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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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0 = 0, Y0 y = 1:

2.1.28			= n:
	Xn x	nXn x = 0:	2:1:31
% 0 = 0 & ' '
	X0 x = A0x + B0:
% n = 2n 2 n =			2.1.31
	1 1

Xn x = Ane x + Bne 2nx

Xn x = An sh 2nx + Bn ch 2nx :

! " ! #							% $ % # "-
& ,						) ! %			2.1.8 , . .
	Xn x = An sh 2nx + Bn sh 2n x :									2:1:32
+ ,	%
	sh 2nx		x=0	= 0		sh 2n x		= 0:
			x=0					x=
+ $ % , !						)
2.1.32		! .
. ", % !							! %/ $ 2.1.27 :
			u0 x y = A0x + B0
un x y = An sh 2nx + Bn sh 2n x cos 2ny									n =		:
un x y = An sh 2nx + Bn sh 2n x cos 2ny									n =	1 1	:
1 $ ! 2.1.7 \| 2.1.9 " # "&

			u x y =			un x y =
						n=0
		1
= A0x + B0 +		An sh 2nx + Bn sh 2n x cos 2ny								2:1:33
		n=1
,	!	)			## &			%:		x y.
4	2.1.33 ) ! %						2.1.8
			1
	'1 y = B0 +		An 0 Bn sh 2n cos 2ny							2:1:34
		n=1

'2y = A0 + B0 +

An shn

0 cos ny :

:1:35

n=1

An Bn

! "-

$ % & ' (0 =2):

+ n0 :

cos

ny cos

ky dy =

% - !

cos

k =

0 1

, !-

"! !

y (0 =2) -

Z '1 y 1 dy =

y cos

ky dy =

Bk shk

k = 1

0 %!

B0 =

Z '1 y dy

:1:36

Bk =

y cos

ky dy

k = 1

shk

2 "-

A0 + B0 =

Z '2 y dy

:1:37

Ak =

Z '

y cos

ky dy

k = 1

shk

! 5 -

" ! .

An Bn , -2.1.36$, 2.1.37$. & ' 2.1.33$

2.1.8$,	2.1.34$ 2.1.35$ '
2 cos 2y = B0	Bn sh 2n $ cos 2ny$	2:1:38$
	n=1
	1
0 = A0 + B0 +	An sh 2n $ cos 2ny$:	2:1:39$
	n=1

/ 2.1.38$ cos 2y = Y1 y$ | . / -' 1 1 1

1 2.1.38$
B1 sh 2 $ = 2 Bn = 0 n 6=:1			2:1:40$
23 2.1.39$
An = 0 n =		:	2:1:41$
An = 0 n =	0 1	:	2:1:41$

& ' 2.1.40$ 2.1.41$ 2.1.33$ uII x y$ 3 ' 2.1.7$| 2.1.9$:

uII x y$ =	2	sh 2x $$ cos 2y:	2:1:42$
	sh 2 $

& ' 2.1.26$ 2.1.42$ 2.1.10$ 1 ' 3 ' 2.1.1$| 2.1.3$.

u = 2 sh3 ch 2 y 2 !! sin 2x sh 22 $ sh 2x $$ cos 2y: 2:1:43$

2.1.1.

5 6 3 '

4u = 0 6

x=0

= 5 cos 3y

= 0 uy

y=0

= 3 sin 2x

y= =2

= 0:

x=0

= 0

x= =2

= 2 cos 2y

y=0

= 3 cos 5x

x=0

= 0

= 5 sin 3y u

y=0

= 0

y= =2

= 3 cos x:

= sin 2y

= 0

= 2 sin 3x:

x=0

x= =2

y=0

x=0

= 0

= 5 cos 2y

= 0

= 3 sin 3x:

y=0

= 0

x= =2

= 2 sin 5y

= 2 cos 3x

= 0:

x=0

y=0

y= =2

= 3 sin y

= 0

= 2 cos 2x

= 0:

x=0

y=0

= 2 cos 3y

= 0 uy

= 0 u

= 3 sin x:

x=0

x= =2

y=0

y= =2

= 0

= 2 sin 3y

= 0 uy

= 3 sin 2x:

x=0

y=0

y= =2

10.

= 0

= 3 sin y

= 3 cos 5x

= 0:

x=0

x= =2

y=0

11.

= 3 cos 5y

= 0

= cos x

= 0:

x=0

y=0

y= =2

12.

= cos 2y

= 0

= 2 sin 3x:

x=0

x= =2

y=0

13.

x=0

= 0

= 3 sin y

= 0

= 2 sin 2x:

y=0

14.

= 0

= 2 cos y

= 2 cos 3x

= 0:

x=0

x= =2

y=0

y= =2

15.

= 2 cos 2y

= 0

= cos x

= 0:

x=0

y=0

16.

= 2 sin 3y

= 0

= 3 sin x:

x=0

x= =2

y=0

y= =2

17.

= 0

= 2 cos 3y

= 3 sin x

= 0:

x=0

y=0

y= =2

18.

= 0

= cos y

= 0

= 2 cos 3x:

x=0

x= =2

y=0

19.

= sin 5y

= 0

= 0 uy

= 2 cos 2x:

x=0

y=0

y= =2

20.

= 0

= 3 sin 2y

= 0

= 2 sin 3x:

x=0

x= =2

y=0

21.

x=0

= 0

= 2 cos 4y

y=0

= sin 2x

= 0:

22.

x=0

= 2 sin 3y

x= =2

= 0

y=0

= 3 cos x

y= =2

= 0:

23.

x=0

= 3 sin y

= 0

y=0

= 0 u

= cos 4x:

24.

x=0

= 0

x= =2

= cos 3y

y=0

= 0

y= =2

= 2 sin x:

25.

x=0

= 0

= 2 sin 5y

y=0

= sin 2x

y= =2

= 0:

26.

x=0

= 3 sin 4y

x= =2

= 0

y=0

= cos x

= 0:

27.

x=0

= 2 cos 3y

= 0

y=0

= 0

y= =2

= cos x:

28.

x=0

= 0

x= =2

= 3 cos 2y

y=0

= 0

= 2 sin 5x:

29.

u x=0= 0

u x= = 3 sin 4y

u y=0= sin 2x

u y= = 0:

30.

x=0

= cos 3y

x= =2

= 0

y=0

= 2 cos x

y= =2

= 0:

2.1.2. ! " #! $! !% !&

4u = f'x y( D = 'x y( : 0 x 0 y 2						'2:1:44(
! ! ! # $ # ! #
u	x=0= 0	u	x= = 0	0 y 2 !		'2:1:45(
@u
@u		@u
@y y=0= 0		@y y= =2= 0			'0 x (	'2:1:46(
$ f'x y( = 4 sin 3x cos 4y:						'2:1:47(

. + # "!#!$ , # |. -!" !& *0 % !! 1 0 !& *0 2 3 % !" ! . " $ *# ! # '2.1.45(, '2.1.46(:

4v + v = 0

D =

x y : 0 x 0 y 2

2:1:48

x=0= 0

x= = 0 0 y 2 !

2:1:49

y=0= 0

@y y= =2= 0 0 x :

2:1:50

2.8.1. ! "

nk = n2 + 2k 2 n =

k =

2:1:51

1 1

0 1

#$% '(

vnk x y = sinnx cos 2ky n =

1 1

k =

0 1

2:1:52

)# ' * + , ! 2.1.44 | 2.1.47 - 0 1 " " '( " vnk x y c '- 2&& ( unk:

1	1	1	1
nX kX		nX kX
u x y =		unkvnk x y =		unk sinnx cos 2ky :	2:1:53
=1	=0	=1	=0

4 0 1 , ! 2 " 1 1 && ( *x y. 6 0 1 '($ f x y " '( " vnk x y:

1	1	1	1
nX kX		nX kX
f x y =		fnkvnk x y =		fnk sinnx cos 2ky :	2:1:54
=1	=0	=1	=0

7 2&& ( 8# * fnk 0#! 0 9' + " ". 6 2.1.54

1	1
nX kX		fnk sinnx cos 2ky	2:1:55
4 sin 3x cos 4y =		fnk sinnx cos 2ky	2:1:55
=1	=0

9 0 , ! sin 3x cos 4y = v32 x y | " '( ".

" ' 2&& ( ' + + '( "+0 , , ! "+ 2.1.55 , 0#!

	= 4		nk = 0		6= 3		6= 2
f32		f		n		k	:

unk = fnk :

2.1.53 2.1.54 2.1.44 :

1
funk4vnk + fnkvnk x y g = 0:	2:1:56
n=1 k=0

vnk x y | " # $ % & % 2.1.48 :

4vnk = nkvnk:

( ) 2.1.56 %

f unk nk + fnkgvnk x y = 0:

n=1 k=0

* # + , " f g | -##$. / 0 % % , " .1 # $,, 0 , . 3. 4 3 % 5

-##$. 2.1.53 , % 5

) 5 .

* ) 5

-##$.

u32 =

unk = 0

% n

6= 3 k 6= :2

2:1:57

32 + 42

2.1.57 2.1.53 % 5

) 1 , 5

2.1.44 | 2.1.47 .

. u x y =

sin 3x cos 4y:

2:1:58

. 7 )

, 5

4u = f x y

D = x y : 0 x 0 y 2

2:1:59

= '1 y

= '2 y

0 y

2:1:60

x=0

@y y=0= 1 x

@y y= =2= 2 x 0 x

2:1:61

f x y = 4 sin 3x cos 4y, '1 y = 2 cos 2y '2 y 0

1 x = 3 sin 2x

2 x 0 2.1.43 , 2.1.58 , !" -" 2.1.1 2.1.2:

2.1.2.

ux y =

ch 2 y

2 !! sin 2x

sh x

cos 2y+

2 sh

sin 3x cos 4y:

y! =

sin nx sinn +

n=1

nX=1 kX=1

sin nx cos

ky sinn cos 2k

n2 + 2k 2

# $

% $ & & % '() *

ux y = Z Z f G x y! d d + Z 1 + 2 G x y! 0 d +

@G x y! 0

+ Z '1

d :

# $ ( ) , - && 4u = fx y % . ,( /, & & 0 . 0&, .

1.	u x=0= u x= = uy y=0= u y= =2= 0					f x y = 2 sin x cos 5y:
2.	ux x=0= u x= =2= uy y=0= uy y= = 0						f x y = cos x cos 4y:
3.	ux x=0= ux x= = u y=0= uy y= =2= 0						f x y = 2 cos 2x sin y:
4.	u x=0= ux x= =2= u y=0= u y= = 0					f x y = 3 sin x sin 4y:
5.	u x=0= u x= = uy y=0= uy y= = 0					f x y = 5 sin x cos y:
6.	ux	x=0= u	x= =2= u	y=0= uy	y= =2= 0		f x y = 3 cos x sin 7y:
					38

x=0

= ux

= u

y=0

= u

= 0

x y = 2 cos x sin 4y:

x=0

= ux

x= =2

= uy

y=0

= u

y= =2

= 0

f x y = 3 sin 3x cos y:

x=0

= u

y=0

= uy

y= =2

= 0

f x y = 2 sin x sin y:

10.

x=0

= u

x= =2

= u

y=0

= u

= 0

f x y = 7 cos x sin 4y:

11.

x=0

= ux

= uy

y=0

= u

y= =2

= 0

f x y = 5 cos 4x cos 3y:

12.

x=0

= ux

x= =2

= uy

y=0

= uy

= 0

f x y = 7 sin x cos 2y:

13.

u x=0= u x= = u y=0= u y= = 0

f x y = 5 sin 2x sin 4y:

14.

x=0

= u

x= =2

= uy

y=0

= u

y= =2

= 0

f x y = 3 cos 5x cos 3y:

15.

x=0

= ux

= uy

y=0

= uy

= 0

f x y = cos 2x cos y:

16.

x=0

= ux

x= =2

= u

y=0

= uy

y= =2

= 0

f x y = 5 sin 3x sin 5y:

17.

x=0

= u

= uy

y=0

= u

y= =2

= 0

f x y = 7 sin 2x cos 5y:

18.

x=0

= u

x= =2

= uy

y=0

= uy

= 0

f x y = cos x cos 2y:

19.

x=0

= ux

= u

y=0

= uy

y= =2

= 0

f x y = 3 cos x sin 3y:

20.

x=0

= ux

x= =2

= u

y=0

= u

= 0

f x y = 2 sin x sin 4y:

21.

x=0

= u

= uy

y=0

= uy

= 0

x y = 5 sin 4x cos y:

22.

x=0

= u

x= =2

= u

y=0

= uy

y= =2

= 0

f x y = 7 cos 5x sin y:

23.

x=0

= ux

= u

y=0

= u

= 0

x y = 5 cos 2x sin 2y:


24.	u x=0= ux x= =2= uy y=0= u y= =2= 0		f x y = 3 sin 5x cos 3y:
25.	u x=0= u x= = u y=0= uy y= =2= 0	f x y = 3 sin 4x sin y:
26.	ux x=0= u x= =2= u y=0= u y= = 0	f x y = 5 cos 3x sin 2y:
27.	ux x=0= ux x= = uy y=0= u y= =2= 0		f x y = 7 cos 2x cos y:
28.	u x=0= ux x= =2= uy y=0= uy y= = 0		f x y = 3 sin 3x cos 4y:
29.	u x=0= u x= = u y=0= u y= = 0 f x y = 7 sin x sin y:
30.	ux x=0= u x= =2= uy y=0= u y= =2= 0		f x y = cos x cos y:

2.2.

2.2.1.							! "
			2
4u = 1r		@		r@u@r ! +	1	@'@ u2 = 0	2:2:1
		@r			r2
" % & " ' &
u	r=a		= g ' = 4 sin2 '				2:2:2

% D = fr ag " (' & '% '" !
r ! 0		ju r ' j 1					2:2:3

% D = fr ag " (' & % '" !

r ! 1	ju r ' j 1:	2:2:4

. )* & " ' 2.2.1 , ' ' ,-

* " ' , ! ' '"
u r 0 = u r 2	@u r 0	=	@u r 2	2:2:5
	@'		@'

u r ' = R r.' :				2:2:6

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