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Национальный исследовательский университет «МЭИ»

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Высшая математика

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УрМатФиз / УрМатФиз с теорией

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| X x + X x = 0


	X x + X x = 0 0 x l	1:1

X0 = X l = 0		1:2
X 0 = X l = 0		1:3
X0 = X l = 0		1:4
X 0 = X l = 0		1:5
% X0 = X2 X 0 = X 2		1:6

. ' , ) *+ , + + + +-+-+ 1.1 , % + ,* + % % -% % |% , , ,+ ,* + + + + +-+ ,

% 1 |2 .

	% 1.1 , 1.2 .	: 1 0,
2 = 0,	3 0.

4 0, % + + + + + % % 55+ +-

4 + 1.1 ) 655 +

2 + = 0

= p

++ %+ + 4 + .

*++ +-+ + + 1.1 7

) 4 %+

X x = C1 chp

x + C2 shp

x :

1:7

% 1.7 + 1.2 , ) % % ,

+ + + + + + 4 C1 C2:

C1 ch 0 + C2 sh 0 = 0

C1 chp

l + C2 shp

l = 0:

9+ + + +-+ 6

+ *+ 4, + ) +-

%+ + 4 + ,:

chp1

l shp0

= 0

shp

l = 0:

1:8

321

1.8	0 , , -
.
= 0, " 1.1 # $
	X x = C1x + C2:	1:9

1.9 " ' 1.2 , )"

C1 0 + C2 = 0 C1 l + C2 = 0:

* $' C1 = C2 = 0 -

= 0 ' ' ' .

0, "

1.1

2 + = 0

1 2 =

. + # $ '

1.1 ,

X x = C1 cosp

+ C2 sinp

1:10

1.10 "

1.2 , )

" C1 C2:

C1 cos 0 + C2 sin 0 = 0

1:11

C1 cosp + C2 sinp

= 0:

' ) . :

1	0
				sinp = 0:
cosp		sinp	= 0		1:12

1.12

qnl = n n = 1 2 ::: :

n =	n	!2	n =		:	1:13
				1 1
	l

! " # $" % . " 1.12 = n. & ' '( " "' , ' '( , " ' ' "* * .

322


		C1 1 + C2			0 = 0:
C1 = 0 C2 \|		.
1.10 = n		! " #$
	nx			l			l
			2		2
			2		2
Xn x = sin	l !	jjXn x jj		= Z	Xn x dx	=		n = 1 1:	1:14
Xn x = sin	l !	jjXn x jj		= Z	Xn x dx	=	2	n = 1 1:	1:14
				0

. ' ! " #$ ( ( ) * (, # # # ( ( ( , #-1.1 , 1.2 .

!	1.1 , 1.3 .	( 1 0,
2 = 0,	3 0.

0, ) / # # ( 1.1

2 + = 0 = p

# . ! , ( 1.1 *1.7 . 1.7 ) ( 1.3 ,/ ) ! # / -C1 C2:

C1 sh 0 + C2 ch 0 = 0

C1 chp l + C2 shp l = 0:

( 2 # :

	chp0	l shp1		= 0 chp	l = 0:	1:15
			l

				, , ! /
4 1.15 #

0 .

= 0, ) ! , ( 1.1 1.9 . 1.9 ) ( 1.3 , -

	C1	= 0

	C1l + C2 = 0

# ( # , (. 6 , = 0 ( ( ( ! .

323

n = 0 1:

	0,			-

"		#$ %$ # -
C1 C2:
	C1 sin0 + C2 cos 0 = 0			:16
	C1 cosp	+ C2 sinp	= 0:

* # % ":

0		1
0		1			cospl = 0:
cosp		sinp	l	= 0	cospl = 0:
,		% #

	q
	q	nl = 2 2n + 1			n = 0 1 2 ::: :

. " $

n = 22n + 12 4l2

1 # " 2 %3 .

:17

:18

=n . ,

, . 4-

C1 0 + C2 1 = 0:

. " C2 = 0 C1 | .

= n, 2 %3

2n + 1x! jjXn x jj

Xn x=cos

Xn x dx=

=0 1:

:19

4 : 1

= 0, 3 0.

%$ #

C1 ch 0 + C2 sh 0 = 0

C1 shp

+ C2 chp

= 0

% $ #, % % %

324

n = 0 1:

1	0
shp	chp	6=:0
shp	chp	6=:0
= 0, 1.1

1.9 . 1.9 1.4 -CC11 =00+ C2 = 0

# # . $ % , = 0% .

0, 1.1 & % -1.10 . 1.10 1.4 ,' ( ) # ) ( -C1 C2:

C1 cos 0 + C2 sin 0 = 0

C1 sinp

+ C2 cosp

= 0:

, ( # ':

= 0 cosp = 0:

sinp

cosp

- 1.21 & # (

q	nl =		2n + 1 n = 0 1 2 ::: :
		2

. ' ) %

n = 22n + 12 4l2

/ ( ' 0 #1 .

1:20

1:21

1:22

=n 1.20. . ',

, . 2-

		C1 1 + C2 0 = 0:
. ' C1 = 0 C2 \|		% .		( %
1.10	= n	0 #1
			l	l
		2	2			:
		2	2
Xn x=sin 2l	2n + 1x! jjXn x jj		=Z Xn x dx=	2 n	=0 1:
Xn x=sin 2l	2n + 1x! jjXn x jj		=Z Xn x dx=	2 n	=0 1:	1 23
			0

325

1.1 , 1.5 . : 1						0,
2 = 0,	3	0.
		0,		!" " ## "-
% "	!" " 1.1 & ""' (## % "
		2 +			= p
				= 0
* ! "'			" . +,- " !" " 1.1 ."
& ! ! 1.7 .			! 1.7 ! 1.5 , & "-
" 0 " *"'			,		!" " * " " C1 C2:

C1 sh 0 + C2 ch 0 = 0

C1 shp

+ C2 chp

= 0:

!" & ( * '

" 0:

shpl = 0:

shp

chp

= 0

1:24

2 !" " 1.24

" * " , ! " , , ! ""'

" " * &

0 " .

,- " !" " 1.1 !

1.9 . ! 1.9 !

! 1.5 , & -

C1 = 0

C1 = 0:

4 " ,

= 0 ! , ! ""' " " ,

! ! 0- , ! "" # " %

X0 x = 1 jjX0 x jj2 = Z

X02 x dx = l:

1:25

& " 0 " " !

" . .

,- " !" " 1.1 !

1.10 .

! 1.10 !

! 1.15 &

C1 sin 0 + C2 cos 0 = 0

1:26

C1 sinp + C2 cosp = 0:

!" & ( * '

" 0:

sinp = 0:

sinp

cosp

= 0

1:27

2 !" " 1.27 " " . ! " *

326

q nl = n n = 1 2 ::: :

n =	n	!2
			n =	1 1	:	1:28
	l

.1.26 = n. #$% %& $ % , %- %& , $ % % $) ) . *$ #$ $ #$

C1 0 + C2 1 = 0:

C2 = 0 C1 | #$%& . # % $%&1.10 #$ = n # %

	nx			l		l
			2		2
			2		2
Xn x=cos	l !	jjXn x jj		=Z	Xn x dx=		n=1 1:	1:29
Xn x=cos	l !	jjXn x jj		=Z	Xn x dx=	2	n=1 1:	1:29
				0
* / 1.1 , 1.6 . *$ $ % : 1									0,
2 = 0, 3	0.
&	0, ) $/ $ 1.1 2 # &

1.7 . 1.7 )$ % 1.6 # %
$ %)$ $ %& C1 C2:
	C11 ch 2 C2 sh 2 = 0	1:30
	C1 sh 2 + C21 ch 2 = 0:

#$% %& 4 $
	1 sh 2 2 sh22 = 2 1 ch 2 6= :0
5% %& , 1.30 %& % $/ C1 = C2 = 0
. #$	0 .
&	= 0, ) $/ $ 1.1
1.9 . 1.9 1.6 # %

C1 2 = 0

1 = 1:

C1 = 0 C2 #$ #$%& , % %& , 0 = 0 % , -

	X00 = 1:	1:31

327

0,

8 C11	cosp		C2 sinp	2
8 C11	cosp	2	C2 sinp	2	= 0
						1:32
						1:32
: C1 sinp 2 + C21 cosp 2 = 0:


:
1 cosp	2 2 + sin2 p	2 = 2 1 cosp			2	= 0:
1 cosp	2 2 + sin2 p	2 = 2 1 cosp			2	= 0:
! " # !
	n = n 2 n =			:		1:33
	n = n 2 n =		1 1	:		1:33

1.33 1.32 ! % : C1 C2# # ! jC1j + jC2j 6=:0 1.10 ! 1.33 ! " ' (

Xn x = An cos nx + Bn sin nx n =	1 1	1:34
8An Bn jAnj + jBnj 6=:0
* " + ' (
2

jjX0 x jj2 = A202 jjXn x jj2 = 0 Xn2 x dx = A2n + Bn2 n = 1 1:

328

D : r a 0 ' 2

@2v

4v r ' + v = r

@r ! +

+ v = 0

2:1

@'2

v a = 0

2:2

= 0

2:3

@v@r

= 0:

2:4

+ hv

r a

# $ %

& ' & &

( &

v r ' = R r)' 6 0:

2:5

& 2.5 ,- 2.1 , &

r dr ! + R

) '

= :

R r =r2

= )'

/ 0& & /,-:

r dr

! +

! R

= 0

r a

2:6

) + ) = 0

0 ' 2 :

2:7

# $ % & 3 ' 2 -& 5 ':

) 0 = ) 2

2:8

) 0 = ) 2 :

7 & 8 | 2.7 , 2.8 : .

1 . &

1.31 , 1.34 , 1.33 . #

n = n2

2:9

n =

0 1

0< $ % 0 &

)n ' = An cos n' + Bn sin n'

n =

0 1

2:10

8An Bn jAnj + jBnj 6= :0

329

		2	2		2	2	2
	0 '			n '				= 1 1:	2:11
			= 2 A0			= An + Bn n
= n 2.9 .26

1	0	n2 1
Rn + rRn + @		r2 A Rn = 0 r a :	2:12

2.2

r = 0:

Rn r ! " |$

2.12		Rn x = 0				2:13
		Rn x = 0				2:13
	jRn x j +1					2:14
' ! ! ( ( x = p						r
' ! ! ( ( x = p						r
2.12 . ) R r = R x=p y x, + :
dy x	dy dx dy p		d2y x	d2y
dy x	dy dx dy p		d2y x	d2y
dr	= dx dr	= dx	dr2	= dx2	:

- . / 0 1 ( 2.12

2 n- -

0d2y	1 dy	0	n2 11	= 0:
@dx2	+ x dx	+ @1	x2 AA	= 0:

4 56 7 1 ! +

y x = CJn x + DNn x

Jn x | 8-9 2 n- -, Nn x | 8-9 : ( n- -. 0 2.14 + 8-9 (

Nn x r ! 0 D = 0:

; ! 6 + - + ( (, 0

r = 0 7 2.12 p r : 2:15=

2.15 2.13 -

5 0/ ! (:

Jn p a = 0:

= > 5 0 ! :

330

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