) ( )( ) ( )
atxatxtxu
2
1
a
2
atx
ξ
ξψ++ϕ+−ϕ= d
,
=
%
),()(==ϕ
txux
txu 2cos
=
1
2
,
0
t
=
),(
txu
1
=
),(
2
1
−
2
)(5
tx
−
+
)(1
tx
−+
, &
′
=ψ
5
x
−
tx
−+
tx
tx
)(5
tx
)(1
−+
),()(
txux
t
4
)(5
+
)(1
+
)(5
tx
+
)(1
tx
++
.
0
=
t
′
t
44
00
==
tt
+−tx
1
tx
2
1
4
2sin
+
)(5
+
tx
44
)(1
++
tx
tx
)(5
44
tx
)(1
++
1
( )
4
1,2cos)0,(,
==
axxu
,
=ξξ+
d
+−tx
=ξ+
tx
)(2sin)(2sin
txtx
−−++
.
5
)1,1( +=u
4sin
.
130
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]