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Владимирский государственный университет им. Столетовых

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zADA^I S KRATKIMI RE[ENIQMI	11
FUNKCII	b
b	b

y^. = xke x[Ps(x) cos x + Qs(x) sin x]:

nAJDENNAQ TAKIM OBRAZOM FUNKCIQ y^. { ISKOMOE ^ASTNOE RE[E- NIE NEODNORODNOGO URAWNENIQ y00+py0+qy = f(x), I TOGDA OB]EE RE[ENIE \TOGO URAWNENIQ ZADAETSQ FORMULOJ yO.N. = yO.O.+y^., GDE yO.O. { OB]EE RE[ENIE ODNORODNOGO URAWNENIQ y00 + py0 + qy = 0.

2.zADA^I S KRATKIMI RE[ENIQMI

nAJTI OB]IE RE[ENIQ ILI OB]IE INTEGRALY DIFFERENCIALX- NYH URAWNENIJ, A TAKVE ^ASTNYE RE[ENIQ, ESLI UKAZANY NA- ^ALXNYE USLOWIQ. w RE[ENIQH PROIZWOLXNAQ POSTOQNNAQ INOGDA DLQ UDOBSTWA PREDSTAWLQETSQ W WIDE ln jC1j, GDE C1 { L@BOE NE- NULEWOE ^ISLO.

2.1. dxdy = xy

(1).

/ rAZDELQQ W URAWNENII (1) PEREMENNYE, POLU^IM dyy = dxx ,

Z dyy =

Z dxx , ln jyj = ln jxj + ln jC1j, jyj = jC1xj, y = C1x, y =

Cx,

GDE

C =

= 0.

pRI

= 0

FUNKCIQ

y = 0

TAKVE QWLQETSQ

RE[ENIEM (1). pO\TOMU y = Cx, C 2 R { OB]EE RE[ENIE (1). .

2.2. dxdy = ;xy

(2), y(1) = 2.

/ rAZDELQQ W URAWNENII (2) PEREMENNYE, POLU^IM dyy

= ;dxx ,

Z dyy = ; Z

dxx , ln jyj = ; ln jxj+ln jC1j, jyj = jC1=xj, y = C1=x,

y = C=x,

GDE

C =

= 0.

pRI

C = 0

FUNKCIQ

y = 0

TAKVE

QWLQETSQ RE[ENIEM (2). pO\TOMU y = C=x { OB]EE RE[ENIE (2),

GDE C

2 R. pODSTAWLQQ W RAWENSTWO y = C=x ZNA^ENIQ x = 1 I

y = 2,

POLU^IM C = 2. pO\TOMU y = 2=x { ^ASTNOE RE[ENIE (2),

UDOWLETWORQ@]EE NA^ALXNOMU USLOWI@ y(1) = 2. .
2.3. y(x2	; 1)dy ; x(y2 ; 1)dx = 0 (3).
/ rAZDELIM URAWNENIE (3) NA (x2 ;1)(y2 ;1), OTMETIW, ^TO x =
1 I y = 1	{ RE[ENIQ URAWNENIQ (3). pOLU^IM	ydy	=	xdx	,
		y2 ; 1		x2 ; 1

dIFFERENCIALXNYE URAWNENIQ

Z d(yy22 ;11) =

d(xx22 ;11), ln jy2

; 1j = ln jx2

; 1j + ln jC1j, jy2 ;

; 2

1)j, y

;

; 1),

T E

. y

; 1 = C(x

; 1) {

1j = jC1(x ;

; 1 = C(x

(3),

0 = C

x =

OB]IJ INTEGRAL

GDE

R fUNKCII

y = 1 TAKVE QWLQ@TSQ RE[ENIQMI (3). .

2.4. y0 = (x + y)2

(4), y(0) = 0.

/ w URAWNENII (4) PEREJDEM K NOWOJ FUNKCII z

= x + y

S PROIZWODNOJ

dz = 1 + dy

= 1 + (x + y)2 = 1 + z2,

= dx,

+ 1

arctg z = x+C, x+y = z = tg(x+C). pO\TOMU y = tg(x+C);x {

OB]EE RE[ENIE (4). pUSTX y(0) = 0. tOGDA 0 = tg C, C = k,

k 2 ZI y = tg(x + k) ; x = tg x ; x. iSKOMOE ^ASTNOE RE[ENIE

y = tg x ; x. .

2.5. y0 = y + sin y

(5), y(1) = =2.

w URAWNENII

(5) PEREJDEM K NOWOJ

FUNKCII t

y=x.

tOGDA y = tx I t0x + t = t + sin t, xdx = sin t, Z

= Z

x ,

sin t

ln tg

= ln jxj + ln jCj, t = 2 arctg(Cx), y = 2x arctg(Cx). tAK

KAK y(1) = =2, TO =2 = 2 arctg C, C = 1. pO\TOMU y = 2x arctg x

{ ISKOMOE ^ASTNOE RE[ENIE. .

2.6. y0 ; xy = 1, y(1) = ln 2.

/ w URAWNENII y0 ; xy = 1 POLOVIM y = uv, y0 = u0v + uv0.

tOGDA

u0v + u v0 ; vx

= 1 (6). eSLI v0 ; xv = 0,

dxdv = xv ,

dvv = dxx , ln jvj =

ln jxj

+ C. pO\TOMU WOZXMEM v =

x. iZ (6)

POLU^AEM u0x = 1, du =

dx, u

= ln x + ln

= ln

, C = 0,

j j

y = uv = x ln jCxj. tAK KAK y(1) = ln 2 = ln jCj,

jCj = 2 I

y = x ln j2xj { ISKOMOE ^ASTNOE RE[ENIE. .

2.7. y0 cos2 x + y = tg x.

/ w URAWNENII y0 cos2 x + y = tg x POLOVIM y = uv, y0

= u0v +

uv0. tOGDA u0v cos2 x+u(v0 cos2 x+v) = tg x (7). eSLI v0 cos2 x+v =

TO dv

, ln v

tg x+C. pO\TO-

;cos2 x

Ztg xv

; Z

cos2 x

;tg x

cos

x = tg x,

MU WOZXMEM

v = e;

. iZ

(7)

POLU^AEM

u0e;

zADA^I S KRATKIMI RE[ENIQMI

du = etg x tg x

= tg xd

etg x

, u

= etg x(tg x

;

1) + C, y =

cos2 x

tg x

(tg x ; 1) + C) e;

; tg x ; 1 + Ce;

2.8. y0 + y = y2, y(0) = 1=2.

/ w URAWNENII y0 + y = y2 POLOVIM y = uv, y0 = u0v

+ uv0.

tOGDA u0v + u(v0 + v) = u2v2

(8). eSLI v0

+ v = 0, TO dxdv

= ;v,

dvv = ;dx, ln jvj

;x + C. pO\TOMU

WOZXMEM

v = e;x. iZ

(8) POLU^AEM u0e;x

= u2e;2x,

duu2

= e;xdxx ,

;d u1 = ;d

e;x ,

= e;x + C,

u =

y = uv

. tAK

e;x + C

1 + Cex

KAK y(0) = 1=2, TO

= 1=2, C = 1 I

{ ISKOMOE ^AST-

1 + C

1 + ex

NOE RE[ENIE. .

2.9. y000 = 24x + cos x.

y000 = 24x+cos x,

;

tAK KAK

y00 = 12x

;sin x+C1, y0 = 4x

cos x+C1x+C2, y = x

+ sin x +

+ C2x + C3, GDE C1 C2 C3 {

PROIZWOLXNYE POSTOQNNYE. .

2.10. xy00 + y0 = 0.

/ tAK KAK URAWNENIE xy00 + y0 = 0 NE SODERVIT QWNO y, TO

POLOVIM y0 = z I POLU^IM xz0 + z = 0, xdxdz = ;z, dzz = ;dxx ,

ln jzj = ; ln jxj + ln jC1j, jzj = ln

Cx1

, y0 = Cx1 , y = C1 ln jxj + C2,

GDE C1 I C2 { PROIZWOLXNYE POSTOQNNYE, PRI^EM PROWERKA PO-

KAZYWAET, ^TO ZNA^ENIE C1 = 0 TOVE DAET RE[ENIE y = C2 URAW- NENIQ xy00 + y0 = 0. .

2.11. yy00 ; (y0)2 = 0.
/ tAK KAK URAWNENIE yy00 = (y0)2, NE SODERVIT QWNO x, TO PO-
	dy		d2y	dz	dz
LOVIM dx		= z(y), dx2 = dx			= dy z I IZ URAWNENIQ yy00 = (y0)2
POLU^IM URAWNENIE yz dz				= z2, RE[ENIE KOTOROGO IMEET WID z =
			dy	dy
C1y.	pO\TOMU		y0 = C1y,		= C1dx, ln jyj = Cx + ln jC2j, y =
				y

C2eC1x, GDE C1 I C2 { PROIZWOLXNYE POSTOQNNYE, KOTORYE MOGUT BYTX RAWNY NUL@, POSKOLXKU W \TOM SLU^AE MY POLU^AEM RE[E-

14 dIFFERENCIALXNYE URAWNENIQ

NIE y = C, KOTOROE MY MOGLI POTERQTX, TAK KAK DELILI NA y I y0 = z. .

2.12. y00 ; 5y0 + 6y = 2ex.

/ tAK KAK KWADRATNOE URAWNENIE t2 ; 5t + 6 = 0 IMEET KORNI t = 2 I t = 1 KRATNOSTI 1, TO OB]EE RE[ENIE LINEJNOGO ODNOROD- NOGO URAWNENIQ y00;5y0+6y = 0 IMEET WID yOO = C1e2x+C2e3x. oB- ]EE RE[ENIE ISHODNOGO NEODNORODNOGO URAWNENIQ y00 ;5y0 + 6y = ex IMEET WID yON = yOO + y^, GDE y^ { KAKOE-NIBUDX ^ASTNOE RE- [ENIE ISHODNOGO URAWNENIQ. tAK KAK 1 { NE KORENX URAWNENIQ

t2 5t+ 6 = 0, TO y^ I]EM W WIDE y^ = Aex. tOGDA y^0	= y^00 = Aex,
y^00; 5y^0 + 6y^ = 2Aex = 2ex, A = 1, y^ = ex, yON = C1e2x + C2e3x +
ex.;	.
2.13. y00 + 4y = 3 cos x ; 6 sin x.
/ tAK KAK KWADRATNOE URAWNENIE t2 + 4 = 0 IMEET KOMPLEKS-

NYE KORNI t = 2i I t = ;2i KRATNOSTI 1, TO OB]EE RE[ENIE LINEJNOGO ODNORODNOGO URAWNENIQ y00 + 4y = 0 IMEET WID yOO = C1 cos 2x + C2 sin 2x. oB]EE RE[ENIE ISHODNOGO NEODNORODNOGO URAWNENIQ y00+4y = cos x IMEET WID yON = yOO +y^, GDE y^ { KAKOE- NIBUDX ^ASTNOE RE[ENIE ISHODNOGO URAWNENIQ. tAK KAK 1 i = i { NE KORENX URAWNENIQ t2 + 4 = 0, TO y^ I]EM W WIDE y^ = A cos x +

B sin x. tOGDA y^0 = ;A sin x + B cos x, y^00 = ;A cos x ; B sin x, y^00 + 4y^ = 3Acos x + 3B sin x = 3 cos x ; 6 sin x, A = 1, B = ;2, y^ = cos x ; 2 sin x, yON = C1 cos 2x + C2 sin 2x + cos x ; 2 sin x. .

2.14. y00 ; 2y0 + y = 6xex, y(0) = 1, y0(0) = 0.

/ tAK KAK KWADRATNOE URAWNENIE t2 ;2t+1 = 0 IMEET ODIN KO- RENX t = 1 KRATNOSTI 2, TO OB]EE RE[ENIE LINEJNOGO ODNORODNO- GO URAWNENIQ y00;2y0+y = 0 IMEET WID yoo = ex(C1 +C2x). oB]EE

RE[ENIE ISHODNOGO NEODNORODNOGO URAWNENIQ y00 ; 2y0 + y = 6xex IMEET WID yON = yOO + y^, GDE y^ { KAKOE-NIBUDX ^ASTNOE RE[ENIE ISHODNOGO URAWNENIQ. tAK KAK 1 + 0i = 1 { KORENX KRATNOSTI

URAWNENIQ t2 ;2t+ 1 = 0, TO y^ I]EM W WIDE y^ = x2ex(Ax+ B) = ex(Ax3 + Bx2). tOGDA y^0 = ex[Ax3 + (B + 3A)x2 + 2Bx], y^00 = ex[Ax3+(B+6A)x2+(4B+6A)x+2B], y^00;2y^0 +y^ = ex[Ax3+(B+ 6A)x2 +(4B+6A)x+2B];2ex[Ax3 +(B +3A)x2 +2Bx]+ex(Ax3 + Bx2) = ex[6Ax + 2B] = 6xex, 6Ax + 2B = 6x, A = 1 B = 0, y^ = exx3, yON = ex(C1+C2x+x3), yON0 = ex(C1+C2x+x3+C2+3x2),

y(0) = C1 = 1, y0(0) = C1 + C2 = 0, C2 = ;1.

zADA^I S KRATKIMI RE[ENIQMI

pO\TOMU RE[ENIE y = ex(1 ; x + x3) UDOWLETWORQET NA^ALX-

NYM USLOWIQM y(0) = 1, y0(0) = 0. .

2.15. y00 + y = sin1 x.

/ tAK KAK KWADRATNOE URAWNENIE t2 + 1 = 0 IMEET KOM- PLEKSNYE KORNI t = i I t = ;i KRATNOSTI 1, TO OB]EE RE[E- NIE LINEJNOGO ODNORODNOGO URAWNENIQ y00 + y = 0 IMEET WID

yOO = C1 cos x + C2 sin x. oB]EE RE[ENIE ISHODNOGO NEODNORODNO-
GO URAWNENIQ
1
y00 + y = sin x IMEET WID yON = yOO + y^	, GDE y^ { KAKOE-NIBUDX
^ASTNOE RE[ENIE ISHODNOGO URAWNENIQ,	KOTOROE MY I]EM ME-

TODOM WARIACII POSTOQNNYH W WIDE y^ = C1(x) cos x+C2(x) sin x.

nEIZWESTNYE FUNKCII C1(x) I

C2(x) UDOWLETWORQ@T

SISTE-

( ;

(x) cos x

(x) sin x

= 0

wY^TEM IZ PERWO-

(x) sin x

C0(x) cos x

= 1= sin x:

GO URAWNENIQ SISTEMY, UMNOVENNOGO NA cos x, WTOROE URAWNE-

NIE, UMNOVENNOE NA sin x. tOGDA C0(x)(cos2 x

+ sin2 x)

;

(x) =

1, C1(x) =

x + C1, C0

C0(x) cos x

;

(x) sin x =

;

= cos x,

(x) = cos x, C2(x) =

cos x dx =

d(sin x)

= ln sin x + C2,

sin x

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

16	dIFFERENCIALXNYE URAWNENIQ

3.zADA^I

nAJTI OB]IE RE[ENIQ ILI OB]IE INTEGRALY DIFFERENCIALX- NYH URAWNENIJ, A TAKVE ^ASTNYE RE[ENIQ, ESLI UKAZANY NA- ^ALXNYE USLOWIQ.

3.1. 2dy;xdx = 0, y(2) = 0. 3.2. (2x+5)dy+ydx = 0, y(0) = 1.

3.3. yx

; y0

= 0, y(0) = 10. 3.4. yy0

= 3, y(6) = 10.

3.5. y0;(2x+2)p

; y2

= 0, y(0) = 1. 3.6. ydy;xdx = 0, y(3) = 5.

3.7. y0p1 + x2

y = 0, y(0) = 4. 3.8. y0x + p4

y2 = 0, y(1) = 0.

;4y = 0, y(0) =

3.9. y0(4+x

)+y

= 0, y(2) = 8= . 3.10. y0(4;x )

5. 3.11. pxdy

;

ydx = dx, y(0) = 0. 3.12. p

;

y0+xp9

;

0, y(0)

3.13.

;

2xy

; y

y(0)

3.15. p

3.14. 3xdx

;

2xdy = dx+dy.

;

x2dx

;

ydx

= 0, y(3=2) =

1. 3.16. dy

)dx =

; 2ydx = dx, y(ln 2) = 5=2. 3.17.

2dy

; (1 + 4y

0, y( =12)

;

1=2.

3.18.

px2

;

+ 8y0

;

3.19. y0p1 + x2

;

y = 0, y(0) = 4. 3.20. xp25

;

0, y(0)

xy0

= y

; y2

3.21.

3.22.

1 + ln x

= x2 ; x.

3.23.

y0 sin x

; y = sin x sin

3.24.

; 5x4y = ex .

2 .

y = xp

xy0

(xy0

3.25.

;

3.26.

;

+ y) = 1.

3.27.

e;x2

3.28.

+ 2

xy0 + y = 1 + ln x.

y0x + y =

;xy1

3.29.

; y tg x = ctg x.

3.30.

3.31.

xy0

;

2y = 2x4.

3.32.

y0 + y tg x =

cos x

3.33.

(xy + ex)dx ; xdy = 0.

3.34.

y = x(y0

; x cos x).

3.35. 2x(x2 + y)dx = dy. 3.36. y0

;

= x. 3.37. y0 +

= x3.

3.38. y0

;

; 1 ; x = 0, y(0) = 0. 3.39. y0

; y tg x =

;

cos x

; e

; xy

y(0) = 0. 3.40. xy0 + y

= 0, y(a) = b. 3.41. y0 =

3.42.

2xy

; y2

+ x = 0.

3.43.

y0 + 2y = y2

ex.

3.44.

xy2y0 = x2

+ y3.

3.45.

y00 = x1 .

3.46.

xy(4)

= 1.

3.47. x4y00

+ x3y0 = 1. 3.48. xy000

+ 2y00

= 0. 3.49. y00 = 1 ;

(y0)2.

3.50. yy00

;

y0(1 + y0) = 0. 3.51. xy00 + y0 = 0. 3.52. xy00

= 0.

3.53. y00

3.54.

y00

9y = 0.

3.55.

y00

= 0.

+ 9y = 0.

;

+ (y0)

3.56.

y00(y

;

4) = (y0)2.

3.57.

y00 cos x + y0 sin x = 1.

3.58.

3.59.

y00

+ x

y00 tg y = 2(y0)

3.60.

xy00 ; y0 = x2ex.;

3.61.

x2y00

= (y0)2 + xy0

; x2.

zADA^I

3.62. xy00

; y0 ln yx0

= 0. 3.63. y00 tg x + y0

= 1. 3.64. xy00

; y0

= x2.

3.65.

xy00 ; y0 = 5.

3.66.

y00

; y0ey = 0.

3.67.

y00

; y0 = 0.

3.68. y00

+ 25y = 0.

3.69.

y00 + 25y0 = 0.

3.70. y00

;

3.71. y00 ;

25y = 0. 3.72. y00

;

2y0 + y = 0. 3.73. y00

;

6y0 + 9y = 0.

3.74. y00

+ 4y0

+ 10y

3.75. y000

;

5y00 + 8y0

;

= 0.

3.76. y00 + 100y = 0. 3.77. y000 + y00 = 0. 3.78. y00

;

20y0

+ 19y = 0.

3.79. y

(4)

;y00

= 0. 3.80. 2y00

;

+6y = 0.

3y0;2y = 0. 3.81. y00+7y0

3.82. y00+6y0+10y = 0. 3.83. y00+3y = 0. 3.84. y00+4y0 +13y = 0.

3.85.

y00

;

2p3y0 + 7y

3.86.

y00

+ y0

;

12y

3.87. y00+4y0+4y = 0. 3.88. y00

;4y0;7y = 0. 3.89. y00

+9y0

;10y =

0. 3.90. y00

+ 10y0

+ 100y

3.91. y00

+ 7y0

+ 2y

3.92. y00

+ 4y0

3.93. y000

+ 3ay00

+ 3a2y0

+ a3y

3.94. y(4)

;

2y0

= 0. 3.95. y(4) + 9y = 0. 3.96. y(4)

;

16y = 0.

3.97. y000

y00 + y0

. 3.99.

y00

+ y = 0. 3.98.

= e

;

4y0

= 4e

3.100. y00

+ 3y0

+ 2y

. 3.101. y00

+ 7y0

+ 20y

3.102. y00

+ y0 + 10y

3x2. 3.103. y00 + y0

+ y

3 cos 2x.

3.104.

y00 + 3y0 + 2y

5e5x. 3.105.

y00

+ y

sin 5x.

3.106.

y00

cos x.

3.107.

y00

cos 3x.

3.108. y00

+ y0

;

2e;2x + e2x. 3.109.

y00

;

e2x.

;

3.110. y00

6y0

+ 9y

. 3.111.

y00

; y0

4 + x.

3.112. y00

;

2y0

;

x . 3.113. y00

+ y

cos x + sin 5x.

3.114. y00

x + e;

. 3.115.

y00 + 100y

sin 2x.

+ 4y0

3.116. y00

+ 2y0

+ y

e;x. 3.117. y00

;

e2x + 3e;2x.

3.118. y00

. 3.119.

+ 9y = x + 4 sin 3x.

;

2y = sin x + x

y00

3.120.

3.121.

(y000)

+ (y00)

= 1.

y00

; 3y0

3.122.

2xy0

= 0,

(x + 1)y00 ; (x + 2)y0 + x + 2 = 0.

3.123.

(1 + x )y00

y(0)

y0(0)

3.124.

1 + (y0)2 =;2yy00,

y(1)

y0(1)

3.125.

yy00 + (y0)2 = (y0)3,

y(0)

y0(0)

3.126.

y00(1 + ln x) +

1 y0

= 2 + ln x,

y(1)

1=2,

y00 = (y0)2

; y,

;

y0(1)

3.127.

y(1) =

y0(1) = 2.

3.128.

2y000

;

3(y0)2 = 0,

y(0)

;

y0(0)

y00(0)

y00

2y0

3y = e

3.130.

y00 + y = 4xe

;

3.129.

;

y00

2y0

3.132.

y00

+ y = x sin x.

3.131.

;

+ y = 6xe

3.133. y00

+ 8y

+ sin 2x. 3.134. y00

5y0

;

4y0

;

3.135.

y(0)

sin 5x.

y00 + 2y0 + 2y = xe;

y0(0)

3.136. y00

;

2y0 = 2ex,

(y(1)

;

1, y0(1) =

y000 ; 3y0

;3,

3.137.

; 2y = 9e

y(0)

(0)

dIFFERENCIALXNYE URAWNENIQ

y00(0)

3.138.

yIV + y00

= 2 cos x,

y(0)

0. 3.139. y00 ; 2y0

y0(0)

y00(0)

y000(0)

+ y =

ex .

y00 + 2y0 + y =

e;x

y00 + y =

3.140.

3.141.

sin x

3.142.

y00

+ 4y = 2 tg x.

3.143.

y00 + y =

cos3 x

3.144.

y00 + y = ctg x.

3.145.

y00

+ 2y =

y(1=2) = 1,

sin x

2=2. 3.146. y00 ; 3y90

y0(1=2) =

+ 2y =

y(0)

; e;x

y0(0) = 0.

3.147. y00 + 9y =

y(0)

= 1, y0(0) =

cos 3x

3.148. y00

+ 4y = 8 ctg 2x, y( =4) = 5, y0( =4) = 4.

oTWETY3.1.

4 + C, C = ;1.

3.2.

+ C, C = p5.

2x + 5

3.3.

y = Cex2=2,

10.

3.4.

y2 = 6x + C,

64.

3.5.

y = sin

x2 + 2x + C

cos(x2

+ 2x).

3.6. y

16.

), C

; x = C, C

3.7.

y = C(x + p1 + x

3.8.

y = 2 sin ln

3.9.

2 arctg

2 + C,

2 + x

y = Ce2px ;x1,2+xC

3.10.

y = C 2 ; x,

3.11.

), C =

3.12. y = 3 sin(C + p1

;

1. 3.13. y

= Ce

, C = p3.

3.14. y =

2 x ; 4 ln jC(2x + 1)j. 3.15. y = Cearcsin 3 , C = e; =6.

3.16. y =

Ce2x

;

=2, C = 3=2.

3.17. y = 1 tg(x

+ C), C

; =3. 3.18.

y =

4 sin ln jC(x ; 2 +

x + 4x + 8)j. 3.19. y =

5 sin(xex ; ex + C), C = 1. 3.20. y =

C + 4x

y = xeCx.

x + 3 . 3.21.

3.22. y =

. 3.23.

2 sin x + C

tg x. 3.24. (x + C)ex4.

1 ; Cx2

3.25. y = 2xp

+ Cx. 3.26.

C ; e2;

(C + arcsin x). 3.27. y =

3.28.

y = ln x +

x .

3.29.

y = 1 +

ln C tg

2 .

cos x

3.30. y =

. 3.31. y = Cx2 + x4. 3.32. y = sin x + C cos x.

x ln Cx

3.33.

+ C) x = 0.

3.34.

y = x(C + sin x).

y = ej

(lnj

3.35. y = Cex

; x2 ; 1. 3.36. y = Cx + x2.

3.37. y =

+ x2 .

zADA^I																						19
				(xp
			1													1 + x		.				y =		x
3.38.	y =		1			1		;		x2 + arcsin x)						1 + x				3.39.				x	.
3.38.	y =					1				x2 + arcsin x)										3.39.					.
			2		x							a			s1 ; x									cos x
				e			a			b e										2
3.40.	y = x +									x; .				3.41.						2
3.40.														3.41.					y(x + Cx) = 1 y = 0.
																			y(x + Cx) = 1 y = 0.
3.42.		y2 = x ln							C .				3.43.				y(ex + Ce2x) = 1 y = 0.
									x
3.44.	y3		= Cx3					;		3x2.			3.45.				y = x ln x					+ C1x + C2.
3.46.								;					3				3				j 2j
3.46.											6y = x			ln jxj + C1x + C2x + C3x + C4.
			1								6y = x			ln jxj + C1x + C2x + C3x + C4.
3.47.			1											3.48.

	y = 4x2 + C1 ln jxj + C2.																y = C1 ln jxj					+ C2x + C3.
	y = 4x2 + C1 ln jxj + C2.																y = C1 ln jxj					+ C2x + C3.
3.49.					2x									3.50.							C2x		1
3.49.	y = ln je + C1j ; x + C2.													3.50.			y = C1e + C2 y = 0.
	y = ln je + C1j ; x + C2.																y = C1e + C2 y = 0.
3.51. y = C1 ln x + C2. 3.52. y = C1x2 + C2. 3.53. y = C1 cos 3x +

C2 sin 3x. 3.54. y = C1e3x + C2e;3x. 3.55. x = ey + C1y + C2.

3.56.

y = 4 + C2eC1x. 3.57. y

;

cos x + C1 sin x + C2.

3.58. y = ;

C1 ln jC2xj. 3.59.

ctg y =

c2 ; C1x. 3.60. y =

3x3

(x ; 1)ex + C1x2 + C2. 3.61. y = ;

ln jC1x2 ; 1j + C2.

2 ;

3.62.

y =

C1x

; 1eC1x+1

+ C2. 3.63.

y = x + C1 ln

tg x

+ C2.

5x + C2.

3.64.

3.65.

y = 3

C1x + C2.

y =

C1x

;

3.66.

c1 + ey = C1(x + C2).

3.67.

C1 + C2ex.

3.68.

y ;= jC1 sin 5jx + C2 cos 5x.

3.69.

+ C2e;25x.

3.70. y = C1e2p2x + C2e;2p2x. 3.71. y = C1e5x +C2e;5x. 3.72. y =

+C1 + C2. 3.73. y = e3x(C1 + C2x). 3.74. y = e;2x(C1 + C2x). 3.75. y = C1ex + e2x(C2 + C3x). 3.76. y = C1 sin 10x + C2 cos 10x.

3.77. y = C1 + C2x + c3e;x. 3.78. y = C1e19x + C2ex. 3.79. y =

C1 + C2x + C3e;x + C4ex. 3.80. y = C1e2x + C2e;x=2. 3.81. y =

C1e;x + C2e;6x. 3.82. y = e;3x(C1 sin x + C2 cos x. 3.83. y =

C1 sin p

x +

C2 cos p

x. 3.84. y = e;2x(C1 sin 3x + C2 cos 3x).

3.85. y = ep

x(C1 sin 2x + C2 cos 2x). 3.86. y = C1e3x + C2e;4x.

3.87. y = e;2x(C1 + C2x). 3.88. y = C1e(2+p

11)x + C2e(2;p

11)x.

C1e;10x + C2ex. 3.90. y

= e;5x(C1 sin 5p

x +

3.89. y =

C2 cos 5p

C1e(p

;7)x=2

+ C2e(p

x).

3.91.

41+7)x=2.

3.92. y = C1 +

C2e;4x.

3.93. y =

e;ax(C1

+ C2x + C3x2).

p2x

+ C3e;

. 3.95. y = C1 + C2x + C3 sin 3x +

3.94. y = C1 + C2e

C4 cos 3x. 3.96. y

C1e2x + C2e;2x

+ C3 sin 2x + C4 cos 2x.

3.97. y = C1e;x + ex=2

x!. 3.98. y =

C2 sin

2 x + C3 cos 2

dIFFERENCIALXNYE URAWNENIQ

+C1 + C2. 3.99. y = +C1 + C2.

3.98. y =

1ex

+ C1 + C2e;x.

1 e;2x

3.99. y = e4x(x + C1) + C2. 3.100. y =

+ C1e;2x + C2e;x.

y = e;7x=2

+ C2 cos

3.101.

C1 sin

y = 0 3x

; 0 06x ; 0 054 + e;

x=2

C1 sin

3.102.

2 + C2 cos

3.103.

y =

sin 2x

;

cos 2x

e;x=2

C1 sin 2

+ C2 cos

2 !. 3.104. y =

e5x + C1e;2x + C2e;x.

3.105.

y = ;24 sin 5x + C1 sin x + C2 cos x.

3.106.

y = x sin x + C1 sin x + C2 cos x.

3.107.

y = 6 sin 3x + C1 sin 3x + C2 cos 3x.

3.108.

y = 4 e2x

;

e;2x + C1e;2x + C2ex.

3.109.

y = ;

5e + C1e + C2e; .

3.110. y = e3x

2 + C1 + C2x!. 3.111. y = ;

2 ; 5x + C1 + C2ex.

3.112.

y =

; 3 ; 9

; 27 + C1e; + C2e .

3.113.

y = ;

24 sin 5x + 2 sin x + C1 sin x + C2 cos x.

3.114.

y = 8

;

4 e;4x + C1e;4x + C2.

3.115.

y = 96 sin 2x + C1 sin 10x + C2 cos 10x.

3.116. y = e;x

+ C2x

. 3.117. y = e2x

+ e;2x

;

2 x2 + C1

4 + C1

4x + C2 .

3.118.

y = C1e2x + C2e;x + 0 1 cos x ; 0 3 sin x ; 2

2 ;

3.119.

y = ;3 xcos 3x + C1 cos 3x + C2 sin 3x.

3.120.

y = ;12 ;

; 9

;

27 x + C1 + C2e .

3.121. y = sin(C1 + x) + C2x + C3.

3.122.

y = (C1ex + 1)x + C2.

3.123.

y = x3 + 3x.

3.124.

y =

(x2 + 1).

y = x + 1. 3.126.

y =

3.125.

2 x .

3.127.

y =

4 ;

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