Литвин І.І., Конопчук О.М., Желізняк Г.О. Вища математика
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171
( ' .
81. a)
3
0
x
x
edx
f
³
; b)
2
3
1
1
x
dx
x
f
³
; c)
2
ln
x
dx
x
f
³
;
82. a)
4
3
9
x
dx
x
f
³
; b)
2
2
0
(1)
dx
x
³
; c)
1
0
ln
x
xdx
³
;
83. a)
2
3
1
1
x
dx
x
f
³
; b)
2
0
4
x
dx
x
f
³
; c)
1
0
ln
x
dx
³
;
84. a)
4
3
0
x
x
edx
f
³
; b)
³
f
1
6
2
1
dx
x
x
; c)
³
f
1
3
2
1
dx
x
x
;
85. a)
³
f
0
2
dxxe
x
; b)
2
22
0
(1)
x
dx
x
³
; c)
6
2
3
2
(4 )
dx
x
³
;
86. a)
0
x
edx
f
³
;b)
2
1
dx
x
f
f
³
; c)
1
0
dx
x
³
;
87. a)
2
2
0
4
x
dx
x
³
; b)
3
1
ln
x
x
f
³
.
8. ^ _
172
;!!< %!= (1)
C;!!!
5!
( F(x)
[a; b].
-!< ;%
/7 .
!= =!+ %!=>!: *
*!! ;!!= %!=*
%
c
i
y=f
1
(x)
173
;!!< %!= (2)
C#!! @:*
[t
1;
t
2
] – !, V(t) – 9(
C#!! #
[, b] – 7, F(x) –
C#!! #’H% % #!!
"#$!% #!! ;!!= %!=*
* * .
b
a
Y
X
174
?CA 9. '?"OAU" ?"E""E
9.1. C!!% &!.
*! ! ) (9 !, 7
) 9() !. #-
! ) (9 (. R7
, < ! (, < , -
) (. ", ! ( ! ' -
(( ’ ( .
/!+%>! %!!!
n- ( ) F(x,y,y
c
,y
cc
...y
(n)
)=0,
( – , & – ( , &
c
, &
cc
,..., y
(n)
– ( ( .
?;’;
( ) '(- ),
( )< .
( ) ( 9 (-
, ( .
"&,
) &
c
+2& = sinx – (;
') &
cc
+&
c
–2 = 0;
&
cc
+ytgx = 0 – (;
) &
ccc
+&
cc
= &
&
c
– (.
I’ ( 9) () .
"&:
) ’ ( (
&
c
– 2 = 0 <:
&
c
= 2,
.2
2
Cxxdxy
³
') ’ ( ( &
cc
– 2 = 0 <:
1
2
2 Cxxdxy
c
³
,
21
3
1
2
3
CxC
x
dxCxy
³
.
C ( ( (, 7 ( ) (,
'! ’, ! !) ( .
(, 7 ’ n- ( ( n (
C
1
, C
2
, ... C
n
.
9. [6*$ "
175
Y
&
0
(
0
, &
0
)
0
0
X
.1
=>! ;’; ( ) ( -
) ), )< (
( , ( .
/ ( ’ ) (-
.
%!, ’ ( ((<
() .
R (, ' ! ’
(
&
c
–x
(.1).
/ (!: ( ’ ( , ,
( . A ' !
( !. ( ( (. 1), !<, 7
( ' '
, ( ! ! (
(
0
; f(x
0
)).
))! (
! .
*
) !-
( ( !
0
.
I’ ( ,
(< ( !
) ;’;
, (!
( ! ’ – ;F
@%.
C( ’ 9 ( (-
. I ( .
9.2. /!+%>!% %!!! ; %F! ;%!!.
C ( :
()V(y)dx+X
1
(x)V
1
(y)dy = 0,
( (),
1
() ( , V(y), V
1
(y) – ( y.
/(9 '( ! ('
1
()V(y), <
( :
0
1
1
dy
yV
yV
dx
.
C < (:
0
1
1
³³
dy
yV
yV
dx
.
9. [6*$ "
176
%&. R7 = , & = " < ’
1
()V(y)= 0,
= , & = " '( ’ ( , 7
! & !< ! . %! ’
() ') , (.
1
. I’ ydy = xdx. C( ! ’
& = 4, 7 = –2.
?;’;!!:
+ () . )! (-
’ :
xy
xdxydy
³³
22
;
22
.
( '9 ) ’ -
! ( ( /2, ( &
2
=
2
+. /(9 -
’ ! & = 4 = –2, < 16 = 4+, ( = 12. ",
! ’ &
2
=
2
+12.
2.
I’ : 2sinydx+(x
2
+3)cosydy = 0; !
’ & =
2
, 7 = 1.
?;’;!!:
/( ! (
2
+3)siny:
0
sin
cos
3
2
2
y
ydy
x
xdx
.
)! ' (, (
1
2
sin
cos
3
2
C
y
ydy
x
xdx
³³
,
1
2
1
2
2
sinln3ln;
sin
)(sin
3
)3(
CyxC
y
yd
x
xd
³³
.
/ ) < (
2
+3)(siny) =
1
c
e
, ' (
2
+3)(siny)=.
", (
2
+3)siny = 0 siny =0, ! ( -
!< ! , siny = 0 – ’ . @ (
( ’ (
2
+3)siny= =0, !,
< ( siny =
3
2
. /(9 ! & =
2
= 1,
< 1= /4, ( = 4. B ’ < (
3
4
2
x
y
.
9. [6*$ "
177
3. I’ (1+
)&&
c
=
. C ! ’
: & = 1, 7 = 0.
?;’;!!:
"
,
dx
dy
y
c
,)1(
xx
e
dx
dy
ye
(
(1+
)ydy = e
x
dx,
)1(ln
2
2
x
e
&
.
/ ) <:
)1(
2/
2
xy
ee
.
/( ’ &=1 =0. "<
eee 2),1(
02/1
.
C(
2
e
. ", ! ’ 9 < (:
xy
e
e
e 1
2
2/
2
, '
xy
eee 12
2/
2
.
9.3. A%!%<!% /!+%>!% %!!! &@= &*.
C ( :
y
c
+f(x)y = g(x) ,
( f(x), g(x) – ( .
/ g(x) =0 ( ) (( (
. 5 < (:
y
c
= f(x)
&,
’< ( ( .
R7 f(x) =0, (
y
c
= g(x) ,
’< .
"( ( ’ ( < -
( ( &() (' ( ( u(x) v(x).
A( ( ’ :
– ( (' u(x)
v(x) :
(uv)
c
+f(x)(uv) = g(x), u
c
v+uv
c
+f(x)uv =g(x);
– < ((:
u
c
v+u(v
c
+vf(x)) = g(x);
– ) v(x) 9 v
c
+vf(x) = 0;
– ( ) v = v(), ( , ( u
c
v()=g(x);
– ( ) u = u(x) +c;
9. [6*$ "
178
– 9 ’ (' ( :
y = v(x)(u(x)+c).
4.
C ’ (
y
c
– yctgx = sinx.
?;’;!!:
< . & y = uv, ( y
c
= u
c
v+uv
c
'( (
u
c
v+uv
c
–uvctgx = sinx,
'
u
c
v+u(v
c
– vctgx) = sinx.
C( ) v v
c
– vctgx = 0. A( ( 9 u <
u
c
v = sinx.
I’ v
c
– vctgx = 0, <:
,sinlnln,ctg,0ctg xvxdx
v
dv
x
dx
dv
v = sinx.
/(9 ! v u
c
v=sinx ’)! , ( u:
u
c
sinx = sinx, u
c
= 1, du = dx, u = x+C.
", ’ < ( y = (x+C)sinx.
5.
C ! ’ (
2
2
22
&&
c
: & = 0, 7 = 0.
?;’;!!:
& & = uv, ( y
c
=u
c
v+uvc. I '( (
2
2
22
xuvuvvu
c
c
, '
2
2
2)2(
xvvvu
c
c
. & v
c
+2xv = 0, (
xdx
v
dv
xv
dx
dv
2,2
.
)!, < lnv = –x
2
, '
2
x
ev
.
/
2
x
ev
(:
dxxdux
dx
du
exeu
xx 222
2,2,2
22
c
, (
Cxu
3
3
2
.
C ’ ( (
¸
¹
·
¨
©
§
Cxey
x 3
3
2
2
.
/(9 ’ ! , <: 0 =
0
(0 + ), (
= 0. ", ! ’ ( (
2
3
3
2
xy
.
9. [6*$ "
179
9.4. A%!%<!% !%!% /!+%>!% %!!! *== &* ; &%<! -
/%+%H!.
( ( ( <-
< (:
&
cc
+py
c
+qy = f(x) ,
( p i q ( !.
R7 f(x) = 0, ( < ((
< (:
&
cc
+py
c
+qy =0.
H( :
7 &
1
&
2
! ’ ((
( ( ( <, !
&
1
/ &
2
z onst, & =
1
&
1
+
2
&
2
(
1
2
– , <
’ .
C( y
1
, &
2
, ( (
( . & '( ( & =
kx
.
, (9 <:
(
kx
)
cc
+(
kx
)
c
p+q(
kx
) = 0,
k
2
kx
+k
kx
p+
kx
q = 0,
kx
(k
2
+kp+q) = 0.
" , 7
k
2
+kp+q = 0.
I k
2
+kp+q = 0 ) :!.
5 !< ! k, & =
kx
< ! ’
(( ( ( (
<. # (:
6.
C ’ ( :
/
!
B ’
(
C ’
(
1
k
1
z k
2
xkxk
&&
21
21
,
xkxk
eCeCy
21
21
2
k
1
= k
2
xkxk
x&&
21
21
,
)(
21
1
xCCey
xk
3
D
r
E
y
1
=
D x
cos
E
x
y
2
=
D x
cos
E
x
y =
D x
(C
1
cos
E
x+C
2
sin
E
x)
&
cc
–5&
c
+6& = 0 .
?;’;!!:
H( ! ( :
9. [6*$ "
180
k
2
– 5k + 6 = 0, D = 25 – 24 = 1,
;2
2
15
1
k
3
2
15
2
k
.
! < ( , &
1
=
2x
, &
2
=
3x
–
! ’, & =
1
2x
+
2
3x
– ’ (
.
7.
C ’ ( :
&
cc
+4&+4& = 0 .
?;’;!!:
O! k
2
+4k+4 = 0, ' (k+2)
2
= 0 < (
k
1
= k
2
= –2, &
1
=
–2
, &
2
=
-2
– ! ’, & =
-2
(
1
+
2
)
’ ( .
8.
C ’ ( :
&
cc
+6&
c
+13&=0 ,
?;’;!!:
H( ! ( :
iki
i
k
Dkk
23;23
2
46
,1613436,0136
21
2
! < – ,
&
1
=
-3
cos2x, y
2
= e
-3x
sin2x – ! ’, y=e
-3x
(C
1
cos2x + C
2
sin2x) – -
’ ( ( .
9.
C ! ’ (
&
cc
–2&
c
+& = 0 ( ! &(0) = 4; &
c
(0) = 2.
?;’;!!:
O! k
2
– 2k + 1=0, ' (k – 1)
2
= 0 < (
k
1
= k
2
=1, &
1
=
, &
2
=
– ! ’, & =
(
1
+
2
) –
’ ( ( . ! !
’ ! ( ( &
c
& =
(
1
+
2
):
&
c
=
(
1
+
2
)
c
+ (
)c(
1
+
2
) =
2
+
(
1
+
2
) =
(
1
+
2
+
2
).
A ( ! ( & &
c
:
°
¯
°
®
)(
21
0
1
0
2
4
'
¯
®
21
1
2
4
, (
1
= 4,
2
= –2 .
/(9 ! ’, ( ! ’
( ( ! :
& =
(4 – 2) .
9. [6*$ "