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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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2.6.1.

4u = 0 u = g D = fr ag

De = fr ag: -cos 2 = 2 cos2 1 cos 3 = 4 cos3 3 cos P0 x = 1 P1 x = x P2 x = 3x2 1 =2 P3 x = 5x3 3x =2 % Pn x |

' .


1.	g = 2 cos + cos 2			De = fr ag:
2.	g = 3 cos2 cos 2			D = fr ag:
3.	g = 2 + 3 cos sin2			De = fr ag:
4.	g = cos2 + 3 sin2		D = fr ag:
5.	g = 3 cos 2 cos 2 De = fr ag:
6.	g = cos 3 sin2		D = fr ag:
7.	g = 3 cos 2	De = fr ag:
8.	g = 1 + sin2 + cos 2			D = fr ag:
9.	g = 2 cos sin2		De = fr ag:
10.	g = 5 cos + 3 cos2 D = fr ag:
11.	g = 1 + cos 3	De = fr ag:
12.	g = 2 cos 2 cos		D = fr ag:
13.	g = cos cos 3 De = fr ag:
14.	g = cos2 + 2 cos 3			D = fr ag:
15.	g = cos 3 + 2 cos		De = fr ag:
16.	g = cos 3 cos 2		D = fr ag:
17.	g = 2 cos 3	De = fr ag:
18.	g = 1 + 2 cos 2 cos D = fr ag:
19.	g = 2 cos 3 cos 3			De = fr ag:
20.	g = 1 + 2 sin2 + cos 3 D = fr ag:

2.6.2.


21.	g = 3 cos 2 cos 3			De = fr ag:
22.	g = 3 cos 2 cos 3			D = fr ag:
23.	g = 2 cos 2 + cos + 1			De = fr ag:
24.	g = 4 cos cos 3		D = fr ag:
25.	g = 3 + cos 3	De = fr ag:
26.	g = 3 cos 3 cos		D = fr ag:
27.	g = sin2 + cos 3		De = fr ag:
28.	g = 2 cos 2 1	D = fr ag:
29.	g = cos 3 1	De = fr ag:
30.	g = sin2 cos 2		D = fr ag:

! ! " ##$! #% # $& # # ' $$ ! r '

2 @u

@2u

4u =

! +

sin

@ ! +

= f r

r2 sin

r2 sin2

@'2

2:6:26

# $ !$ $ ! ' ( ! ! '

# $ '

= 0 :

2:6:27

r=a

		1 ! D = fr ag # *$ ! ' $( ! !!$# !
		juj 1 + r ! 0	2:6:28
		f r = r2 cos2 ,	2:6:29
		f r = r2 cos2 ,	2:6:29
		2 ! D =fr ag # *$ ! ' ( !$# ! 2.12
		u r ' = O 1r! + r ! 1	2:6:30
		f r = r 4 cos2 :	2:6:31
		f r = r 4 cos2 :	2:6:31
		. - + # ./ 2.6.26 ! # $ + -

' !!$& ', ! 2 ! # $ ', . . u r . -$(! ! 2.6.26 + '

2 @u

4u =

! +

sin

@ ! = f r :

2:6:32

r2 sin

!" # $ D D % " 2.6.27 , ) !" # * . . 3 .

+" -

!" # , $,

, , !" # -" | $" 2.6.8 , 2.6.11 , , 2.6.1 . 2.6.14 2.6.15 .

2 , 2.6.32 , 2.6.27 3 !"#- % 4!! # Rn r n = 0 1

1
u r = nX=0 Rn r Pncos	2:6:33

% , % !! # r . 2 ") !" # ) f r 6" -

				1
			f r =	X	fn r Pncos	2:6:34
				n=0
% 4!! #		:
	2n + 1
fn r =	2n + 1		Z f r Pncos sin d n = 0 1:			2:6:35
fn r =	2		Z f r Pncos sin d n = 0 1:			2:6:35
			0

8 2.6.33 , 2.6.35 9: 2.6.32 " , %-$ Pncos ) !" # 2.6.8 , 2.6.11 , . .

sin1 dd sin dd Pncos ! = n n + 1Pncos : ; " "

1		1 d			2 dRn r		n n + 1	=
X				@		A	r2

n=0	8r2		dr 0r		dr 1			Rn r + fn r 9 Pncos = 0:
	:

, -, -Pn cos ! n = 0 1 , $ f g :

2 dRn r!

n n + 1!

r! = fn r! n = 0 1:

2:6:36!

dr @

2.6.33! - 2.6.27!

- $ Rn r!:

Rn a! = 0

n =

2:6:37!

0 1

D -- 2.6.28!

jRn r!j 1		r ! 0 n =	0 1		:		2:6:38!
0 D -
- 2.6.30!
Rn r! = O 1r!		r ! 1 n =				:	:6:39
Rn r! = O 1r!		r ! 1 n =		0 1		:	:6:39
							-
Rn r,	! ! "						#
$ " .

% $ ! ! ! &'' ( fn r , ) * + ! *

	-
	%		#
		r2 cos2 = fn r Pn		:		:6:40
			0
	1 #			1 :
2		+BP1	+CP2 = A 1+B cos +C		3 cos2	1	:
cos	= AP0	+BP1	+CP2 = A 1+B cos +C		2		:

5 ! &'' ( ! " +" cos , # 5167

8 A	C		= 0	8 A =	1
		2			3
B = 0				B = 0
3C		= 1		C =	2	:
2		= 1		C =	3	:
:				:

2.6.40 :

r13P0cos + 23P2cos ! =

fn r:

f0 r =	r2	f2 r =	2r2	fn r
	3		3

fn r Pncos :

0 n 6=f0 2g:

2.6.36 , 2.6.37 , 2.6.38 # $% &-

( n 6=f0 2g:
		Rn r		0		n 6=f0 2g:			2:6:41
+ & % $# $ %( R0 r:
			2	R0 =	r2			9	2:6:42
		R0 +	r		3				2:6:42
		R0 +


		R0 a = 0						=	2:6:43
		R0 a = 0							2:6:43
		jR0 r j 1					r ! 0:		2:6:44

-./ & 0 12 2.6.42 3 %.									4 . .
n = 0 44.6 :		R0 r = C0			+ D0r 1:
		R0 r = C0			+ D0r 1:
-./ & 0 12 2.6.42 12 %5									6 -
	& 3 3, 3 .$ 6
		(
		R0 r = C0 r + D0 r r 1							2:6:45
0 0 r D0 r

C00 r + D00 r r 1 = 0

D00 r r 2 = r2=3:

0	r3
C0 r =	3
0	r4
D0 r =	3

			r4	~
				~
C0 r =	12 + C0
		r5		~
				~
D0 r =		15 + D0:

2.6.45 -

2.6.42

~	~	1 r4		2:6:46

			60:
r = C	+ D r

% 2.6.46 & 2.6.43 , 2.6.44 -

% ) * 2.6.42 | 2.6.44

R0 r =	1	a4	r4 :
	60

. ) % ) * R2 r:

R2 +	2	R2	2 3	R2 =	2r2	9
R2 +	r	R2	r2	R2 =	3
	r		r2		3
R2 a = 0
R2 a = 0						=
jR2 r j 1 r ! 0:

2 &				2.6.48 ) .

n = 2 4.6 :

R2 r = C2r2 + D2r 3:

2:6:47

2:6:48 2:6:49 2:6:50

4 .

2 & 2.6.48 3 %- 455 6 ) * ), ) 6 % 3

6 7

R2 r = C2 r r2 + D2 r r 3

2:6:51

& 2 r D2 r 7 *

= 0

8 C2 r r

+ D2 r r

=3:

2C2 r r

3D2 r r

= 2r

2 8 7

C2 r =

+ C2

2r6

2r7

D2 r =

105 + D2

2.6.51 -

2.6.48

2:6:52

21:

r =

C r r

+ D

r r

% 2.6.52 & ' 2.6.49 , 2.6.50 -

% + , 2.6.48 | 2.6.50

R2 r =

a2r2

2:6:53

% 2.6.47 , 2.6.53 , 2.6.41 2.6.33 -

+ , 2.6.32 , 2.6.27 , 2.6.28 , 2.6.29

u r =

r4 P0cos

a2r2

r4 P2cos =

3 cos

2:6:54

0 1

2.6.32 ,

2.6.27 ,

2.6.30 ,

2.6.31 . 2 2.6.34 2.6.31

r4 cos2 = fn r Pncos :

n=0

3 4 1 ,				,				3
r 4 1P0cos + 2P2cos ! =						1
						X	fn r Pncos :
	3	3
						n=0
5 6 % 788 9 fn r:
f0 r =	r 4	f2 r =	2r 4		fn r		0		n 6=f0 2g:
	3			3

2 % , 2.6.36 , 2.6.37 , 2.6.39									4 -
' n 6=f0 2g:

Rn r 0		n 6=f0 2g:
; % 4 , ' R0 r:
00	2 0		r 4		9
R0	+ rR0 =			3	9
					=
R02 = 0
R0 r = O1=r					r ! 1:
		97

2:6:55

2:6:56 2:6:57 2:6:58

2.6.56

r = C r + D r r 1

2:6:59

0 r D0 r !

8 C0 r + D0 r r

= 0

r r

=3:

r 3

r 2

C0 r =

6 + C0

r 2

r 1

D0 r =

+ D0:

& '() * 2.6.59 '() -

2.6.56

1 r 2

2:6:60

R0 r = C0 r + D0

r r

& ' , 2.6.60 2.6.57 2.6.58 '() ,- / ! ) 2.6.56 | 2.6.58 :

				a 1r 1 r 2				2:6:61
	R0 r =				6	:		2:6:61
					6
3 / , / ! ) ' R2 r:
00	+	2	0	2 3	R2 =	2r 4	9	2:6:62
R2	+	r	R2	r2	R2 =	3	9	2:6:62
		r		r2		3
R2 a = 0								2:6:63
R2 a = 0							=	2:6:63
R2 r = O1=r					r ! 1:			2:6:64

2.6.62


R2 r = C2 r r2 + D2 r r 3	2:6:65
2 r D2 r !
98

8 C r r

+ D0 r r 3 = 0

= 2r

=3:

2C2 r r 3D2 r r

2r 5

r 4

r =

r = 30 + C2

D2 r =

+ D2:

#$ -

& '(

r 2

R2 r =

C2 r r

+ D2

r r

6 :

:6:66

)

& ) ,

R2 r =

r 2 ar 3

:6:67

1 & & , - )

ur = 1

a 1r 1 r 2 P0

r 2 ar 3 P2

= 1a 1r 1 r 2 +

r 2 ar 3

cos2 1:

:6:68

. 1 1 & , -:

ur =

a4 r4 +

a2r2 r4

cos2 1 3

2 & & , -:

ur =

1 a 1r 1 r 2 +

r 2 ar 3 cos2 1:

2.6.2.

.
	4u = f r D = fr ag

u= g

f r = r2 cos2

g = cos 2 + cos

2.6.21 2.6.54 , ! "# ! # 2.6.1 2.6.2:

u r = 3 + a

r cos +

3 cos

1 +

a2r2

r43 cos2

4u = f r De = fr ag

u r=a= g

f r = r 4 cos2 g = cos 2 + cos 2.6.25 2.6.68 :

u r = 13ar 1 + a2r 2 cos + 23a3r 33 cos2 1 + 16 a 1r 1 r 2+ +121 r 2 ar 33 cos2 1:

) * 4u = f r " " u r=a= 0 D = fr ag De = fr ag:


1.	f r = 5r	3 cos 3	De = fr ag:
2.	f r = 5r2 cos 3		D = fr ag:
3.	f r = 3r	4 cos 2	De = fr ag:
4.	f r = 3r2 cos 2		D = fr ag:
5.	f r = 5r	3cos + cos 3 De = fr ag:
6.	f r = 5r2cos cos 3			D = fr ag:
7.	f r = 9r	62 + cos 2		De = fr ag:

100

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