Answers

Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный авиационный университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

DDR 4 p.190-249.pdf

Скачиваний:

Добавлен:

19.02.2016

Размер:

469.97 Кб

Скачать

☆

<<< < Предыдущая 1 2 3 4 5 6 7 89 / 169 10 11 12 13 14 15 16 > Следующая >>>

			19.														1																														20.					y								10e		y
									y								y .																																															.
									y								y .																																															.
				Find out the linear dependency of the given system of functions
			21.							y				1,					y			2			sin 2					x,					y		3		cos 2x .
												1										2															3
			22.						y					x2				x 3,										y		2		2x2 x,														y	3	2x 4 .
									1																					2																	3
			23.						y1 x 2,																y2		x 2 .
													x 2									x 2							x 2								Answers
				1. y									x 2				ln x					x 2				C			x 2					C		2			x C				3	.				2. y e3x x6 C C																									2	x C				3	x2 .
				1. y													ln x									C				2				C					x C					.				2. y e3x x6 C C																										x C					x2 .
														2									4				1			2																																						1
3.				y					1				e5x				sin x C										x2	C				2		x	C .										4.			y					1 cos2x sin x C																	x2			C		2		x C .
3.				y			125						e5x				sin x C										2	C						x	C .										4.			y					1 cos2x sin x C																		2		C				x C .
							125																		1		2											3															8															1			2								3
5. y ex											x4			C				x2		C						x	C			. 6. y									x5					1	sin 3x C												x2			C	2		x C .							7. y						x2		ln x
									24								1	2							2			3											60				27										1					2			2							3									2
	x2							x3			C x C												.		8.		y					5 x						x3 C x										C					.					9.		y						3x			sin x C										x2
											C x C												.		8.		y			(ln 5)2								x3 C x										C					.					9.		y				(ln 3)3					sin x C
			4						6						1					2										(ln 5)2															1						2													(ln 3)3													1 2
C x C .																10. y									4x					1			sin3x								1			cos5x C										x2					C x			C .							11.					y ln2 x
C x C .																10. y								(ln 4)3						27			sin3x							125				cos5x C											2				C x			C .							11.					y ln2 x
			2						3															(ln 4)3						27										125											1				2					2						3												2
C1 ln x C2 .																		12.							y C1 x e x C2 .																						13.											y arcsin2 x C1arcsin x C2 .
14.					y C e									5x C					2		.				15.			ln			C ( y 1) 1													C (x C												2	), C 0;										y C ;						y x C .
14.					y C e									5x C							.				15.			ln			C ( y 1) 1													C (x C													), C 0;										y C ;						y x C .
									1																								1														1													1									3
16.				e y C										(x C							2		)2 .				17.					1 C y 2										C x C								2		. 18.								y C					2		eC1x .					19.				x C
									1												2																1								1					2															2														2
		2																																													1										20e y C												C		1
		2		(			y 2C )												y C							).				20.						x C						2					1		ln																1						1	.			21.Linear
3														1											1																	2					C1										20e y C												C
3																																															C1										20e y C												C		1
																																																																	1						1
dependency.														22.				Linear dependency																			23.					Linear independency .
														Individual tasks
						T 2.								Individual tasks
					2.1. Integrate the given equations of the second order
			2.1.1.																sin x																																															2		) 2 .
			2.1.1.											y			cos2 x .																															2.1.2. y (4 x																				) 2 .
					2.1.3. y											4cos2 x .																																2.1.4. y 2x arctg x .
					2.1.5. y												1 x2								x 0 .																							2.1.6.											y arctg x
218

2.1.7. y xex . 2.1.9. y x ln x .

2.1.11. y (1 x2 ) x3 .

2.1.13.y x2 1 x .

2.1.15.y 6x arctg x .

2.1.17.y cos3 x .

2.1.19.y 4 sin 2 x .

2.1.21.y 1 x x .

2.1.23.y cos3 x sin x .

2.1.25.y x ln x .

2.1.27.x2 y ln x .

2.1.29.(x 1)2 y x 2 2x .

2.1.8. y

1 x2

2.1.10.y x sin 2 x .

2.1.12.y 3x2 ln x .

2.1.14.y x sin 2x .

2.1.16.y sin3 x .

2.1.18.y xe2x .

2.1.20.xy 1 x2 .

2.1.22.y x x 1 .

2.1.24.y ex (x 1) .

2.1.26.y cos4 x .

2.1.28.y (x 3)e x .

2.1.30.2 x ( x 1)2 y 1 .

2.2. Integrate the following equations using substitution

y z(x)

2.2.1. (9 x2 )y 2xy 0 .

2.2.2. xy y x .

2.2.3.

ctg x

cos x .

(4x 2) y

2.2.4. y (x

2.2.5. y 2 ctg x y sin3 x .

2.2.6. (2 x2 ) y 2xy x2

2.2.7. xy

0 .

2.2.8. y

sin x y

cos x sin x .

x( y )

2.2.9. xy y e x x2 .

2.2.10. y 4(tg x)y cos2 x .

2.2.11.

2.2.12. xy

( y )

2.2.13.

2.2.14. y

2(tg x) y

cos

x .

y x ln x y

2.2.15. y 2xy 4x .

2.2.16. xy y x2 cos x .

2.2.17. y ¢¢-2 y ¢ctg x = 0 .

2.2.18. xy y xex .

2.2.19. (x 2 1) y

2x( y 1) .

2.2.20. xy y x2 sin x .

2.2.21. (x2 1)y

4x( y 1) .

2.2.22. x(ln x 2) y

y .

2.2.23. xy

y ln(y / x) .

2.2.24. (1 x2 ) y

2xy .

2.2.25. xy

y x2 .

2.2.26. (1 x2 ) y

2xy .

2.2.27. xy

y x .

2.2.28. y 2(tg x) y cos x .

219

2.2.29. x2 y xy 1.																			2.2.30. y y /(x 1) x2 x .
2.3. Integrate the following equations using substitution																																		y p( y) .
2.3.1.	yy						2			.									2.3.2. 2yy														2			0 .
2.3.1.	yy	( y )								.									2.3.2. 2yy						( y )											0 .
2.3.3. yy		2y										2	.						2.3.4. yy														2			0 .
2.3.3. yy		2y								( y )			.						2.3.4. yy				2( y )													0 .
2.3.5. yy									2		2y			3		y		.	2.3.6. y		3	y				2y							0
2.3.5. yy		2(y )									2y					y		.	2.3.6. y			y				2y							0
2.3.7. yy							2			.									2.3.8. y											2		2y								.
2.3.7. yy			3(y )							.									2.3.8. y					(y )								2y								.
2.3.9.		3	2y							.									2.3.10.		3			y			2		y		y								.
2.3.9.	y y		2y							.									2.3.10.		3			y					y		y								.
2.3.11. y 2 yy .																			2.3.12.	2y 2 y											y .
2.3.13. y3 y 6 .																			2.3.14. y 2 y											y .
2.3.15.	y	2 yy							0 .										2.3.16. y					y															3	.
2.3.15.	y	2 yy							0 .										2.3.16. y					y						(y )										.
2.3.17. 4		y y		1 .															2.3.18.		2yy															2			y			.
2.3.17. 4		y y		1 .															2.3.18.		2yy								( y )										y			.
2.3.19. yy								2			1	0 .							2.3.20.		2yy														2			y				0 .
2.3.19. yy			(y )								1	0 .							2.3.20.		2yy							( y )										y				0 .
2.3.21. yy											2						3	.	2.3.22.		y
2.3.21. yy			2(y )									( y )						.	2.3.22.		y							2y ( y 1) .
2.3.23. yy								2			y	2		y			.		2.3.24. yy									2yy					ln y										2	.
2.3.23. yy			( y )								y			y			.		2.3.24. yy									2yy					ln y								(y )			.
2.3.25. 3y			y	5 / 3							.								2.3.26. yy															2			y
2.3.25. 3y			y								.								2.3.26. yy									2(y )									y
2.3.27. y						2				y		.												3			9 0 .
2.3.27. y		( y )								y		.							2.3.28. y y								9 0 .
2.3.29. y					2					2e		y			.																									2	y		.
2.3.29. y		(y )								2e					.				2.3.30. y ( y 1) ( y )																						y		.

Тopic 3. Linear differential equation with constant coefficients

Linear differential equation with constant coefficient Euler’s method. Nonhomogeneous linear equations with constant coefficients. Method of independent coefficients and method of variation of arbitrary constants (Lagrange’s method).

Literature: [2, chapter 3, sec.3.3], [3, chapter. 8, §4], [4, chapter 8, §26], [6, chapter 11, sec..11.3, 11.4.], [8, chapter 13, §§21–28], [10, §§4–5].

T3. Main theoretical information

3.1. Homogeneous linear equations.

220

Homogeneous linear equation of the n’th order with constant coefficients is written below

		a0 y(n) +a1 y(n-1) +...+an-1 y ¢+an y = 0,	(3.24 )
where a0 ,	a1 , , an –any real numbers, a0 0 .
General solution of homogeneous linear equation:
		y =C1 y1 +C2 y2 +...+Cn yn ,
where y1 ,	y2 , ..., yn – linearly independent particular solutions		of the

homogeneous equation (3.24), C1 , C2 , ..., Cn –any constants.

Euler’s method. Particular solution of the equation (3.24) is evaluated in the form y = ekx , where k –unknown constant. Using the substitution, we will have:

ekx (a0 kn +a1kn-1 +...+an-1k +an ) = 0 ,

Therefore

a kn +a kn-1		+...+a	n-1	k +a	n	=0.	(3.25 )
0	1		n-1		n

Equation (3.25) is called Auxiliary equation. This equation is obtained from (3.24) with the help of substitution of the derivatives of the evaluating function by using the powers of k and y equals to one. It means:

y(n) kn , y(n-1) kn-1 , ..., y ¢ k, y 1 .

Auxiliary equation (3.25) is algebraic one of the nth order and has n number of roots (real or complex). A particular solution of the equation (3.24) depends on the roots of the auxiliary equation (3.25). Correspondence between roots of the auxiliary equation and the particular solutions are given in table3.2.

			Table3.2
		The number of
№	Characteristic of the	the linear	Particular solutions which
	root in the Auxiliary	independent	corresponds to the given root
	equation	solutions which
		corresponds to
		the given root
1	k –real simple root	one	ekx
	divisible by 1	one	ekx
2	k –real simple root	m	ekx , xekx , ..., xm-1ekx
2	divisible by m	m	ekx , xekx , ..., xm-1ekx
3	k =a bi – pair	two	eax cos bx, eax sin bx
3	of complex	two	eax cos bx, eax sin bx
			221

compared simple

roots

k =a bi – Pair

eax cos bx, eax sin bx,

of complex

compared simple

roots divisible by

xm-1eax cosbx, xm-1eax sin bx

Let’s consider the equation of the second order

a0 y a1 y a2 y 0 .

(3.26 )

Its auxiliary equation has the following structure

a k2

+a k +a

= 0 .

(3.27 )

Dependent on the denominator D a2 4a a

we have 3 such conditions,

according to table. 3.3.

Table3.3

№

Roots of the auxiliary equation (

General solution of the

п/п

27 )

equation

k1 and k2 – real and different

y C ek1x C

ek2x

k1 and k2 –real and different

y ek1x (C

k1 i , k2 i –complex

y e x (C

cos x C

sin x)

roots

3.2. Nonhomogeneous linear equation

The following equation:

a0 y(n) +a1 y(n-1) +...+an-1 y ¢+an y = f (x),

( 3.28 )

where

a0 , a1 , ..., an – constants, f (x) – continues on

(a,b) function is a

linear nonhomogeneous differential equation with constant coefficients. General solution of non homogeneous equation (3.28) looks like the

following

y = y + y* ,

where y is the general solution of the corresponding homogeneous equation (3.24),

y is a particular solution of the nonhomogeneous equation ( 3.28 ). General solution of y equation (3.24) is considered before.

To find particular solution of the equation (3.28) such methods are used: 1) independent coefficients;

222

<<< < Предыдущая 1 2 3 4 5 6 7 89 / 169 10 11 12 13 14 15 16 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
30.08.201981.92 Кб2Conflict. Lecture 7..doc
#
29.10.2018459.26 Кб5Cost Estimation book rus 20071210.doc
#
19.02.20161.06 Mб5DC14Sample.pdf
#
19.02.2016556.67 Кб10DDR 2 p.64-131.pdf
#
19.02.2016511.86 Кб8DDR 3 p.132-189.pdf
#
19.02.2016469.97 Кб6DDR 4 p.190-249.pdf
#
17.08.2019147.46 Кб2default.doc
#
19.02.2016131.58 Кб4DIDAKTIChNIJ_PLAN.doc
#
19.02.201677.85 Кб18Diplomna (1).docx
#
19.02.2016320.35 Кб11Diplomna (2).docx
#
19.02.2016195.35 Кб16DIPLOM_2.docx

Answers

Table3.2