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Файл:Сборник задач по высшей математике. Тестовые методы контроля знаний. В 3 томах. Т.3
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5. "& &
[
]
∈
(
)
l
π
3
=
2=l
7
l
l
π
l
a
k
π
π
(
)
=
ϕ
′=ψ
(
)
=
ϕ
=
2=l
2
3
π
π
π
π
2
2
3
a
k
()(
)
7≠k
7=k
7≠k
(
)
(
)
7=k
)7(
k
−
π
π
a
a
21
21
x
sin
7
, 0),(
txu ,
2
′′
=
tt
′′
uau
,
xx
lx ,0
,
txu
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0==t
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,
1
=
tt
=
0
0==x
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=lx
-
1
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.
. "& &
=
2
a
k
+
'
b . %
k
k
l
0
∞
=
1
ϕ=
atxu
( )
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sin
k
1
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sin
π
l
2
atk
l
0
),()(
txux
7
π
atk
+
b
kk
l
xk
dx
,
b
),(
txux
,
0
=
t
0
x
,
, 0
=ψ ,
)(xx
π
sinsincos),(
xk
,
l
xk
ψ
sin)(
x
.
0
=
tt
a k.
k
dx
,
=
2
=
b
k
2
1
cos
0
ak
π
3
12
0
7
2
sin
π−
7
sin
cos
−
xkx
=
dx
7
xkxk
π+
=
2
) I, :
.
=
I
3
= k
3
%
,
, :
π+
1 xk
2
−ππ
( )
7
1
kak
2
( )
7
−ππ
kak
7
sin
( )
π−
xk
2
−
+π
( )
2
7
k
7
sin
2
7sin
−−π
k
( )
7
+π
k
( )
+π
, -
, .
%
Idx
.
2
=
2
0
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.
=
2
1
=
I
( ) ( )
0
7cos0cos
dxx
131
=π−
1
2
=π−
dxx
0
7cos1
=
=
7=k
),(tx
u
7=k
π
π
π
π
7
[
]
∈
l
π
=
9
16=t
4=l
l
π
l
l
()(
)
4=l
4
l
π=π
()(
)
3≠k
3=k
π
x
7
7sin
π
1
−
x
π
21
a π
2
0
=
21
2
.
a
*,
0
b k,
k
, &
,
7
( )
=
btxu
sin,
7
2
sin
7
2
:
,
xat
2
=
π
21
a
7
sin
=
7
21
2
. ' &-
abπ
7
sin
xat
.
2
1
2
=x
!
1 at
,
tu
7
=
21
2
π
a
sin
7
at
2
ππ
sin
2
=
21
2
π
a
sin
7
π
.
2
6. "& & -
x
′
2
uau′′=
,
xxt
lx ;0
,
txu
=
t
=
0
3
sin8),(
,
2
txu
0),(
=lx
&
,
=x
,
.
. "& & -
2
ak
π
t
l
⋅=
sin,
txux
3
.
0,==ϕt
x
sin8
3
4
xk
π
,
. '
π−
cos
−
3
C
4
.
k
π+
xkxk
C
l
C
2
k
0
4
2
=
k
0
4
−
eCtxu
k
1
( )
∞
k
=
xk
ϕ=
sin)(
x
,
3
sin8
4
( )
sin
dx
,
3
=
x
4
cos2
0
=ϕ
x
sin8
ππ
xkx
dx
4
txu
=
0),(
,
0==x
.
Idx
) I, :
.
132
3≠k
(
)
(
)
3=k
(
)
−
=
3≠k
=
),(tx
u
3=k
4
4
9
16=t
+
=
′
′
2
6
+
=
′
′
6
322
=
′
′
)()(yxxy
ψ+ϕ
+
4
3
+
,
4
xk
3
0
=
.
0
k
-
%
I
3
( )
π−
k
4
π+
2
=
sin
3
π−
xk
−
4
4
3
k
( )
3
sin
π+
4
,
, , . %
4
I
0
,
&
−=
3
x
π
cos0cos2 dx
dx
2
C
k
, &
,
−
( )
eCtxu
3
4
−=
0
0
3
x
π
cos12
2
, 8
=
2
& .
3
2
−xx
3
π
3
sin
π
2
4
=
0
,
a
π
t
3
x
π
sin,
⋅=
−
e
22
33
a
π
t
44
3
x
π
sin8
⋅=
.
2
%
,
=x
&
8
.
2
3
a
π
16
⋅
−
2
9
4
π
sin816,
⋅=
=
4
6
22
9
a
π−
eeu
.
644 – 648 & !!
.
23
644.
645.
646.
647.
648.
xx
2 .
xx
′′
yy
xy
′′
xy
1
u .
yxu
.
xyyu
yxu
42+=
.
:
:
:
2
yx ψ+ϕ++
:
22
xyyxu
−+=
.
:
yxx
yyx
ψ+ϕ++
3
yx
322
yyx
ψ+ϕ++
2233
yxxyyx
)()(
)()(
yyx
)()(
xxy
)()(
yx
ψ+ϕ+−
649 – 653 , !-
& !! .
133
649.
=
′
′
311
(=,
u
211
(=,
u
=′′
411
(=,
u
011
(=,
u
211
(=,
u
=′′+′′−′′
)()2(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)()3(yxgyxf
u
+++
=
=′′+′′−′′
)()4(yxgyxf
u
+++
=
xx
xu
6
,
.
650.
651.
652.
,)
/)
′′
,)
)
u
yy
,)
/)
′′
2 yuxx=
2 yxyxy −+
6
3 xy −
3yuxy=
323
3yyxx +−
; 0)
322
yxyyx ++
; #)
yxyx ++
yxyx −−
33
; )
33
.
33
yxyx +−
;
: 0
2
,
.
2222
2xyyxyx +−
4222
; /)
; 0)
2 xyyxy −+
422
; #)
22322
yyxyxx −++
;
433
23 xxyyx −+
.
: /
,
3
3xxy +
; 0)
32
; #)
.
3 xy +
3
32
; )
32
32
26 xy −
;
yxy −+
.
: 0
2
,
.
,)
)
22223
yyxyxxy −+−
; 0)
32 xyyx +−
32
; /)
yxyxyyx −+−
223
; #)
22232
yyxxyx −−+
;
:
653.
′′
xy
,)
)
yxu
232−=
,
yxyyxxy ++−
yyxxyyx +−+
222
; /)
.
22222
; 0)
yxyxyyx −++
223
; #)
22222
yyxxyyx ++−
;
: #
654 – 665 & !!
.
654.
uuu
023
yyxyxx
.
:
433
2xxyyx −+
.
423
2xxyyx +−
.
655.
656.
uuu
uuu
034
yyxyxx
yyxyxx
.
045
.
:
:
134
657.
=′′+′′−′′
)5()(yxgyxf
u
+++
=
=′′+′′−′′
)6()(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)7()(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)8()(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)9()(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)10()(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)()11(yxgyxf
u
+++
=
=′′+′′−′′
)()12(yxgyxf
u
+++
=
=
′
′+′
′−′
′
)()13(yxgyxf
u
+++
=
0≠a
0=∆
u
=′′+′′+′′
0≠a
=′′+′′+′′
0=∆
u
0≠a
0=∆
u
=
′
′+′
′+′
′
uuu
056
yyxyxx
.
:
658.
659.
660.
661.
662.
663.
664.
665.
uuu
uuu
uuu
uuu
067
yyxyxx
yyxyxx
yyxyxx
yyxyxx
uuu
uuu
uuu
uuu
.
078
.
089
.
0910
.
01011
yyxyxx
yyxyxx
yyxyxx
yyxyxx
.
01112
.
01213
.
01314
.
:
:
:
:
:
:
:
:
666. . (
,)
′′
2
uau′′=
xxtt
; 0)
; )
′′
tt
):
+
2
u
∂
;
2
y
∂
2
; /)
∂
t
∂
=
2
a
2
uau
∆=
u
u
∂
2
x
∂
#)
′
t
2
fuau
+∆=
; )
uuu
0
zzyyxx
.
667. . (
=
2
2
a
fuau
+∆=
; 0)
2
∂
∂
x
∂
u
+
2
∂
′′
2
u
; #)
2
y
,)
/)
′
t
∂
u
∂
t
668. . 5 (
,)
/)
; 0)
=
2
a
u
∂
t
∂
′′
2
u
∂
+
2
x
∂
2
uau′′=
xxtt
2
u
∂
; #)
2
y
∂
; )
2
uau′′=
; )
xxtt
′′
tt
2
uuu
zzyyxx
):
′
t
′′
tt
2
2
uau
∆=
+∆=
; )
fuau
∆=
0
;
uau
;
; )
: ,,
):
.
: ,, /
uuu
0
zzyyxx
.
: ,,
135
669 – 676 & 3& -
11(,u
2
sin24
+
2
cos
5
+
2
sin5,
2
2
cos
5,05,
3
−
1
+
x
2
sin5,04,0
+
1
+
x
2
cos
5,05,
1
−
,0),(),0(],;0[
==∈
tlutulx
=
′
l
π
6
6=l
3
6
π
669.
670.
671.
672.
673.
674.
′′
=
′′
=
′′
=
′′
=
′′
=
′′
=
.
22
′′
=+∞−∞∈
′
txxtt
axxuxxuxuau
.1,6)0,(,)0,(),,(,
==
: 8
′′
=+∞−∞∈
′
txxtt
322
axxuxxuxuau
.1,)0,(,)0,(),,(,
==
: 4
22
′′
=+∞−∞∈
′
txxtt
axxuxxuxuau
.1,cos4)0,(,2)0,(),,(,
==
:
′′
=+∞−∞∈
′
txxtt
22
axxuxxuxuau
.1,3)0,(,cos2)0,(),,(,
==
:
2
′′
=+∞−∞∈
′
txxtt
axxuxxuxuau
.1,cos3)0,(,sin2)0,(),,(,
==
:
2
′′
=+∞−∞∈
′
txxtt
axxuxxuxuau
.1,sin4)0,(,cos3)0,(),,(,
==
:
675.
2
′′
=
′′
)0,(),,(,
xuxuau
2
=+∞−∞∈
x
′
txxtt
2
axxu
:
676.
2
′′
=
′′
)0,(),,(,
xuxuau
9
=+∞−∞∈
x
′
txxtt
3
axxu
:
677 – 682 & &
.
xu
2
uau′′=
xxtt
1
=2sin
)0,(
,
x
,
;
3
2
tu ,
=?
xu
t
,0)0,(
677.
′′
.1,cos)0,(,
==
.1,sin)0,(,
==
136
:
1 at
cos
678.
,0),(),0(],;0[
==∈
tlutulx
=
′
l
π
5
2=l
5
,0),(),0(],;0[
==∈
tlutulx
=
′
l
π
4
5=l
8
,0),(),0(],;0[
==∈
tlutulx
,0)0,
(=x
u
l
π
3
2=l
a
π
6
,0),(),0(],;0[
==∈
tlutulx
,0)0,
(=x
u
l
π
4
1=l
a
π
20
,0),(),0(],;0[
==∈
tlutulx
,0)0,
(=x
u
l
π
2
3=l
a
π
8
′′
2
uau′′=
,
xxtt
xu
t
,0)0,(
679.
680.
xu
1
=4sin
)0,(
x
,
;
1
4
=?
tu ,
1
:
2
′′
=
xu
′′
uau
,
xxtt
sin
5
,
1
=
)0,(
x
;
1
6
=?
tu ,
xu
t
,0)0,(
at
π
2cos
1
:
2
′′
=
′
xu
t
′′
uau
,
xxtt
sin
4
,
1
)0,(
=
x
;
1
4
=?
tu ,
cos
π
at
1
at
π
:
2sin
681.
′′
′
t
=
xu
2
′′
uau
,
xxtt
sin
5
,
1
=
)0,(
x
;
1
10
=?
tu ,
1
:
682.
′′
′
t
=
xu
2
′′
uau
,
xxtt
sin
6
,
1
)0,(
=
x
;
1
12
=?
tu ,
1
:
683 – 688 & &
.
at
π
5sin
at
π
2sin
137
683.
,0),(),0(],;0[
==∈
tlutulx
l
π
2=l
1=a
e
,0),(),0(],;0[
==∈
tlutulx
l
π
2=l
1=a
e
,0),(),0(],;0[
==∈
tlutulx
l
π
1=l
1=a
e
,0),(),0(],;0[
==∈
tlutulx
l
π
5=l
1=a
2π−e
,0),(),0(],;0[
==∈
tlutulx
l
π
2=l
1=a
e
,0),(),0(],;0[
==∈
tlutulx
l
π
3=l
=
e
x
;
2
4 π−
;
2
uau′′=
u
,
xxt
1
2
=?
4,
=2sin)0,( ,
xu
′
:
684.
685.
686.
x
2
;
′′
uau
,
xxt
1
1
u
=?
,
2
4
xu
=
′
=
:
2
;
′′
uau
,
xxt
1
1
u
10
,
=?
5
=
xu
′
=
:
2
;
′′
uau
,
xxt
1
u
=?
1,
=
xu
′
=
4
sin3)0,(
sin5)0,(
sin4)0,(
,
2
23π−
x
5
,
2
55π−
x
5
,
;
;
;
6
2
:
687.
688.
x
2
;
′′
uau
,
xxt
1
1
u
=?
,
3
6
=
xu
′
=
:
2
;
′′
uau
,
xxt
1
1
u
12
,
=?
2
xu
=
′
=
:
138
6
sin3)0,(
sin2)0,(
,
2
33π−
x
6
,
2
2 π−
;
;
" 4 – . !
[3, 5, 10, 12, 19, 25, 35, 36, 50].
% «. !». 9 , . 3 5 & – . $ & 1 ,
& , 0 .
, 9,
– 0 .
139
"* 1%&. 2(%
=+′′−′′
=′+′+′′
=+′′+′′
=+′′−′′
=′−′+−′′+′′
=′′−′′+′′
=′+′′+′′+′′
=′+′′+′′+′′
=′′+′′−′′
),(),,(yxy
x
ψ
ϕ
),(),,(yxy
x
ψ=ηϕ=
ξ
=
′+′′+′′+′
′
yxyxxyy
x
+=ψ−=
ϕ
),(;5),
(
yyxxyyx
=ψ+=ϕ
),(;5),
(
yyxxyyx
=ψ−=ϕ
),(;5),
(
xyxxyyx
=ψ+=ϕ
),(;5),
(
yyxxyyx
=ψ+=ϕ
),(;5),
(
=′−′′+′′−′′
=′+′+′′
=′+′+′′
=′+′+′′
=′+′+′′
=′+′+′′
.
1 2 3 4 5
. , :
0
1
,)
)
uuu
ξξξη
uuu
ηηξη
0
; 0)
; /)
uuu
ηηξξ
0
0
;
;
,, 0, # 0, /, # 0, ,, , # , /
ηξξη
uuu
#)
. :
,)
2
)
. !
140
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yyxyxx
3
0)
)
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#)
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ξ
uuu
787
4
)
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uuu
787
uuuuu
ξηξξξη
uuuu
xyyxyxx
0394
0694
0
0918
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yyyxyxx
02510
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; 0)
; /)
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; /)
;
;
;
uuu
, -
;
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ξ
uuu
ξ
039
787
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;
0364
787
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08118
,, # 0, / ,, /
0334
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#)
ξ
uuu
0432
787
140
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