Добавил:

Chupapi_Munyanya Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный исследовательский университет «МЭИ»

Предмет:

Высшая математика

Файл:

УрМатФиз / УрМатФиз с теорией

.pdf

Скачиваний:

Добавлен:

16.05.2024

Размер:

4.39 Mб

Скачать

☆

<<< < Предыдущая 8 9 10 11 12 13 14 15 16 17 18 1920 / 3620 21 22 23 24 25 26 27 28 29 30 31 32 > Следующая >>>

cos 2x = X0 x cos 6x = X1 x | - . ! ! !! 3.1.23 , %

		g0 = 2				g1 = 3 gn = 0			n 6= 0 1:	3:1:24
( % 3.1.18 3.1.19 3.1.13 :
	8dun t		2	2	2	2n + 1	2		9
		dt	+ a	2		2n + 1		un t fn t = cos 2 2n + 1x = 0:
n=0	:	dt
-		f g \| . / 0 1

% ! , / , 2.

3 2 % 345 % ! un t
dun t	+ a24 2n + 12un t = fn t n =		:	3:1:25
dun t	+ a24 2n + 12un t = fn t n =	0 1	:	3:1:25
dt
( % 3.1.18 3.1.22 / 3.1.15				%
1
Xfun0	gng cos 2 2n + 1x = 0 un0 = gn:			3:1:26
n=0

7 % un t 2 8 % 0 9 8 3.1.25 , 3.1.26 345. - 8 % , 3.1.21 , 3.1.24 , 0 -9 8 3.1.25 , 3.1.26 n 6= 01 2 % 2 8 un t 0fn t 0 n 6=2 / -

gn = 0 n 6= 01 . : 9 8				n = f0 1 2g % 2
2; :
du	t	+ a24u0 t = 0	u00 = 2<		3:1:27
0
dt
du	t	+ a236u1 t = 0	u10 =	3<	3:1:28
1
dt		+ a2100u2 t = cos 2t u20 = 0:
du	t	+ a2100u2 t = cos 2t u20 = 0:			3:1:29
2

> 8 % 0 3.1.27 . 3 ; 8 45 u0 t = A0e a24t

% / % u00 = A0 = 2:

191

3.1.27
u0 t = 2e	a24t:	3:1:30
! "	3.1.28 :
u1 t = 3e	a236t:	3:1:31

% & '( " )* 3.1.29 " , &. -'( " .( " )*

u t = A2e a2100t:

-'( " " )* ' /

u2 t = A2 t e a2100t:

3:1:32

0 3.1.32 )* 3.1.29 ,

t = ea

100t cos 2t:

-. !

a2100t

A2 t =

10a 4 + 4

100 cos 2t

+ 2 sin 2t + A2:

-'( " " )* 3.1.29

a2100t

u2 t =

10a 4 + 4

100 cos 2t + 2 sin 2t + A2e

% &

A2, , , 5" 6 /

u20 = 0.

3.1.29 ,"

u2 t =

a2100 cos 2t + 2 sin 2t

a2100e a2100t :

3:1:33

10a 4

+ 4

0 3.1.30 , 3.1.31 , 3.1.33 un t 0 ," n 6=f0 1 2g 3.1.18 , " ! & .

192

. u x t = 2e a24t cos 2x						3e	a236t cos 6x+
		1						2
+				a2100 cos 2t + 2 sin 2t				a2100e a	100t cos 10x:
+	a	4 + 4		a2100 cos 2t + 2 sin 2t				a2100e a	100t cos 10x:
	a	4 + 4
.
Gx y! t = 8					e a24 2n+1 2 t cos			n + 1 x cos n + 1 y
		n=0
" # $ # %&' (
		=4					t =4
ux t = Z g y G x y! t 0 dy + Z Z f y G x y! t dy d :
		0				0	0
3.1.2.				) " & $& - ' $ -
+ #$ #
	@u	= a2	@2u		+ f x t D = fx			: 0 x lg 0 t
	@t	= a2	@x2		+ f x t D = fx			: 0 x lg 0 t
	@t		@x2

- + - $ - $& - $ -

	u t=0= g x x l :
	u t=0= g x x l :
1.	u x=0= u x= = 0
	fx t = sin x cos t	g x = 3 sin 2x sin 3x:
2.	u x=0= ux x= =2= 0
	fx t = 2 sin 3x cos t	g x = 2 sin x 3 sin 5x:
3.	ux x=0= u x= =2= 0
	fx t = cos 5x sin t	g x = 2 cos x + 3 cos 3x:
4.	ux x=0= ux x= = 0
	fx t = 2 cos 3x sin t	g x = 3 + cos x 5 cos 2x:

5.u x=0= u x=2 = 0

fx t = sin x cos t g x = 2 sin x=2 sin x:

193

6.	u x=0= ux x= = 0
	f x t = sin 5x=2 sin t		g x = 3 sin x=2		sin 3x=2 :
7.	ux x=0= u x= = 0
	f x t = cos 3x=2 cos t		g x = cos x=2		2 cos 5x=2 :
8.	ux x=0= ux x= =2= 0
	f x t = 2 cos 2x cos t		g x = 1 + cos 2x		2 cos 4x:
9.	u x=0= u x=2 = 0
	f x t = 2 sin x sin 3t		g x = sin x=2 3 sin x:
10.	u x=0= ux x=2 = 0
	f x t = sin x=4 sin 2t		g x = 2 sin x=4		sin 3x=4 :
11.	ux x=0= u x=2 = 0
	f x t = cos x=4 sin t		g x = 3 cos x=4		cos 3x=4 :
12.	ux x=0= ux x=2 = 0
	f x t = 2 cos 2x cos t		g x = 2 cos x=2		cos x:
13.	u x=0= u x= = 0
	f x t = sin 2x t	g x = sin x		2 sin 3x:
14.	u x=0= ux x= =2= 0
	f x t = sin x sin t	g x = 3 sin x		sin 3x:
15.	ux x=0= ux x= =2= 0
	f x t = cos 3x sin t	g x = cos x		3 cos 5x:
16.	ux x=0= ux x= = 0
	f x t = cos x cos t	g x = 2 cos x + 3 cos 2x:

194

17. u x=0= u x=2 = 0

f x t = sin x t g x = sin x=2 + 3 sin 2x:

18.u x=0= ux x= = 0

f x t = 2 sin 3x=2 sin t g x = sin x=2 + 2 sin 5x=2 :

19.ux x=0= u x= = 0

f x t = cos x=2 sin 2t g x = 2 cos x=2 + cos 3x=2 :

20.ux x=0= ux x= =2= 0

f x t = cos 4x sin t g x = 2 cos 2x + cos 6x:

21.u x=0= u x=2 = 0

f x t = 3 sin 2x sin t g x = 3 sin x=2 + sin x:

22.u x=0= ux x=2 = 0

	f x t = sin x=4 cos t		g x = sin x=4 sin 5x=4 :
23.	ux x=0= u x=2 = 0
	f x t = 2 cos 3x=4 cos t g x = cos x=4 2 cos 5x=4 :
24.	ux x=0= ux x=2 = 0
	f x t = 3 cos x sin t		g x = cos x	3 cos 2x:
25.	u x=0= u x= = 0
	f x t = sin x sin t	g x = sin 2x		sin 5x:
26.	u x=0= ux x= =2= 0
	f x t = sin 5x t	g x = sin x + 2 sin 3x:

27.ux x=0= u x= =2= 0

f x t = cos x cos t g x = cos x + 2 cos 3x:

195

3.2.1.

28.	ux x=0= ux x= = 0
	f x t = 3 cos 2x sin 2t	g x = 2 cos x	cos 3x:
29.	u x=0= u x=2 = 0
	f x t = sin 2x sin t g x = sin x 3 sin 5x:
30.	u x=0= ux x= = 0
	f x t = sinx=2 cos t	g x = sin 3x=2	sin 5x=2:

3.2.,

, ,

1 -! " # D

@u	= a24u + f x y t D					= f x y : 0 x 0 y =2g			0 t
@t									3:2:1
! # &" ! " @D									3:2:1
! # &" ! " @D
	u		= u		=	@u	= @u	= 0 0 t	3:2:2
		x=0		x=		@y y=0	@y y= =2
&" ! "
	u t=0= g x y = 2 sin x cos 2y							x y 2 D	3:2:3
#			f x y = 3 sin 2x cos 4y cos t:						3:2:4
			f x y = 3 sin 2x cos 4y cos t:						3:2:4

2 - - ! # " ! D

@u	= a24u + f r ' t D = f r ' : 0 r b 0 ' =2g	0 t
@t		3:2:5
		3:2:5

! # &" ! " @D

196

@u	@u
@' '=0= @' '= =2= u r=b= 0 0 t			3:2:6

	0	2 r 1
u t=0= g r ' = J2 B		1b C cos 2'	3:2:7
	@	A


0	20 r 1		3:2:8
f r ' t = 2J0 B		C sin t	3:2:8
@	b	A
k2n \| k- J2n = 0		J2n x \| !
2n- " .

. 1 % & ' () |* , ) " ) ' ) ) ' " ,3.2.1 " f x y t 0 -3.2.2 . -& .) ' " " 3.8.1. % / )-' ) ) ) 01 / ) ' 0)

3.8.19 , 3.8.20
nk = n2 + 2k 2 vnk x y = sinnx cos 2ky jjvnkjj2	=	2 k0	+ 1
nk = n2 + 2k 2 vnk x y = sinnx cos 2ky jjvnkjj2	=	8

n = 1 1 k = 0 1:

-& , ' / ) ' 4 "

.) / ) , " " ' 4 ) -) ' " " ) ) " x y ' " " t:

1 1 1 1

u x y t = X X unk t vnk x y = X X unk t sinnx cos 2ky : 3:2:9
n=1 k=0	n=1 k=0

6' ) f x y t g x y 3.2.3 3.2.4 ) 4 '- 4 " ) / ) , vnk x y

1 1

f x y t = 3 sin 2x cos 4y cos t = X X fnk t vnk x y =

n=1 k=0

1	1
= X X fnk t sinnx cos 2ky		3:2:10
n=1 k=0

197

g x y = 2 sin x cos 2y =

gnkvnk x y =

1 k=0

gnk sin nx cos

:2:11

n=1 k=0

fnk

t =

ZZ f x y t vnk x y dxdy

jjvnkjj

gnk =

g x y vnk x y dxdy

n = 1 1

= 0 1:

jjvnkjj

fnk

t gnk

, " # $ % &.

( , v22

x y = sin 2x cos 4y v11 x y = sin x cos 2y %-

" % + # % + $+ %

f22t = 3 cos t

fnk

t 0 " n k 6=2 2

:2:12

g11 = 2 gnk = 0 "

n k 6=1 1:

, %

% ./

1 1

8dunk t

t f

x y = 0:

+ a

t = v

n=1 k=0

" # % +

% D " 1./

dunkt

n = 1 1

k = 0 1:

:2:13

+ a

nkunk t = fnk

, " %

% %

X Xfunk

gnkgvnk x y = 0 unk

= gnk

:2:14

n=1 k=0

n = 1 1 k = 0 1:

198

3.2.13 , 3.2.14 , : n k = 1 1

n k = 2 2 , !"

# %

3.2.12 , # #$, unk

0 n k

6= 1 1 n k

6= 2 2:

' % #$

+ a2 11u11 t = 0

u110 = 2)

+ a2 22u22 t = 3 cos t

u220 = 0

% #% % * +

u11 t = 2e

a2 t = 2e

a25t

3:2:15

u22 t = 3

sin t + a220 cos t a220e

a220t

3:2:16

a4400 + 1

3.2.15 , 3.2.16

unk t

n k

6= 1 1

n k

6=2 2 3.2.9

3.2.1 | 3.2.4 :

u x y t

= 2e

a25t sin x cos 2y+

sin t + a220 cos t

a220e a220t

sin 2x cos 4y

3:2:17

a4400 + 1

2 $ % & |' (, ) ( - ( + ( ,- 3.2.5 f r ' t 0 , . , ( 3.2.6 . / 0 % 3.8.1. $2 , - ( 34 2 , 5 )6 3 ( 5 3.8.28 , 3.8.29

0 2n 12

nk = B k C vnk r ' = J2n q nkr cos 2n'

@ b A

0 2n

n0 + 1

jjv

n = 0 1 k = 1 1

= bJ2n k

| k-% ) + ( J2n

= 0 J2n x | 5 )6 ( 7 (

. k

2n-. ( ) .

199

, -!r '"t:

u!r ' t" =

0k=1

n=0 k=1

unk!t"vnk!r '" =

0 2n r 1

unk!t"J2n B k C cos!2n'": !3:2:18"

@ b A

) f!r ' t" g!r '" !3.2.8", !3.2.7"vnk!r '"

0	20 r 1	fnk t vnk r ' =
f!r ' t" = 2J0 B	C sin t	fnk t vnk r ' =
@	b A	n0 k=1

					0	k		r 1
						2n
=			fnk t J2n		@	b		A	cos n'
	n=0 k=1				B	b		C
gr ' =	J2	0	12 r	1 cos 2'		= nX kX gnkvnk r ' =
gr ' =	J2	0	b	1 cos 2'		= nX kX gnkvnk r ' =
		B		C			=0 =1
		@		A

r 1

= nX kX gnkJ2n

cos

fnk

t =

f r ' t vnk

r ' r drd'

jjvnkjj

ZZ g r ' vnk r ' drd'

gnk =

n = 0 1 k = 1 1:

jjvnkjj

:2:19

:2:20

fnk						t gnk
! , #$ % & '.
) ,	v02 r ' = J0 0	20 r	1	v11	r ' = J2	0	12 r	1 cos 2':
) ,	v02 r ' = J0 0		1	v11	r ' = J2	0		1 cos 2':
	B	b C				B	b C
	@		A			@		A
* & # & + $& + %+ &
	#

200

<<< < Предыдущая 8 9 10 11 12 13 14 15 16 17 18 1920 / 3620 21 22 23 24 25 26 27 28 29 30 31 32 > Следующая >>>

Соседние файлы в папке УрМатФиз

#
16.05.20249.25 Mб1Bilety_po_urmatfizu_v0_9.docx
#
16.05.202417.63 Mб0Kachestvennaya_podgotovka_k_ekzu_21_goda_1.docx
#
16.05.2024207.5 Кб0Задача Коши для ДУ 1-го пор.docx
#
16.05.2024416.66 Кб0Общ. реш. ДУ 1-го пор..docx
#
16.05.2024147.34 Кб1Программа по Матфизике.pdf
#
16.05.20244.39 Mб9УрМатФиз с теорией.pdf