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Владимирский государственный университет им. Столетовых

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kONTROLXNYE WOPROSY I ZADANIQ

y(x + 2) = ;x ; 6.

y = C1e;

+ C2e

3.128.

3.129.

5 e

3.130.

y = C1 cos x + C2 sin x + (2x

;

2)ex.

y = (C1 + C2x + x

C1 ;

! cos x

C2 +

sin x.

3.131.

3.132.

3.133.

y = e

(C1 cos 2x + C2 sin 2x) +

4 e

cos 2x +

sin 2x.

3.134.

y = C1 + C2e

; 5 x

;

125

x +

(cos 5x ; sin 5x).

3.135.

y = e;x(x

sin x).

3.136.

y = e2x;1

;

2ex + e

;

3.137.

y = (x

;2x

3.138. y

= x

;

1)(e

;

x sin x

2 cos x.

; x

ln jxj.

3.139. y = e (x ln jxj + C1x + C2). 3.140. y = (C1

+ C2x)e;

+ xe;

3.141.

y = (C1 + ln j sin xj) sin x + (C2 ; x) cos x.

3.142.

= sin 2x ln j cos xj ; x cos 2x + C1 sin 2x + C2 cos 2x.

y = C1 cos x + C2 sin x

;

cos 2x

3.143.

cos x .

3.144.

y = C1 cos x + C2 sin x + sin x ln j tg x=2j.

3.145.

y = (1 + ln j sin

xj) sin x ; xcos x.

3.146.

y = ex ln

+ e2x ln

1 + e;

1 + e

3.147.

y = (1 + ln j cos 3xj) cos 3x + 3x sin 3x.

3.148. y = (5 + 2 ln j tg xj) sin 2x.

4.kONTROLXNYE WOPROSY I ZADANIQ

dIFFERENCIALXNYE URAWNENIQ PERWOGO PORQDKA: ZADA^A kO[I I TEOREMA SU]ESTWOWANIQ I EDINSTWENNOSTI EE RE[ENIQ, METOD IZOKLIN, URAWNENIQ S RAZDELENNYMI I RAZDELQ@]IMISQ PERE- MENNYMI, URAWNENIQ WIDA y0 = f(y=x) I SWODQ]IESQ K NIM, LINEJNYE URAWNENIQ PERWOGO PORQDKA, URAWNENIQ bERNULLI, URAWNENIQ W POLNYH DIFFERENCIALAH, URAWNENIQ S INTEGRIRU@- ]IM MNOVITELEM.

dIFFERENCIALXNYE URAWNENIQ WTOROGO PORQDKA: URAWNENIQ WIDA y00 = f(x), F (x y0 y00) = 0, F (y y0 y00) = 0. oDNORODNYE I NEODNORODNYE LINEJNYE URAWNENIQ WTOROGO PORQDKA.

nAJTI OB]IJ INTEGRAL DIFFERENCIALXNOGO URAWNENIQ ILI RE[ITX ZADA^U kO[I:

4.1.

(1)

20xdx ; 3ydy = 3x2ydy ; 5xy2dx

dIFFERENCIALXNYE URAWNENIQ

(2)

= x2 + 2xy

;

5y2

(3)

;

2=x2,

y(1)

(4)

y00

2x2 ;

6xy

; x

5y0 + 6y = e;

;

1, y(1) =

;

1, y0(1) =

;

1 (5) y00

;

y(0) =

y000

10x

0, y0(0) =

1 (6)

; 100y0 + y = 100 cos 10x + 20e

(7)

y00 + y =

y(0)

= 1,

y0(0)

(8)

y + 2

cos x

2x + y

;

y(;1)

(9)

dx + (2x + sin 2y

2 cos

y)dy = 0,

(10) y000

; 5y00 + 6y0

= 6x

+ 2x ; 5.

4.2. (1) 4xdx ; 3ydy = 3x2ydy ; 2xy2dx (2) y0 =

x2 + 4x

+ 2

(3)

y0 ; xy = x2,

4y3y00

= y4 ; 1,

y(1)

(4)

y(0)

+ 2y = e;x, y(0) = 0, y0(0) = 1

y0(0) = 1=

(5) y00 + 3y0

(6)

y00 ; 2y0

= e2x + e;2x (7) y00

+ 2y =

, y(0) = 3, y0(0) =

cos x

(8)

y0 =

x + 2y ; 3

(9)

y2dx +

x + e2=y

dy = 0,

y(e)

;

(10) y000

+ 3y00

+ 2y0

= 1 ; x .

3y3 + 2yx2

4.3.

(1)

xq1 + y2 ; yy0p1 + x2 = 0

(2)

xy0

2y2 + x2

(3)

;

y ctg x = 2x sin x,

y( =2)

(4)

y00 = 128y3

, y(0)

(5) y00

+ 3x,

y(0) = 0,

(0)

y0(0)

;

= 6x

9e3x

(6)

y00

+ y = 2 sin x ; 6 cos x + 2ex (7) y00

+ 3y0

, y(0) =

1 + e3x

ln 4, y0(0) = 3

;

3 ln 2 (8) y0 =

x + y

; 2

(9)

y4ey + 2x

= y,

;

y(0) = 1 (10) y000 ; y00

= 6x2 + 3x.

x + y

4.4.

(1)

q41

+ y2dx ; ydy = x2ydy

(2)

; y

(3)

+ y cos x =

2 sin 2x, y(0)

(4)

y3y00 =

;64,

y(0) = 4, y0(0) = 2 (5) y00

y = sin 2x, y(0)

0, y0(0) =

(6)

y000

; y0

= cos x + 2ex (7);y00

+ 4y = 8 ctg 2x,

y( =4) =

y0( =4)

(8)

3y ; x ; 4

(9)

y2dx + (xy

;

1)dy

= 0,

3x + 3

y(1) = e (10) y000 ; y0

= x2 + x.

ydy = x2ydy

4.5.

(1)

3 + y2dx

;

(2)

xy0 = y +

x2 + y2

(3)

1=2

(4)

y00

+ y tg x = cos

y( =4)

;

2 sin y cos

y(0) = 0, y0(0) = 1 (5) y00

; 3y0 + 2y = e

y(0) = 0, y0(0) = 1

(6) y00 ; 3y0

= e3x + e;3x (7) y00

; 6y0 + 8y =

, y(0) = 1 +

1 + e;2x

kONTROLXNYE WOPROSY I ZADANIQ

ln 4, y0(0) = 6 ln 2 (8)

;

(9) 2

4y2 + 4y

;

= 1,

x + y

;

y(0) = 0 (10) y000 ; 3y00 + 3y0 ; y = 2x.

4.6.

(1)

6xdx ; 6ydy = 2x2ydy ; 3xy2dx

2y0

= x2,

(2)

= x2 + 6x

+ 3

(3)

; x

y(1)

(4)

y00

= 32 sin3 y cos y,

y(1)

=2,

y0(1)

(5)

y00

;

= 5(x + 2)2,

y(0)

y0(0)

(6)

y00

+ 4y =

;8 sin x + 32 cos x + 4e

(7)

y00 ;

9y0 + 18y =

y(0)

y0(0)

1 + e;3x

x + y

(8)

(9)

cos 2y cos2 y ; x

= sin y cos y,

;

; IV

y(1=4) = =3 (10) y

; y000 = 5(x + 2)

dx + yp

dy = 0 (2) xy0 =

3y3 + 4yx2

4.7. (1) x

3 + y2

2 + x2

2y2 + 2x2

(3)

y0 ;

= ex(x + 1),

y(0)

(4)

y00

= 98y3,

x + 1

y(1)

y0(1)

(5)

y00

;

2y0

+ y = ex,

y(0)

y000 ; y0

y0(0)

(6)

= 10 sin x + 6 cos x + 4e

(7)

y00 +

2y =

y(1=2)

y0(1=2)

2=2

sin x

(8) y0

= 2x + y

;

3 (9)

x cos2 y

;

= y cos2 y, y( ) = =4

x ;

(10) y

; 2y000

+ y00

= ;2x

+ 2x

x + 2y

4.8.

(1)

(e2x + 5)dy + ye2xdx = 0

(2)

; y

(3) y0

; x = x sin x,

y( =2)

(4)

y3y00 =

;49,

y(3)

;7,

y0(3)

(5)

y00 + 2y0

+ y = cos 2x,

y(0)

y0(0) =

(6) y00 ; 4y0 = 8e4x + 8e;4x (7) y00 + ;2y =

, y(0) = 2,

2 cos(x= )

x + 7y

;

y0(0)

(8) y0

= 9x

;

(9)

(dx ; 2xydy) = ydy,

y(0) = 0 (10) y

+ 2y000 + y00

= x

+ x ; 1.

4.9.

yy0s1 ;; y2 + 1 = 0

(1)

(2)

xy0 = 2qx

+ y

(3) y0 + x

= sin x, y( ) = 1= (4) 4y3 y00 = 16y4

; 1, y(0) = p2=2,

y0(0)

p2=2

(5)

y00

;

y0 = ex,

y(0)

y0(0)

(7) y00 ; 3y0 = 9e;

(6) y00 + 9y = ;18 sin 3x ; 18e

=(3 + e;

dIFFERENCIALXNYE URAWNENIQ

y(0)

4 ln 4,

y0(0)

9 ln 4

; 3

(8)

+ 3y + 4

;

(9) (104y3 ; x)y0 = 4y, y(8) = 1 (10) yV ; yIV

= 2x + 3.

4.10.

(1)

6xdx ; 6ydy = 3x2ydy ; 2xy2dx

(2)

3y0

+ 8 y

+ 4

(3)

y0 +

= x2,

y(1)

(4)

y00

;

8 sin y cos3 y, y(0) = 0, y0(0) = 2 (5) 3y00

+ y0 = 6x

; 4y0 = 8 sin 2x

;2x

y(0) = 0, y0(0) = 1 (6) y000

; 4 cos 2x + 24e

(7)

y00

+ y = 4 ctg x, y( =2) = 4, y0( =2) = 4 (8) y0

3y + 3

2x + y

; 1

)dy = 0, y(;1) = 0 (10) 3y

(9)

dx + (xy ; y

+ y000 = 6x

; 1.

3y3

+ 6yx2

4.11. (1) xq5 + y2dx + yp4 + x2dy = 0 (2) xy0

2y2 + 3x2

(3)

2xy

2x2

, y(0)

2=3 (4) y00

= 72y3, y(2)

x + 1

+ 1

+ y = 4x2,

y0(2)

(5)

y00

+ 2y0

y(0)

y0(0)

(6)

y00 ; 5y0

= 25e5x + 25e;5x

(7)

y00 ; 6y0 + 6y =

2 + e;2x

y(0)

1 +

3 ln 3,

y0(0)

10 ln 3

(8)

x + 2y

;

(9)

(3y cos 2y

sin 2y

2x)y0 = y,

y(16)

4x ; y ;

;

(10) y

+ 2y000 + y00

= 4x

4.12.

(1)

exdx

;

y(4 + ex)dy = 0 (2)

x2 + xy

; y2

x2 ;

2xy

(3)

;

2x ; 5 y = 5,

y(2)

(4)

y3y00

;

36,

y(0)

y00 + y0

= e;x,

y0(0)

(5)

y(0)

y0(0)

y00 + 16y = 16 cos 4x ;

16e4x

(7) y00 + 6y0

4e;2x

(6)

+ 8y =

2 + e2x

y(0) = 0, y0(0) = 0 (8) y0 =

x ; 2y + 3

(9) 8(4y3 + xy

;

y)y0 = 1,

;2x

; 2

y(0) = 0 (10) y000 + y00 = 5x

; 1.

y0p

+ xy2

4.13.

(1)

+ x = 0 (2)

xy0

2x2

+ y2 + y

x + 1 ;

(3)

ex,

y(1)

(4) y00

18 sin3 y cos y,

x2 + x,

y(1) =

, y0

(1)

(5)

y00

+ 4y0

+ 4y =

;

y(0)

y0(0)

(6)

y000

; 9y0 = 18 sin 3x ; 9 cos 3x ; 9e

(7)

y00 + 9y =

y( =6)

4, y0( =6)

= 3 =2

sin 3x

(8) y0

x + 8y

;

(9) (2 ln y

;

ln2 y)dy = ydx

;

xdy, y(4) = e2

10x ; y ; 9

kONTROLXNYE WOPROSY I ZADANIQ

(10) yIV + 4y000 + 4y00

= ;x2 + x.

2xdx ; 2ydy = x2ydy

; 2xy2dx

4.14.

(1)

(2)

+ 6

(3)

ln x

y(1)

; x

(4) 4y3y00

= y4 ; 16, y(0) = 2p2, y0(0) =

2 (5) 7y00 ; y0

= 12x,

y(0) = 0, y0(0) = 1 (6) y00 ; y0

= ex

+ e;x (7) y00 + 9y =

cos 3x

y(0)

1, y0(0)

(8) y0

2x + 3y

; 5 (9) 2(x + y4)y0 = y,

y(;2) = ;1 (10) 7y000 ; y00 = 12x.

5x ; 5

dx + yp

dy = 0 (2) xy0

3y3 + 8yx2

4.15. (1) x

4 + y2

1 + x2

2y2 + 4x2

q 12

y00 = 50y3,

(3)

y0 ; x

;x3

y(1)

(4)

y(3)

1, y0(3)

(5)

y00 + 3y0 + 2y = e;2x,

y(0)

y0(0)

(6)

y00

+ 25y =

;

10 sin 5x + 20 cos 5x + 50e5x

y00 ; y0

e;x

(7)

y(0)

ln 27,

y0(0)

ln 9 ; 1

2 + e;x

(8) y0 =

4y ;

(9) y3(y

;

1)dx + 3xy2(y

;

1)dy = (y + 2)dy,

;

3x + 2y

+ 2x.

y(1=4) = 2 (10) y000 + 3y00 + 2y0 = 3x

4.16.

(1)

(ex + 8)dy

;

yexdx = 0

(2)

x2 + 2xy ; y2

2x2 ;

2xy

(3)

y0 +

= x3,

y(1)

;5=6

(4)

y3y00 = ;25,

y(2)

;

y0(2)

;

(5)

y00

;

= 3x2

;

2x, y(0)

y0(0)

(6) y000 ; 16y0

;64 sin 4x + 64 cos 4x + 48e

(7)

y00 + 4y

= 4 ctg 2x,

y( =4)

3, y0( =4)

(8) y0 =

x + 3y

;

(9)

2y2dx + (x + e1=y)dy = 0, y(0)

; y

(10) y000 ; y0

= 3x

; 2x.

5 + y2 + yy0p1

;

4.17.

(1)

x2 = 0 (2) xy0 = y + 3

x2 + y2

(3) y0 + x

= 3x,

y(1) =

1 (4)

y00 = ;18 sin y cos

y, y(0) =

y0(0)

(5)

y00 ; y0 = 4x2 ; 2,

y(0)

y0(0)

(6)

y00 + 2y0

= e2x ; e;2x

(7)

y00 ; 3y0 + 2y =

3 + e;x

y(0)

8 ln 2,

y0(0)

14 ln 2

(8)

y0 = ;2x + y + 3

; 2

(9) (xy + py)dy + y

dx = 0, y(;1=2) = 4 (10) y000 ; y00 = 4x ; 2.

dIFFERENCIALXNYE URAWNENIQ

4.18.

(1)

6xdx ; ydy = x2ydy

; 3xy2dx

2xy

= x2 + 1,

(2)

2y0 = x2 + 8x

+ 8

(3)

;

x2 + 1

y(1)

(4)

y00 = 8 sin3 y cos y,

y(1)

=2,

y0(1)

(5)

y00

;

3y0 + 2y = ex,

y(0)

y0(0)

(6)

y00

12 cos 6x + 36e

+ 36y = 24 sin 6x

;

y00 ;

6y0 + 8y =

4e2x

(7)

y(0)

y0(0)

1 + e;2x

(8)

y0 =

x + 2y

; 3

(9)

sin 2ydx = (sin2 2y

;

2 sin2 y + 2x)dy,

x ;

; 3y000

y(;1=2) = =4 (10) y

+ 3y00 ; y0 = x ; 3.

3y3 + 10yx2

4.19.

(1)

y ln y + xy0

= 0

(2)

xy0

1 ; 2x

+ 5x

(3) y0 +

y = 1,

y(1)

(4) y00

= 32y3,

y(4)

y00 + 2y0 + y = e;x,

y0(4)

4 (5)

y(0)

y0(0)

(6)

y000

;

25y0

= 25(sin 5x + cos 5x)

;

50e5x

(7)

y00

16y

16= sin 4x,

y( =8)

= 3,

y0( =8)

(8)

3x + 2y ; 1

x + 1

(9) (y2 + 2y ; x)y0

= 1, y(2) = 0 (10) yIV + 2y000 + y00

= 12x2

; 6x.

4.20.

(1)

(1 + ex)y0

= yex

(2)

x2 + 3xy

; y2

y0 + 3xy =

y3y00

3x2

; 2xy

(3)

y(1)

(4)

= ;16,

y(1)

y0(1)

(5)

y00

4y0

;

x2 + 2,

y(0)

y0(0)

(6) y00 + 3y0

;3x

+ 16y =

16= cos 4x,

y(0)

= e

; e;

(7) y00

5y + 5

y0(0) = 0 (8) y0

(9) 2ypydx ; (6xpy + 7)dy = 0,

4x + 3y

;

y(;4) = 1 (10) y000

+ 32.

; 4y00 = ;384x

4.21. (1) y0p

+ xy2 + x = 0 (2) xy0

; x2

= 3

2x2 + y2 + y

(3)

y0 + 2xy =

;

2x3,

y(1)

e;1 (4)

y00 =

q32 sin y cos y,

y(0)

(5) y00

+ 2y0

+ y =

;

y(0)

(0)

;

0, y0(0)

(6)

y00 + 49y = 14 sin 7x + 7 cos 7x + 98e

(7)

y00 ; 2y0 =

4e;2x

y(0)

ln 4,

y0(0)

ln 4

; 2

1 + e;2x

(8) y0 =

x + 4y

;

(9) dx = (sin y + 3 cos y)dy = 0, y(e =2) = =2

6x ; y

;

(10) y

+ 2y000

+ y00

= ;3x

+ 2.

6xdx ; 2ydy = 3x2ydy ; 3xy2dx

4.22.

(1)

kONTROLXNYE WOPROSY I ZADANIQ

(2)

y0 =

+ 8

+ 12

(3)

= x=2,

2(1

; x2)

y(0)

y00 = 50 sin

y cos y,

=2,

2=3

(4)

y(1)

y0(1)

(5)

y00 + y0

;

x2,

y(0)

y0(0)

(6)

y000 ; 36y0

;72 sin 6x

;

72 cos 6x

+ 36e

(7) y00 + y

1 ctg x

, y( ) = 2, y0( ) = 1=2 (8) y0

x + y + 2

x + 1

(9)

2(cos2 y

cos 2y

;

x)y0 = sin 2y,

y(3=2)

5 =4

(10) y000 + y00

= ;24x

+ 49.

3y3 + 12yx2

4.23.

(1)

y(1 + ln y) + y0x = 0

(2)

xy0

2y2 + 6x2

(3)

y0 + xy =

;

x3,

y(0)

(4)

y00 = 18y3,

y(1)

y0(1)

y00 ; 2y0

y(0)

(0)

(5)

= 3x + x,

(6)

y00 + 4y0 = 8e4x ; 8e;4x

(7)

y00 ; 3y0 + 2y =

2 + e;x

y(0)

3 ln 3,

y0(0)

5 ln 3

(8)

y0 = 2x + y

; 3

(9)

ch xdx + (1 + x sh y)dy = 0,

y(1)

4x ; ln4 2

(10) y000 ; 2y00

= 3x2 + x.

= x2 + xy ;

3y2

4.24.

(1)

yy0 (3 + ex) = ex

(2)

x2 ;

4xy

(3)

;

= (x + 1)2 ex,

y(0)

(4)

y3y00 = ;9,

x + 1

y(1)

= 1,

y0(1) =

(5) y00

;

13y0 + 12y = ex,

y(0)

y0(0)

(6)

16 cos 8x

y00 + 64y = 16 sin 8x

;

64e

y00 + 3y0

e;x

(7)

+ 2y =

y(0)

y0(0)

2x + y

; 3

2 + e

(8)

(9)

(13y3

;

x)y0 = 4y,

y(5)

;

; 2

= x ; 1.

(10) y000

13y00

+ 12y0

4.25. (1) 3 + y2 + yy0p1

;

x2 = 0 (2) xy0

= 2

3x2 + y2 + y

y0 + 2xy = x e;

y00

(3)

sin x,

y(0)

(4)

= 4y

;

y00

+ y0 = 2x,

y(0)

y0(0)

(5)

y(0)

y0(0)

(6)

y000 ; 49y0

= ;49 sin 7x ; 49 cos 7x + 14e7x

(7)

y00 + 4y =

y( =4)

y0( =4)

sin 2x

(8)

(9)

y2(y2 + 4)dx + 2xy(y2 + 4)dy = 2dy,

2x + 2y ;IV

y( =8) = 2 (10) y

+ y000 = x.

4.26. (1) xdx ; ydy = x2ydy ; xy2dx (2) 4y0

= x2 + 10x

+ 5

dIFFERENCIALXNYE URAWNENIQ

(3)

y0 ;

= (x + 1)3, y(0)

1=2

(4) y00 = ;50 sin y cos3 y,

+ 1

y(0) =

(0)

(5)

y00 ; y0 = 6x + 5,

y(0)

y0(0)

(6)

y00 + 5y0 = 25e5x ;

25e;5x

(7)

y00 + 4y =

y(0)

cos 2x

x + 5y

(x + ln2 y ; ln y)y0 = y=2,

y0(0)

(8)

y0 =

;

y ; 6

(9)

;

y(2) = 1 (10) y000 ; y00 = 6x + 5.

4.27.

(1)

5 + y2dx + 4(x2y

+ y)dy = 0

xy0

3y3 + 14yx2

(2)

2y2 + 7x2

(3)

; y cos x = ; sin 2x, y(0)

(4)

y00 = 8y3,

y(0)

= 1, y0(0)

2 (5) y00

+ 3y0 + 2y = x2 + 3,

y(0) = 0, y0(0) = 1 (6) y00

+ 81y = 9 sin 9x + 3 cos 9x + 162e9x

(7)

y00 + y0 =

y(0)

ln 27, y0(0)

; ln 9

2 + ex

(8)

y0 = x + y

; 4 (9) 2y2dx + (2xy + p

)dy = 0,

1=2) = 1

;

(10) y000

+ 3xy00;+22y0

= x2 + 2x + 3.

4.28.

(1)

yy0 (ex + 1) = ex

(2)

x2 + xy ;

5y2

(3)

4xy

y(0)

1=2

(4)

x23; 6xy

;

4x ,

;

y y00 =

y(0)

y0(0)

y00

5y0

;3x

;

(5)

;

+ 6y = e

y(0) =

(0)

(6)

y000 ;

64y0 = 128 cos 8x ; 64e

(7)

y00 + y = 2 ctg x, y( =2) = 1, y0( =2) = 2 (8) y0 =

2x + y ; 1

(9)

ydx + (2x

;

2 sin2 y

;

y sin 2y)dy = 0,

y(3=2)

=2x ;

2=4

(10) y000

; 5y00 + 6y0

= (x ; 1)

4.29.

(1)

3y(x2 + 1)dy +

2 + y2dx = 0

ln xq

xy0 = 4

+ y2 + y

(2)

(3)

y(1)

; x

(4)

y cos y,

y(1)

y00 = 2 sin

=2,

y0(1)

(5) y00 ; 6y0 + 9y = e2x, y(0) = 0, y0(0) = 1 (6) y00 + y0 = ex ; e;x
(7)	y00 ; 3y0 + 2y =				1	,	y(0)	= 1 + ln 4, y0(0) =	3 ln 2
				1 + e;x
(8)	y0 = ;	2x + 3y + 1			(9)		dx = 2(y3 ; y + xy)dy, y(;2)		= 0
		3x + 3
(10) yIV ; 6y000 + 9y00 = 3x ; 1.								2xdx ; ydy = x2ydy ; xy2dx
	4.30.		2	(1)
		y		y				y0 ; 3x2y = x2(x3 + 1)=3,
(2)	3y0 = x2 + 10x				+ 10		(3)

sPRAWO^NYJ MATERIAL

y(0)

(4)

y3y00 = y4 ; 16,

y(0) 2

=2,

y0(0)

(5)

y(0)

y00

13y0 + 12y = 3x

y0(0)

(6)

;y00

+ y = 2 sin x ; 3 cos x ; 2ex

(7)

y00 ; 3y0 + 2y =

y(0)

0, y0(0)

1 + e;x

(8)

;

(9) dx = (2y + x tg y

;

y2 tg y)dy, y(0) =

;

5x + 4y

; 39.

(10) y000 ; 13y00 + 12y0

= 18x

2x + 2xy2 + y0p

4.31.

(1)

2 ; x2

= 0

xy0 = 4

2x2 + y2 + y (3)

y0 ; y cos x = sin 2x,

(2)

y(0)

(4)

y0(

= 1

(5) y00 + y0 = 12x + 6,

y00 = 2y ,

;

y(0)

y0(0)

(6) y000 ; 81y0 = 81 sin 8x + 162e

(7)

y00 + y =

y( =2)

y0( =2)

sin x

(8)

x + 6y ;

(9) 4y2dx + (x + e1=(2y))dy = 0, y(e) = 1=2

(10) y

; y ;

+ y000

= 12x + 6.

5.sPRAWO^NYJ MATERIAL

5.1.sHODIMOSTX I RASHODIMOSTX NEKOTORYH RQDOW

5.1.1.dLQ WSEH x WERNY RAWENSTWA:

ex = 1 + x + x2 + x3 + : : : =

1 xn

n=0

sin x = x

x3 + x5

: : : = 1 (;1)kx2k+1

; 3! 5! ; 7! ;

k=0

(2k + 1)!

cos x = 1

+ x

+ : : :

(;1) x .

;

4! ;

k=0

(2k)!

5.1.2. pRI jxj < 1 WERNY RAWENSTWA

;

= 1 + x + x

+ x + : : :

n=0

= 1 ; x + x2 ; x3

+ : : : =

(;1)nxn

1 + x

n=0

(1 + x)a = 1 +

+ a(a ; 1)

; n + 1) xn + : : :.

NESOBSTWENNYJ INTEGRAL OBA RASHODQTSQ.

sPRAWO^NYJ MATERIAL

( 1)n;1xn

ln(1 + x) = x ; 2 + 3

;

4 + : : : =

;

n=1

arctg x = x

+ x5

+ : : : =

(;1)nx2n+1 .

; 3

5 ; 7

2n + 1

n=0

5.1.3. oBOB]ENNYJ GARMONI^ESKIJ RQD 1+1=2p+1=3p+1=4p+

: : : =

1=np

SHODITSQ PRI

p > 1 I RASHODITSQ PRI p

1. w

n=1

^ASTNOSTI, GARMONI^ESKIJ RQD 1+1=2+1=3+1=4+: : : =

1=n

n=1

RASHODITSQ.

5.1.4. nEOBHODIMYJ PRIZNAK. eSLI RQD

1 an SHODITSQ,

n=1

lim an = 0, A ESLI

lim an

= 0,

TO RQD

RASHODITSQ.

n!1

n=1

eSLI

lim an = 0, TO RQD

MOVET KAK SHODITXSQ, TAK I

n!1

n=1

RASHODITXSQ.

5.1.5.

pERWYJ

PRIZNAK SRAWNENIQ. pUSTX

n=1

{

TAKIE RQDY, ^TO

bn DLQ WSEH n

NA^INAQ

NEKOTOROGO NOMERA. tOGDA IZ SHODIMOSTI RQDA

bn SLEDUET

n=1

ABSOL@TNAQ SHODIMOSTX RQDA

an, A IZ RASHODIMOSTI RQDA

n=1

SLEDUET RASHODIMOSTX RQDA

bn.

n=1

5.1.6.

wTOROJ

PRIZNAK

SRAWNENIQ. pUSTX

n=1

{

RQDY S

POLOVITELXNYMI

^LENAMI I

SU]ESTWUET

n=1

KONE^NYJ

NENULEWOJ

PREDEL

lim

tOGDA

RQDY

n!1 bn

n=1

bn LIBO OBA SHODQTSQ, LIBO OBA RASHODQTSQ.

Pn=15.1.7. pRIZNAK dALAMBERA. pUSTX DLQ RQDA

an SU]E-

an+1

n=1

STWUET KONE^NYJ ILI BESKONE^NYJ PREDEL lim

= q. tOGDA

PRI q < 1 RQD

n!1

q > 1 RQD

ABSOL@TNO SHODITSQ,

PRI

n=1

an RASHODITSQ, A PRI q = 1 \TOT RQD MOVET KAK SHODITXSQ,

n=1

TAK I RASHODITXSQ.

5.1.8. rADIKALXNYJ PRIZNAK. pUSTX DLQ RQDA

n=1

1=n

SU]ESTWUET KONE^NYJ ILI BESKONE^NYJ PREDEL nlim!1 janjP

= q.

tOGDA PRI q < 1 RQD

an ABSOL@TNO SHODITSQ, PRI q > 1 RQD

n=1

1 an RASHODITSQ, A PRI q = 1 \TOT RQD MOVET KAK SHODITXSQ,

n=1

TAK I RASHODITXSQ.

5.1.9. iNTEGRALXNYJ PRIZNAK. eSLI PRI x 1 FUNKCIQ

f(x) NEPRERYWNA, UBYWAET I POLOVITELXNA, TO RQD P1 f(n) I

n=1

Z +1 f(x) dx LIBO OBA SHODQTSQ, LIBO

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