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ФЕДЕРАЛЬНОЕ АГЕНСТВО ПО ОБРАЗОВАНИЮ РОССИЙСКОЙ ФЕДЕРАЦИИ

КАЗАНСКИЙ ГОСУДАРСТВЕННЫЙ АРХИТЕКТУРНО-СТРОИТЕЛЬНЫЙ УНИВЕРСИТЕТ

Кафедра высшей математики

ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ

Задания типового расчета для студентов второго курса

Казань

2009

5354.ru

Составители: Леонов А.И., Шмагин Ю.А. УДК 512

Дифференциальные уравнения. Задания типового расчета для студентов второго курса. Казанский государственный архитектурно-строительный университет. Составители: Леонов А.И., Шмагин Ю.А.

Казань,2009. - 16c.

Задания типового расчета предназначены для самостоятельной работы студентов второго курса.

Редактор: Н.Х. Михайлова Корректор: М.А. Рожавина

Редакционно-издательский отдел Казанского государственного архитектурно-строительного

университета

Подписано в печать		Формат 60х84/16
Заказ	Тираж 300экз.	Усл.печ.л.1,0
Бумага тип.№1	Печать RISO	Учетн.изд.л.1,0

Печатно-множительный отдел КГАСУ 420043, Казань, Зеленая, 1.

c Казанский государственный архитектурно-строительный

университет, 2009

5354.ru

				Вариант 1
1)	y0 ctg x + y = 2; yjx=0 = 1:				2)	y0(2 x2)=x = 2y2:
3)	(x + y)dy = (y x)dx:				4)	xy0 y = x tg(y=x):
5)	y0 + y=x + sin 2x = 0:				6)	y0 = 2y 4x; yjx=0 = 1:
7)	x2y00 + xy0 = 1:				8)	y00 sin x = sin 2x:
9)	4y00 4y0 + y = 0; yjx=0 = 1; y0jx=0 = 0:
10)	y00	+ 3y0		+ 2y = 2 cos 2x:	11)	y0 + 4y = e 2x:
12)	y00	+ 4y0		+ 3y = e 3x:	13)	7y00 y0 = 14x:
	8	dx	+ y = 0;
	8		+ y = 0;
	>	dt
	>
	>
	>
	>
14)	< dy
14)	>		+ 3x + 3y = 0:
	>		+ 3x + 3y = 0:
	>
	> dt

Вариант 2

xy0 + y = y2; yjx=1 = 2:

2) y(2 + x)dy + (1 + y2)dx = 0:

(y 3x)y0 = x 3y:

xy0

= y x e y=x:

3x3 + 2x = 0:

y0+

+3 sin x = 0; yjx= =2

= 0:

y00 ctg x + 2y0 = 0:

3yy00 + (y0)2 = 0:

y00 8y0 + 16y = 0; yjx=0 = 0; y0jx=0 = 1:

10)

y00 + 3y0 = 9x:

11)

y00

+ 9y = 4 sin 2x:

12)

y00

+ 7y0 = e 7x:

13)

y00

+ 6y0 + 13y = e 3x cos 2x:

= x + 3y;

14)

< dy

= 3x + y:

> dt

5354.ru

			Вариант 3
	r		dx = xydy; yjx=1 = 1:		y2xy0 + 1 = x:
1)		y2 + 1		2)
3)	(3x + y)y0 = x + 3y:			4)	y2 + y0(x2 xy) = 0:
5)	xy0 + y + 2 e x = 0; yjx=1 = 2: 6)				y0 + y + 2x 3 = 0:
7)	2xy00 = y0:			8) y00y3 + 9 = 0:
9)	y00 4y0 + 3y = 0; yjx=0 = 1; y0jx=0 = 3:

10)	y00	+ 2y0 + 2y = x2 + 2:			11)	y00 4y0+4y = x2 e x:
12)	y00	+ y = 3:			13)	y00 6y0+9y = sin 3x:
	8	dx	= 3y	8x;
	8	dt	= 3y	8x;
	>	dt
	>
14)	>
14)	>
	>
	>
	< dy		= 2x + 3y:
	>		= 2x + 3y:
	>	dt
	>	dt
	>

Вариант 4

(x2 1)y0

+ 2xy2 = 0:

xydy +(y + 1)dx= 0; yjx=1 = 0:

y0 = (4x

y)=(y

x):

(x2 + y2)y0 = xy:

y0 = 3y + 3x:

6) y0 + xy x = 0; yjx=0 = 0:

(y0)2 + (y00)2 = 1:

y00 tg x y0 + 1= sin x = 0:

y00 + 6y0 + 25y = 0; yjx= = 0; y0jx= = 1:

10)

y00 + 8y0

= 8x:

11)

27y00 12y = 8x:

12)

y00 4y0

+ 4y = sin 2x:13)

y00 + 4y0 + 4y = 8 e 2x:

> dt

14)

< dy

= x

> dt

5354.ru

Вариант 5 p

	x2yy0+1 = y; yjx=1 = 0: 2)
1)	x2yy0+1 = y; yjx=1 = 0: 2)							x2 + 1dy = xydx:
3)	y0 = (2x + y)=(x + y): 4)						(y2 2xy)dx + x2dy = 0:
	y0 y = 2x + 1:						xy0+y+xp			= 0; yjx=1 = 1:
5)	y0 y = 2x + 1:					6)	xy0+y+xp		x2 + 2	= 0; yjx=1 = 1:
7)	y00x ln x = y0:					8)	y00 tg y = 2(y0)2:
9)	y00 + 2y0 + 3y = 0; yjx=0 = 1; y0jx=0 = 2:
10)	y00	+ 2y0		+ y = e x:		11)	y00 y0 = 12x2 + 6x:
12)	y00	+ 4y = 25x e x:				13)	y00 5y0 + 6y = 13 sin 3x:
	8	dx	=		y;
	8		=		y;
	>	dt
	>
14)	>
14)	>
	>
	>
	< dy
	>		+ 3x + 2y = 0:
	>
	> dt

					Вариант 6
1)	xydx + (x + 1)dy = 0:				2)	(y2 1) + 2x2yy0 = 0; yjx= 1 = 2:
3)	2x3y0 = y(2x2 y2):				4)	(2x + y)dy = (2y 3x)dx:
5)	xy0 + y e x = 0:				6)	y0 y + 3x 1 = 0; yjx=0 = 3:
7)	xy00 + y0 = 1:				8)	2yy00 = (y0)2:
9)	y00 6y0 + 34y = 0; yjx=0 = 0; y0jx=0 = 1:
10)	y00	2y0 + 2y = x e x:			11)	y00	+ y0 + 0; 25y = 25 cos 2x
12)	y00	y = e x:			13)	y00	+ 4y0 2y = 8 sin 2x:
	8	dx	= y	x;
	8		= y	x;
	>	dt
	>
14)	>
14)	>
	>
	>
	< dy			y:
	>		= x	y:
	>
	> dt

5354.ru

1) y0(x x2) = y:

3) y0 = (x + 4y)=(y + 4x):

Вариант 7

2) 2x2yy0 + y = 2; yjx=1 = 0:

4) xy0 y = y ln (y=x):

5)xy0 + y = 2x=(x2 + 3): 6) y0 + 3 y2 + 2x3 1 = 0; yjx=1 = e : x x

7) tg x y00 = 2y0:

8) yy00 2(y0)2 = 0:

9)64y00 + 16y0 + y = 0; yjx=0 = 1; y0jx=0 = 0:

10)y00 + y = x cos x: 11) y00 + 4y0 + 5y = 5x2 32x + 5:

12) y00		+ 3y0 = 4:		13) y00 + 3y0 + 2y = 2 sin 2x + 4 cos 2x:
	8	dx	= 2x + 8y;
	8	dt	= 2x + 8y;
	>	dt
	>
	>
	>
	>
14)	< dy
14)	>		= x + 4y:
	>		= x + 4y:
	>
	> dt

Вариант 8

1) yy0=p

+ 3 = y2:

2) y0

xy2 = 2xy; y

jx=1

= 1:

3) (y

3x)dx = (x + y)dy:

4) (y + p

)dx = xdy:

5) y0 + 2y 3 + x = 0:

6) xy0 + y

= 0; yjx=1 = 4:

x 3

7) xy00 + y0 = x + 1:

8) y00 y3 = 1:

9) y00 + (2=3)y0 + y=9 = 0; yjx=0 = 1; y0jx=0 = 2:

10) y00

+ y = 1:

11) y00 + 3y0

= 3x e 3x

12) y00

5y0 + 6y = 13 sin 3x: 13) y00

+ 5y0

+ 6y = 10(1 x) e 2x:

= x + 5y;

14)

< dy

= 7x + 3y:

: dt

5354.ru

						Вариант 9
1) y0(xy2 + 2xy) = 1:							2) yy0 = (1 x)(y + 2); yjx=0 = 1:
3) (x + 2y)y0 = x + y:							4) xy0 = r			+ y:
3) (x + 2y)y0 = x + y:							4) xy0 = r		x2 y2	+ y:
			p				6) y0 3y + 2 x = 0; yjx=0 = 4:
			p	2	+ 1:
5) y0 = y=x + 1= x					+ 1:
7) x3y00 + x2y0 = 1:							8) 2y00y2 = y0:
9) y00 3y0 = 0; yjx=0 = 1; y0jx=0 = 2:
10) y00		+ 4y0 + 4y = e 2x:					11) y00	7y0 = (x 1)2:
12) y00		+ y0 = 2 sin 2x:					13) y00	+ 3y = e x:
	8	dx	= 2x + 3y;
	8		= 2x + 3y;
	>	dt
	>
	>
	>
	>
14)	< dy
14)	>		= x + 2y:
	>		= x + 2y:
	>
	> dt


Вариант 10
1) 2 + y2 = xyy0:	2) xdx +(y x2y)dy = 0; yjx=p				= 1:
				2
3) y0 = (2y 4x)=(2x + y):	4) (p		x)dy + ydx = 0:
		xy

5) xy0 + y 2x + 1=4 = 0: 6) y0 + xy 2x2 = 0; yjx=1 = 1; 5:

7) x4y00 + x3y0 = 1:			8) 2yy0 (y0)2 + 1 = 0:
9) y00 + y0 y=4 = 0; yjx=0 = 1; y0jx=0 = 2:
10) y00	+ 25y = cos 5x:		11) y00	+ 4y0 = 4 sin 2x:
12) y00	2y0 = x2 x:		13) y00	+ y = sin x cos x:
8	dx	= 4x + 6y;
8		= 4x + 6y;
>	dt
>
>
14) >
>
>
< dy
>		= 4x + 2y:
>
> dt

5354.ru

Вариант 11

1) (1 + 2y)xdx +(1 + 2x2)dy = 0: 2) 3x2y2y0 + 2 = 3y; yjx=1 = 1:

3) (3x + y)y0 + 2x				3y = 0:	4) (x2 + xy + y2)dx = x2dy:
							2x+1
5) y0 + 3y + e 2x = 0:					6) xy0		2x+1
						+y +		= 0; yjx=2	= 1:
						+y +	x2	= 0; yjx=2	= 1:
7) xy00 + 2y0 = 0:					8) yy00 = (y0)2 + (y0)3:
9) y00 + 2y0 48y = 0; yjx=0 = 0; y0jx=0 = 14:
10) y00		+ 9y0 = 3 cos 3x:			11) y00	+ 2y0 + 2y = 1 + x:
12) y00		2y0 + y = 5x2 1:			13) y00	+ y0 + y = x e x:
	8	dx	= x + 4y;
	8		= x + 4y;
	>	dt
	>
	>
	>
	>
14)	< dy
14)	>		= 2x + 3y:
	>		= 2x + 3y:
	>
	> dt

		Вариант 12
1)	2xydx = (2x 1)dy:	2)	xy(1 + x)y0						= 1+y2; yjx=1 = 0:
3)	(5x + y)y0 x 5y = 0: 4)			dy	=	x	+	y	:
				dx		y		x
5)	y0 + y=x = 3 cos x:	6)	y0 + 2y e x = 0; yjx=0 = e :
7)	(1 + x2)y00 + 2xy0 = x3: 8)		y00 = (y0)2:
9)	2y00 9y0 = 0; yjx=0 = 1; yjx=0 = 0:

10)	y00 + 6y0 + 9y = 10 sin x:			11)	y00	+ 2y0 + 5y = e x:
12)	y00	4y0 + 5y = e 2x:		13)	y00	+ y = 4x cos x:
	8	dx	= 5x + 8y;
	8	dt	= 5x + 8y;
	>	dt
	>
14)	>
14)	>
	>
	>
	< dy		= 3x + 3y:
	>		= 3x + 3y:
	>
	> dt

5354.ru

									Вариант 13
1) x3ydy = dx=r									:	2) x2y0 + 2y = y2; yjx=1 = 1:
1) x3ydy = dx=r						1 + y2			:	2) x2y0 + 2y = y2; yjx=1 = 1:
3) y0		= (3x + 3y)=(3x + y):								4) (y2 2x2)dy = 2xydx:
5) y0					p					6) y0 + y=x 2 cos 2x = 0; yjx=1 = 2:
5) y0		+ 4y=x					x	= 0:		6) y0 + y=x 2 cos 2x = 0; yjx=1 = 2:
7) x5y00 + x4y0 = 1:										8) 3yy00 = (y0)2:
9) y00 + 26y0 + 169y = 0; yjx=0 = 0; y0jx=0 = 1:
10) 4y00 3y0					= x e 3x=4					11) y00+5y0+6y = 12 cos 2x 8 sin 2x:
12) y00			10y0		+ 25y = e 5x:					13) y00 4y0 + 5y = e 2x:
	8		dx	= 3x + y;
	8			= 3x + y;
	>		dt
	>
	>
	>
	>
14)	< dy
14)	>			= 8x + y:
	>			= 8x + y:
	>
	> dt

Вариант 14

1)(y

x)dy +(y + 1)dx= 0: 2)

dy = 0; y x=4 = 0:

(y + 1)

3) (y

4x)dy + (x + 4)dy = 0: 4) y0 = y2=x2

+ 2 = 0:

5) xy0 + y + x

6) y0 4y=x 2px = 0:

7) xy00 y0 + 1=x = 0:

8) (y0)2 + 2yy00 = 0:

9) 4y00 + 12y0 + 9y = 0; yjx=0 = 1; y0jx=0 = 0:

10) y00

+ 16y = 2 cos x:

11) y00 4y0 5y = 2x2 3x+1:

12) y00

2y = e 2x:

13) y00

4y0

= x e 4x:

= 4x + y;

14)

< dy

= 8x + 9y:

> dt

5354.ru

				Вариант 15
1)	(2 + y2)dx xydy = 0:				2)	y0 tg x + 3y = 1; yjx= =4
3)	y0 = (x + y)=(x y):				4)	y0 = (2xy y2)=x2:
5)	y0 = y + 1 3x:				6)	xy0 + y e x = 0; yjx=1
7)	xy00 + y0 x = 0:				8)	(y0)2 = (y2 + y)y00
9)	y00 + 22y0 + 121y = 0;				yjx=0 = 0;		y0jx=0 = 1:
10)	y00	+ 5y0 6y = e 2x:			11)	y00	4y0	= cos 8x:
12)	y00	+ 10y = 3 sin 5x:			13)	y00	+ 2y0	+ y = 2:
	8	dx	= 2y + 3x;
	8	dt	= 2y + 3x;
	>	dt
	>
	>
	>
	>
14)	< dy
14)	>		= 5y 4x:
	>		= 5y 4x:
	>
	> dt

Вариант 16

=0:

=3:

1) y2xy0 + 1 = x; yjx=1 = 0:						2) r	y2 + 1	dx = xydy:
3) x + 2y + y0(y 2x) = 0:						4) x2dy + (y2 2x2)dx = 0:
5) xy0 + y + 2 e 3x = 0:						6) y0 y = 2x+1; yjx=0 = 1; 5:
7) xy00		+ y0 = p				8) yy00 = (y0)2:
7) xy00		+ y0 = p			x:	8) yy00 = (y0)2:
9) 9y00 30y0 + 25y = 0; yjx=0 = 0; y0jx=0 = 1:
10) y00		2y0		+ y = sin 3x:		11) y00 14y0+13y = x2 7x+2:
12) y00		6y0		+ 73y = 2 sin 5x:		13) y00 y0 = x e x:
	8	dy	= 3z + y;
	8		= 3z + y;
	> dx
	>
	>
	>
	>	dz
14)	<	dz
14)	>		= z + 3y:
	>		= z + 3y:
	>

: dx

5354.ru

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