Сборник задач по высшей математике 2 том

2.1.50. In Jtg ~J =

C -

2 sin~. 2.1.51. (1 + e"')(l + eY ) = C.

",2

1.

2.1.55.

J(3 + x 2 )3

!

2.1.54. Y = Ce 2

-

2+y

2

= C. 2.1.56. Y = a + Ce"'.

2.1.57. y = Csinx -

a. 2.1.58. y =

Co~x -1. 2.1.59. Y = Csin2 x - ~.

2.1.60. Y =

1 + C~l' 2.1.61. Y -

x + C = Inx 2 + In Iyl·

2.1.62. 4.JX+1 +x+ = c, y = 1.

2.1.63.2";2 -

x 2 - axctgy = C.

In

y

2.1.64. Y =

~1' 2.1.65. x 2 -

2y2 = 2. 2.1.66. 1 + y2 = ~.

x+

I-x

2.1.67. y =

x 2 - 2. 2.1.68. Y =

x, Y = 27r -

x. 2.1.69. y2 =

(x + 3)3.

2.1. 70. 2'" -

axctg 2Y

= 1 - ~. 2.1.71. Y = In x. 2.1.72. 2y + 1 = 4 sin2 x.

2.1.73. Y =

e" + 8

y = 2x

3

; 6) y =

16

4

- 9 - ' 2.1.74. a)

-16x _ 33.

2 .1.75. a) y = x;

6) -IX = 2y.

2.1.76. Y

~

2.1.77. y =

2

2

= Ce 2 .

-3-

HJIH Y = - 1 - '

- x

+x

2.1. 78. Y =

2

- 11- 2.1. 79. ~ 817 JIeT. 2.1.80. N =

t

C + In Ix

a . 3:{.

t

-

tl

2.1.81. 0,2 KM/'Iac. 2.1.82. 45°C. 2.1.83. x = ml

. (:~) t2 -

tl

1'.

cIIoco6aMH, IIOJIy'IHM ypaBHeHHe: -S(h) dh =

v(h) dt· S, r.n;e S -

IIJIOIIIa.n;b

OTBepCTHH, S(h) -

IIJIOIIIa,!l;b IIOIIepe'IHOrO Ce'IeHHH cocy.n;a, h -

YPoBeHb

)l{H.n;KOCTH, v(h) -

CKOPOCTb ee HCTe'IeHHH, t

-

15

BpeMH.) 2.1.89. 11 M/C.

2.1.90.200;v=

k

1

_

(

C+

1 ~3 1 _/r.--=3')

,

. 2.1.91. y-lntg

3y (x+1)

-3y(x-1)

mt +0,1

x ~ 1. 2.1.92. a) y = Cefsin,,2dx; 6) 5x + lOy + C = 31n 110x -

5y + 61;

1, Y = -1, x ~+ yv'f=X2 =

1;

y-x

x = 27rk, k E Z.

,Il;) 4y - 6x + 1 = Ce- 2,,; e) ctg - 2 - = x + C, y -

2.2.2. Y

2

+ 2xy = C. 2.2.3. Y

2

1l.

= 2x axctgx.

e'" = Cx. 2.2.4. Y

2.2.6. y = 1 +

x - I

1)1, C i= O. 2.2.8. x + yIn Cy = O.

-2-1n IC(x -

2

.2.9.

(

x -

1)2

+ y

2

=

1

.2.2.1

0

• Y =

Cx2

1

2.2.11. Y

2

=

2

x

21

C

-2- -

2C'

n x'

2.2.12. e- ~ + In Cx = O. 2.2.13. Y =

xeC,,+l. 2.2.14. y = x In2 Cx.

2.2.15. s ~ t = InC(s -

t). 2.2.16. x 2 - y2 = Cx. 2.2.17. Y = xln2

2.2.21. Y = xeGx . 2.2.22. y2 -

Y

x 2 = C y3. 2.2.23. ex = -x2 .'

-x

2.2.24. x 3 + y3 = Cxy. 2.2.25. Y = x In x. 2.2.26. y = x In ey.

~-y

x 2

1

y

lX+y-C

x .

229

22-C

.

2.2.27. n2yx- + y- - x arctg x' 2.2.28. n - x - -

..2 . xe

Y -

2.2.30. Y = xtg(lnCx). 2.2.31. y = x -

-x1 .2.2.32. Y = -x.

nex

2.2.39. a) x -

y + C = In 13x -

4y + 11; 6) (y -

2x -

2

= C, Y = x.

x)e Y -

x

2.2.40. x 2 = (x 2 - y)lnCx, y = x 2. 2.2.41. a) Cx = r.p (~);

6) y + y'2x2 + y2 = Cx 5 ;

B)

Y +

2X

= Cx. 2.2.42. y2 = 2Cx + c2, KpHBbIe

5

y+

x

(ce'leHHH)-

napa60JIbI, nOBepxHocTb 3epKaJIa -

napa60JIOH,ll; Bpam;eHHH.

2.2.43. a -

(3

= a(3.

§ 3. J1MHeiiiHble ypaBHeHMH. YpaBHeHMH 6epHynnM

2.3.2. Y = (x + C) eX

2

.2.3.3.

Y = x

2

C

1.

2, Y = O.

+ X. 2.3.4. x = Ce Y -

7(x + 1)2

x

C

-2x

x

2.3.5.y=

C - (x + 1)7.2.3.7.y=e

-x-1.2.3.8.y=

e

+e.

2.3.9. Y =

x3 + C

4

2

,

y = O. 2.3.11. x =

In y + C

, Y = O.

- 2 -- '2.3.10. x

= Y + Cy

Y

x

+

1

2.3.12. Y = 1

+

~

x +

l' 2.3.13. Y =

. 1

+

, Y = O.

nx

y'Ce2x2

1

2.3.14. Y = Ce- sin x

+ 2 sin x -

2. 2.3.15. Y = x 3 + ¥.

2.3.16. y2(2x + C) = ex2 ,

Y = O. 2.3.17. Y =

o/3x + C

2

= C -

Y

x

. 2.3.18. x

cosy ' Y = O.

2.3.19. y = eX (C + Inx + ~2). 2.3.20. Y = VI - x2(2arcsinx + C).

2.3.21. Y = (x + C) tg~. 2.3.22. Y = ~ x 3. 2.3.23. Y = 2sinx -

cosx.

1.

2x

(C + x tg x + In cos x ) 2

2.3.24. x = eY. 2.3.25. y =

- 2 -- '2.3.26. Y =

x

' y = O.

x

-3

2.3.27. x = Cy2 + Iny2 -

y + 1, y = 0.2.3.28. Y = e- x (1 + arctgx).

2.3.29. y = 4 -

4cost + e

I-cos t

.2.3.30.

r

=

e'"

+ 2a2

-

eO<

r.p

2

2

1

~7

+11

)4

2.3.31. x = Y

-

_lL

(3VX

'

2 + e

2. 2.3.32. Y = x2

14

1(3#-17)4

gm

_'!'t

y = x 2

14

. 2.3.33. C1 + C2 = 1. 2.3.34. v = T(1 -

em).

2x

6

)

I =

kt

kL (

e

- B. t

-

)

2.3.35. Y = 2x _

1 - nmep

OJIa. 2.3.36. a

R + R2

L

1 j

6) 1=

2 A 2

2 (R sin wt -

wL cos wt + wLe-¥ t).

R

+w L

2.3.37. Y = Ce-<p(x) + rp(x) -

1. 2.3.38. a) eY = 2 C

, 0603Ha'lHTb eY = Zj

x

+Cx

x

2.4. 7. ~x 2 -

~x 3y2 + y3 = C. 2.4.8. ~2 cos 2y -

3x = C. 2.4.9. x 2 + y + eXY = 2.

2.4.10. y'x 2 + y2 + xy = 4. 2.4.11. x 3 + y3 + 3x 2y + 3x - 3y = C.

2.4.12. x sin(x + y) = C. 2.4.13. x 3 + x3 In y -

y2 = C.

2.4.14. x 3y -

cos x - siny = C. 2.4.15. x 3 + xy2 + xy + eY = 1.

2.4.16. x 3 + 3x 2y -

y3 + 1 = O. 2.4.17. ~In(x 2 + y2) - arctg ~ = C.

2.4.18. x 2 -

x + ¥ - y2 = C. 2.4.19. x 2 + ye ~ = C.

2.4.20. sin ¥ - cos ~ + x -

~ = C. 2.4.21. xsiny -

ycosx + Inxy = C.

x

2

2

= C. 2.4.24. xy -Iny = C,

2.4.22. 2

eY + tgy - Y = C. 2.4.23. yshx + xshy

8Q

8P

1

x

3

= C, t

1

8x - 8y

,lJ;OJDKHO 6blTb

t(y) = -yo

2.4.25. - . - + x

= -.-. 2.4.26.

P

-

Q

SIllY

SIllY

<PYHKU:Helt OT x + y. 2.4.27. a) x 2 + y2 = Cj 6) xy = C, T. K. d(xy) = O.

§ 5. YpaBHeHIIIH llarpaH)I(a III Knepo

2.5.2. Y = Cx + C - C 2, Y = (X ~ 1) 2. 2 .5 .3 . Y = Cx -

3C3, oco6oe pemeHHe

~

= O. 2.5.5. Y = (v'X+I+ C)

2

, Y = O. 2.5.6. Y =

x 2

- C 2

9y ± 2x2

2C .

X

= Inp -

arcsinp+ C,

c

= xInx - x.

2.5.8. {

r1"-::::i

2.5.9. Y = Cx -

e

, y

y=p+yl- p2.

{x = (p + l)eP + C,

2.5.10. Y = Cx

2

+

1

±2x. 2.5.11.

Y = O.

-C' y =

2

Y = peP,

2.5.12. {

Y = Xp2 -

p,

1

P -Inp + C

2.5.13. Y = Cx + C + .,;0, y = - ( ).

X=

2 •

4x+1

(p -

1)

2.5.14. Y = Cx -InC, y = Inx + 1. 2.5.15. Y = Cx + h, y2 = 4x.

1 .

x 3

3x

2

+ Clx

+ C2.

2.6.2. Y = -16 sm4x +

3

- 2""

6 3

1

5",

1 5

+

C

IX+

C

2·

2... y= 25e

-cosx-lQx

_I!

e

",2

dx +

-32-'"-

+ Clx + C2.

2.6.4. Y -

-2

In

3

4 y(

_

5

2

-

COS 2x

cos 4x

+ cosx

+

x + 2

)5

+

C

IX +

C

2.6.5. Y -

4'x

- 2 - -

32

15

2·

y... = (x + 3) Inx + xsinx -

(4 + sin 1 + cos l)x + 5 + cos 1.

2.6.10. Y = (x 3 + x 2 -

2x + 1)e2'" -

x cos 3x + CIX + C2,

x 5

+ CIX3 + C2.

y... = (x3 + x 2 - 2x + 1)e2'" - x cos 3x + 2x. 2.6.12. Y = 5"

2.6.13. Y = CI(x + 1)2 + X + C2. 2.6.14. Y = ~ + C1lnx + C2.

2.6.15. Y = CI sin x -

x -

~ sin 2x + C2. 2.6.16. Y = (x - l)e'" + C1X2 + C2.

2.6.17. Clx(lnx -1) + C2. 2.6.18. Y = -x -

C1cos x + C2.

2.6.19. y =

x 2

+ C1lnx

x

3

x

-4

+ C2. 2.6.20. Y = 12 -

'4+ Cl axctgx + C2.

2

- 1 = (Clx + C2)

2

CI

X + C 2

+ e

X + C 2

)

2.6.23. Cly

(

--

---

1

1

•

. 2.6.24. y = 2' e

C

C

2.6.25. (x -

C1)2 = 4C2(y - C2). 2.6.26. ycos2(x + Cd = C2.

2.6.27. Y = ±axcsine",+c 1

+ C2 II Y = C. 2.6.28. Y = x.

2.6.33. Y = C2xe

_ C 1

2

x . 2.6.34. Y . cos (x + Cd = C2.

2.6.36. Y = l6 cos 2x + Cl X3 + C2X2 + C3X + C4.

2.6.37. Y =

~eb'" + CIXS + C2X7 + ... + Csx + C9.

b

2.6.38. lox5 + CIX2 + C2X + C3.

2 .6.39. Y =

sin

3x

3'

X4

+

C

IX

2

+

C

2X

+

C

3·

-~ + smx -

24

2.6.40. Y =

1

1

1

+ sin2x)

2

- 32 (64 sin8x + 27 sin6x + l sin4x

+ C1X + C2X + C3.

_ X5

X4

X3

2

_ X5

X4

x 3

3 2

,

2.6.41. Y -

60 +

"8 - "6

+ Clx

+C2X + C3, y.. -

60

+ "8 - "6

+"2 x

+ 2x + L

2.6.42. Y = -32 cos ~ + C1X4 + C2X3 + C3X2 + C4X + C5,

y.. = -32 cos ~2 + x 3 + X + 33. 2.6.43. Y =

1

2 + C1X 2 + C2X + C3,

171

(X + 2)

Cl = -8'C2 =

4'C3 = 4'

)) + C2X + C3.

2.6.45. Y = 48 X4 -

~X2 + C l (x arctg x -lIn(l + x

1

2

2.6.46. -Clln Ixl -

1

2x + C2X + C3.

2.6.47. Y = -lIn IxlCl(xln Ixl- 1) + C2X2 + C3X + C4.

2.6.48. Y = -Cl cos X + C2X -

x 2

1

"2

+ 4 cos 2x + C3.

2.6.49. Y = C1X3 + C2X + C3 + (x - 2)ex.

2.6.50. y = iCl sin 2x + l(1 + 2Ct}x3 + C2X 2 + C3X + C4.

2.6.53. Y = - sin(x + Ct) + C2X + C3. 2.6.54. x = Cly2 + C2Y + C3.

2.6.55. Y = 112 (x 3 + 6x2) + Clx In Ixl + C2X + C3.

X + C l = lIn It I + ~

t = y".

2.6.56. {

1

3 4t

y+C2 = -4t + 3'

4t

2.6.57. Y = C2 (xe C1X -

61 eC1X )

+ C3. YlCa3axue. I10JIOlKHTb y = e!pdx.

4

2) 2' 2.6.59.

{X=1+t(2Int-1)

"

2.6.58. y =

(x -

Y = t

2 In t,

t

= Y .

2.6.60. Y = C3e

x (! In2 X+C1 In X+C2)

.

!

d

YlCa3axue. I10JIOlKHTb y = ePa: .

2

+ Cd -

2 .:£...+1

+ C2.

2.6.61. y = (1 + C l ) In Ix

C1X + C2. 2.6.62. Y = (C1X + C l )e C1

2.6.63. Y =

~

~

3

+ 3x.

12 + "2 + C1X In Ixl + C2X + C3. 2.6.64. Y = x

2.6.65. Y = ~X2. 2.6.66. Y = lx2. 2.6.67. x = Cly2 -

ylny + C2.

2.6.68. Y = !(Cl -

2X)3/2 + C2X + C3. 2.6.69. Y = (Cle X+ l)x + C2.

2.6.70. x = -~2(Y + 2)2/3.2.6.71. Y = eX. 2.6.72. y = ~.

,

cos

X

2.6.73. Y =

(C;+1+x)24

!!

+ C2, oco6oe pellleHHe y = C.

2

+ aCl (x + 1) 2

2.6.76. x 2 + y2 + Clx + C2Y + C3 = O. 2.6.77. 2(C1Y -

1)3/2 = 3C1X + C2.

2.6.78. In y = Clex + C2e- x. 2.6.79. x = Cl + In I:~~:I·

2.6.80. Y = l(l + x 2).

2.6.81. Y = Cl ~2

+ (Cl -

C;)x + C2, oco6oe pellleHHe y = l2 (x + 1)2 + C.

1

11 + C2X + C3.

2.6.82. Y = "2x In Ixl + Clln Ix -

2.7.9. 3aBHCHMbl. 2.7.11. Y" + Y = O.

2.7.12. Y" - Y = O. 2.7.13. y" = O.

2.7.14. y" - 2y' + y = O. 2.7.15. y" -

3y' + 2y = O. 2.7.16. y" - 4y' + 4y = O.

.

7

2.7.21. Yoo = Cle

(-2+v'7)x

+ C2e

(-2-v'7)x

.2.7.22. Yoo = Cle

-x

~x

+ C2e 3 .

9

2.7.23. Yoo = CI + C2e- 25x . 2.7.24. Yoo = CI + C2e4"'.

2.7.26. Yoo

= (CI +C2x)e3x . 2.7.27. Yoo = (CI +C2x)e2X .

= (CI

3

=

2

2.7.28. Yoo

+ C2x)e'Ix. 2.7.29. Yoo

(CI + C 2 x)e-3"x.

2.7.31. Yoo

= CI cos 2x + C2 sin 2x. 2.7.32. Yoo = C I cos ~x + C2 sin ~x.

.. 33 .

y'3

C· y'3 )

- ~2.

2 7

Yoo

= (CI cos TX + 2sm

TX

e

2.7.34. Yoo = (CI cos ~x + C2 sin ~x) e ~.

2.7.35. Yoo

= (C

J3f

+ C·

J3f

)

i!.x

I cos -4-x

2sm -4-x

e4 .

3

1

2

.

7

.45. YOH =

(C

+

C)

3x

2 2

+

5

11

I

2X

e

+ gX

27 x +

27'

2 7 46

C

Ie

x

+

C

2e

2x

3

9

2

21

15

•• • YOH =

+ x +

2"x

+ 2

X - 4'

2.7.47. YOH = (CI cos X + C2 sinx)eX + x + 1.

2.7.48. YOH = Clex + C2e- 5x -l.

2.7.49. YOH = Clex + C2e-!x + (~x-

;~) e 2x .

5

2 cos x. YICa3atme. y... = A sin x + B cos x.

2.1.59. YOH = Cl + C2e - 2 x + 5 sin x -

2.1.60. YOH = Cl e-

2x

+

C2e

2x

1

2x

.

- (20 sin 2x + 1~ cos x) e

2.1.61. YOH = Cl e- 2x + C2e4x

+ ~ cos 2x + tsin 2x

YICa3anue. y...

= A cos 2x + B sin 2x.

2.1.62.

YOH

=

(C

l

2

2x

6

57

8

1~5) cosx.

+ C x)e-

+ (25 x + 125) sin x -

(25 x +

i

2

2.1.61. Yon = Cl COSX + C2sinx -

~ cos x + i

sinx.

2.1.68. YOH = Cl cos 3x + C2 sin 3x -

~~xcos3x - 112x2 sin3x.

2.1.69. YOH = (Cl cos 2x + C2 sin 2x)eX + iex sin 2x.

YICa3anue. y... = x(Acos 2x + Bsin 2x)ex.

2 110

4

C·

4

) ~x

1

~x

4

••

• YOH = (C1 cos 5x +

2sm

5X

e5 -

gXe 5 cos

5x.

2.1.12. YOH = (Cl + C2x)e'" + x 2 + 3x + 7 -

~ cos x -

~(x + 1) sinx.

2 1 13

'1/08=

C

le

-2x + C

2e

-3x + 1 2 1

65 +

x

(

1 + 1) -3x

••

fiX

+18 X -108

-2x

e .

2.1.15. YOH = Cl cos3x + C2sin3x + C18x -

55 )

e- 3x

+ 7x2.

4

2.1.16. YOH = (Cl + C2x)e X + ~ cos x + ~x2eX -le- x.

2 118

• YOH =

C

le

x

+

C

2e

-x

1

x I eX - 1

1

-X(

e

x

-

I (X

1))

.

••

+ 2 e

n - 2 - -

2e

n e -

2.1.19. YOH = Clex + C2xe x + x(ln Ixl- l)ex.

2.1.80. YOH = Cl e3x + C2xe3x -

xe3x + xe3x In Ixl.

(x - ~ tg x) sin 2x.

2.7.91. Y = xe'" + x

2

6

2

'" +

1~ sin3x -

6

+ 2.2.7.92. Y = 13 e-

13 cos3x.

Cl

C Inx

Y =

C

1 +

C

I

2.7.98. Y = x +

2-X' 2.7.99.

2 nx.

Cl

C2

HCrrOJIb30BaTb H,ll;eIO rrpe,ll;bI,ll;Jw;ero

2.7.104. Y = x

+ 2'.7.107. YlCa3a1tUe.

x

Cl

2C2

3C3_

l+x

1+4x

1+9x

alek1 '" + a2ek2 '" + a3eks ", == 0, r,n;e, HarrpHMep, al "# 0, TO

1 + (32e(k2-kd", + (33e(ks -kd '" == 0 ((32 = ~~ , (33 = ~~ , k2

- kl "# 0, k3 - kl "# O), a

rrOCJIe ,n;H<p<pepeHIJ;HpOBaHHH (k2 - kI)(32e(k2-kd", + (k3 -

kI)(33e(ks-kd", == O.

2.7.114. cos2 x -

sin2 x - cos2x + O· sin2x == O.

2.7.115. sin3a -

3sina + 4sin3 a + 0·1 == O.

2.7.116. 3cosa -

4cos3 a + cos3a + 0·5 == O.

2.7.118. e'" sin2 x + e'" cos2 x - e'" + 0 . e- 2",

== O. 2.7.119. YlCa3aHue. COCTaBHM

,ll;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe BTOpOro rropH,n;xa C 06w;HM pellleHHeM

y =Cl sin x + C2 sin 2x. ,Il;.rrH aToro H3 CHCTeMhI

{ Cl cos x + 2C2 cos2x = y'

I 2sil2x

sin x

+

-Cl sin x - 4C2 sin 2x = y"

cos 2x

cos x

RaxO,n;HM

C _

2y' sin 2x + y" cos 2x

y' sin x + y" cos x

1 -

2 cos x sin 2x - sin x cos 2x ' C2 = - 2(2

.

2

X -

.

2 }

COSXSlD

SlDXCOS

X

,

C2"

2C2

X3 y",

xy"

Y = C1 -

X2' Y

= 7·OTCIO,n;a C2 = -2-'C1 = Y

+ -2-·I10cJIe