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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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. | -, -


4.1.17$ f x t$ 0
		4.1.18$ .	4.1.1, \|						4.1.7$,
4.1.8$$		X	x$ + X x$ = 0

		X0$ = 0 X =2$ = 0:
*		+ ,		. 1 . $. - -
	.		- / 0					. ,
11.22$, 11.23$ l = =2:
		n = 2n + 1$2 n =
					0 1
									4:1:22$
		Xn x$ = sin 2n + 1$x		n =		0 1	:

* , 4.1.17$| 4.1.21$ - - 4 / 0 , - / 0 4.1.22$+// 0 un t$

	1
u x t$ =	un t$Xn x$ = un t$ sin 2n + 1$x	4:1:23$
0	n=0

, 4 // 0 4 t x. * 4 / 0 f x t$ g x$ p x$ - -

/ 0 4.1.22$

1			1
f x t$ =	fn t$Xn x$ =		fn t$ sin 2n + 1$x		4:1:24$
n=0			n=0
1		1
g x$ =	gnXn x$ =	gn sin 2n + 1$x			4:1:25$
n=0		n=0
1		1
p x$ =	pnXn x$ =	pn sin 2n + 1$x:			4:1:26$
n=0		n=0
7 +// 0 fn t$ gn				pn . *-
4.1.24$, 4.1.25$ 4.1.26$
		1
3e t sin x =		fn t$ sin 2n + 1$x			4:1:27$
		n=0
		1
2 sin 3x + sin 5x =		gn sin 2n + 1$x			4:1:28$
		n=0
		1
	0 =	pn sin 2n + 1$x			4:1:29$
		n=0

241

sin x = X0 x sin 3x = X1 x sin 5x = X2 x |. -4.1.27 , 4.1.28 , 4.1.29 , (

f0 t = 3e t	fn t 0			n 6= 0	4:1:30
g1 = 2 g2 = 1	gn = 0			n 6= 1 2	4:1:31
					4:1:32
pn = 0	n =	0 1	:		4:1:32

* ( 4.1.23 4.1.24 4.1.17 :

	8d2un t		2		2	9
		2	+ a	2n + 1		un t fn t = sin 2n + 1x = 0:
n=0	:	dt
-		/ 0 1 2 .

( ., 0 , 3. 4-

3 ( 456 ( un t
d2un t	+ a22n + 12un t = fn t n =		:	4:1:33
		0 1
dt2

* ( 4.1.23 , 4.1.25 , 4.1.26 0 4.1.19 4.1.20 (

4:1:34

X fun0 gngsin 2n + 1x = 0

un0 = gn

n =

0 1

n=0

8dun0

dun0

= pn

n = 0 1:

4:1:35

pn=sin 2n+1x = 0

n=0

9 ( un t 3 : ( 1 ; : 4.1.33 , 4.1.34 , 4.1.35 . - 1 ; : n 6=0 1 2 ( 3 : un t 0 56 fn t 0 n 6= 0 0gn = 0 pn = 0 n 6= 12 .

< ; : n 6= 01 2 ( 3 3= . :

2			t
		0
d u					+ a2u0 t = 3e t	u00 = 0	u00 0 = 0	n = 08	4:1:36
	dt2				+ a2u0 t = 3e t	u00 = 0	u00 0 = 0	n = 08	4:1:36
	dt2
2				t
		1			+ a29 u1 t = 0	u10 = 2	u10 0 = 0	n = 18	4:1:37
d u					+ a29 u1 t = 0	u10 = 2	u10 0 = 0	n = 18	4:1:37
		dt2
2			t
		2			+ a225 u2 t = 0	u20 = 1	u20 0 = 0	n = 2:	4:1:38
d u					+ a225 u2 t = 0	u20 = 1	u20 0 = 0	n = 2:	4:1:38
	dt2

> : ( 1 ; : 4.1.36 . 4 = : 56 (

242

t = A0 cos at + B0 sin at + u : :

u : = Ae t:

! A = 3=

+ a2 : "#

t = A0 cos at + B0 sin at + 3e t=

+ a2 :

$%&& ' ( A0 B0

) , % ( +

,--

(

/ 0 , $

u0t = 3 a1 sin at

cos at + e t! =

+ a2 :

:1:39

/ 0 ,

"# :

u1 t = A1 cos

at + B1 sin

at :

,- (

) A1 = 2 B1 = 0:

/ 0 , $

& '

t = 2 cos at :

:1:40

6 ,

! 0 , $

t = cos at :

:1:41

un t 0 n 6= 0 1 2

, ! ) 0 , .

u x t = 3 1a sin at

cos at + e t! sin x=

+ a2+

+2 cosat

sin 3x + cos at sin 5x:

Gx y t =

sin a

n+1 t sin

n+1x sin

n+1y

a nX

n + 1

! " # "$%& '

t	=2	=2	@G x y t 0
ux t = Z Z		f y G x y t dy d + Z g y	@G x y t 0	dy+
ux t = Z Z		f y G x y t dy d + Z g y	@t	dy+
0	0	0

+ Z p y G x y t 0 dy:

243

4.1.2. - -

	@2u	2	@2u
	@t2	= a	@x2 + f	x t D = fx : 0 x lg 0 t

		u 0 = g x			@u
		u 0 = g x			@t t=0= p x 0 x l :
1.	u x=0= ux x= =2= 0
	f x t = sin 5xe t t				g x = 0	p x = sin x 2 sin 3x:
2.	ux x=0= u x= =2= 0
	f x t = 2 cos 5xt2				g x = 0	p x = cos x 3 cos 3x:
3.	ux x=0= ux x= = 0
	f x t = 2 cos 3x sh t				g x = 0	p x = cos x + 3 cos 2x:
4.	u x=0= u x=2 = 0
	f x t = sin xe tt			g x = 0 p x = sin x=2 + 2 sin x:
5.	u x=0= ux x= = 0
	f x t = sin 5x=2 sh t g x = sin x=2 + 2 sin 3x=2						p x = 0:
6.	ux x=0= u x= = 0
	f x t = cos 3x=2 3t2				g x = 2 cos x=2 cos 5x=2		p x = 0:
7.	ux x=0= ux x= =2= 0
	f x t = cos 2x ch t				g x = cos 2x + cos 4x p x = 0:
8.	u x=0= u x=2 = 0
	f x t = sin x t2 + t				g x = 0	p x = 2 sin x=2 + sin x:
9.	u x=0= ux x=2 = 0
	f x t = sin x=4 e t				g x = 0	p x = sin x=4 + 2 sin 3x=4 :
10.	ux x=0= u x=2 = 0
	f x t = cos x=4 t2				g x = 0	p x = 2 cos x=4 + 3 cos 3x=4 :
					244

= ux

= 0

x=2

f x t = 2 cos 2xte

g x = 0

p x = cos x=2 + 2 cos x:

12.

u x=0= u x= = 0

f x t = sin 2xe 2t

g x = 3 sin x + sin 3x

p x = 0:

13.

u x=0= ux x=

2= 0

f x t = sin x t2

g x = sin x + 3 sin 5x

p x = 0:

14.

ux x=0= u x=

2= 0

f x t = cos 3x sh t

g x = 3 cos x + cos 5x

p x = 0:

15.

ux x=0= ux x= = 0

f x t = 2 cos xt2

g x = cos x

cos 2x

p x = 0:

16.

u x=0= u x=2 = 0

f x t = sin x ch t

g x = 0 p x = 3 sin x=2 sin 2x:

17.

u x=0= ux x= = 0

f x t = sin 3x=2 e

g x = 0

p x = 2 sin x=2 sin 5x=2 :

18.

ux x=0= u x= = 0

f x t = cos x=2 t

t2 g x = 0

p x = cos x=2

cos 3x=2 :

19.

ux x=0= ux x=

2= 0

f x t = cos 4xte 3t

g x = 0 p x = cos 2x

cos 4x:

20.

u x=0= u x=2 = 0

f x t = sin 2xt3

g x = 2 sin x=2

3 sin x

p x = 0:

21.

u x=0= ux x=2 = 0

f x t = 2 sin x=4 e

g x = 3 sin x=4 + 2 sin 3x=4

p x = 0:

22.

ux x=0= u x=2 = 0

f x t = cos 3x=4 e

g x = 5 cos x=4 + cos 3x=4

p x = 0:

23.

ux x=0= ux x=2 = 0

f x t = cos xte t

g x = 3 cos x + 2 cos 3x

p x = 0:

245

24.	u 0 = u x= = 0
	f x t = 2 sin xe t	g x = 0	p x = 2 sin x	sin 3x:
25.	u x=0= ux x= 2= 0
	f x t = 3 sin 5xt2	g x = 0	p x = 5 sin x	sin 3x:
26.	ux x=0= u x= 2= 0
	f x t = cos x sh t	g x = 0	p x = 2 cos x	cos 3x:
27.	ux x=0= ux x= = 0
	f x t = cos 2xte 2t	g x = 0	p x = cos x + cos 3x:
28.	u x=0= u x=2 = 0
	f x t = sin 2xte 3t	g x = sin x 3 sin 3x		p x = 0:
29.	u x=0= ux x= = 0
	f x t = 2 sin x=2 t2	g x = 2 sin x=2 + 3 sin 3x=2 p x = 0:
30.	u x=0= u x= = 0
	f x t = sin x sh t	g x = sin 2x + sin 3x p x = 0:

4.2.,

, ,

4.2.1.				1 -							! ! "
# $ " % "& $# D
@2u	2	4u + f x y t D					= f x y : 0 x 0 y =2g 0 t
2	= a	4u + f x y t D					= f x y : 0 x 0 y =2g 0 t
@t											4:2:1
( # )& $( "&							@D				4:2:1
( # )& $( "&							@D
		u		= u		=	@u	= @u		= 0 0 t	4:2:2
			x=0		x=		@y y=0	@y y=	2
)& $( "&
				u t=0= g x y = 2 sin x cos 2y							4:2:3

246

@u	x y 2 D	:2:4
= p x y = 4 sin x cos 4y	x y 2 D	:2:4
@t 0

f x y t = 3 sin 2x cos 4ye	t :	:2:5

2 - -D

@2u	= a24u + f r ' t D = f r ' : 0 r b 0 ' =2g
2	= a24u + f r ' t D = f r ' : 0 r b 0 ' =2g
@t
# @D
	@u	@u
	@' '=0=	@' '= =2= u r=b= 0 0 t
#
		0	2 r1
	u t=0= g r ' = J2 B		1	C cos 2'
		@	b	A

@u	0	24 r1
@t	t=0= p r ' = 2J4 B	b	C cos 4'
	@		A

	0 r1
0	0 r1		2t
f r ' t = 2J0 B	2	C e	2t
@	b	A

0 t :2:6

:2:7

:2:8

:2:9

:2:10

2kn | k- J2n = 0 J2n x | ) * + 2n- , .

. 1 . / |0 ,, , #2

	, f x y t 0 # # -
	3 4 , , 2.8.1. .5-

# 67 5 # ) * 6 )

nk = n2 + 2k 2 vnk x y = sin nx cos ky jjvnkjj2 =					2 n0 + 1
					8
n =	1 1	k =	0 1	:

3 2 5 8 , 4 5 # ) * , , , 8 ))- * 8 # , , # x y t:

247

11 1

ux y t =	unk t vnk x y =	unk	t sin nx cos ky :
1 k=0	n=1 k=0
			:2:11
f x y t, g x y, p x y
! " vnk			x y:
	1	1
fx y t = 3 sin 2x cos 4y e t =		fnk t vnk x y =

n=1 k=0

fnk

t sin nx cos

:2:12

n=1 k=0

g x y = 2 sin x cos 2y =

gnkvnk x y =

n=1 k=0

gnk sin nx cos

:2:13

n=1 k=0

p x y = 4 sin x cos 4y =

pnkvnk x y =

n=1 k=0

pnk sin nx cos 2ky

:2:14

n=1 k=0

fnk

t =

f x y t vnk

x y dxdy

jjvnkjj

gnk =

g x y vnk x y dxdy

jjvnkjj

ZZ p x y vnk x y dxdy

pnk =

n = 1 1

k = 0 1:

jjvnkjj

' ( ) * + , fnk t, gnk pnk" , $ - $ .. / , ) v22 x y =

= sin 2x cos 4y	v11 x y = sin x cos 2y		v12 x y = sin x cos 4y -
* % ! ! -!
)
f22	t = 3e t	fnk t 0 n k 6=2 2
	g11 = 2	gnk = 0	n k 6=1 1

248

p12 = 4 pnk = 0 n k 6= 1 2

4.2.11 , 4.2.12 4.2.1 ,

1 8d2unk t

dt2 + a2 nkunk t

n=1 k=0 :

fnk t 9= vnk x y = 0:

! " D

d2unk t	2
d2unk t	2	nkunk t = fnk t n = 1 1 k = 0 1:	4:2:15
dt2	+ a	nkunk t = fnk t n = 1 1 k = 0 1:	4:2:15
dt2

4.2.11 , 4.2.13 4.2.14 % &

4.2.3 , 4.2.4

X Xfunk0 gnkgvnk x y = 0

unk0 = gnk

4:2:16

n=1 k=0

1 1

8dunk0

dunk0

= pnk

4:2:17

X X

pnk= vnk x y = 0

n=1 k=0

n =

1 1

0 1

4.2.15 | 4.2.17 , : n k = 1 1 ,

n k = 1 2 n k = 2 2

&$' ( 4.2.16 , 4.2.17 ,

$'$% , unk t

n k

6=11 , n k 6=12 n k 6=22: + ( $%

d2u11 t

11u11 t = 0

u110 = 2

du110

= 0-

dt2

+ a

d2u12 t

12u12 t = 0

u120 = 0

du120

= 4-

dt2

+ a

d2u22 t

22u22 t = 3e

u220 = 0

du220

= 0

dt2

+ a

('$( ( /& 0

u11 t = 2 cosaq

t = 2 cosap

4:2:18

u12 t =

sinaq 12t =

sinap17t

4:2:19

ap 12

ap17

249

u t =

sin aq

t cos aq

t + e

# =

1 + a

ap5t

cos ap5t + e

# :

sin

:2:20

1 + 20

unk

n k

6=1 1 n k 6=

2 n k 6=2 2

u x y t = 2 cos ap

4 sin ap

t sin x cos 4y+

t sin x cos 2y +

ap17

sin 2x cos 4y

+3 "

ap5t

ap5t + e

2ap

sin

cos

1 + 20a

2 " # $|% &, ( & - & * & +,

f r ' t 0 + . + & / 0 # 2.8.1. " 1 + - & 23 1 + 4 (5 2 & 4

nk = 0

vnk r ' = J2n

nkr cos n'

n0 + 1

jjvnkjj

bJ2n

n = 0 1 k = 1 1

| k-# ( * & J2n = 0 J2n x | 4 (5 & 7 &

. k

2n-. & ( .

/ , # 1 ( * 8 & & 0 1 + 4 (5 & , . & 8 * . 44- 5 * + r ' t:

ur ' t = X X unk t vnk r ' =

n=0 k=1

	1 1				unk t J2n 0	2n	r 1 cos
=					unk t J2n 0	k	r 1 cos	n' :
	n	=0	k	=1	B	b	C
		=0		=1
	X		X
					@		A
9 + 4 (5 f r ' t, g r ' p r '
8 1 +, 4 (5 # vnk								r ':

:2:21

250

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