II

10 2011 .

3

1 4

2 26

3 35

4 45

5 54

6 65

7 74

8 79

92

(, ) . :nike32@abv.bg, justmathbg.info.

(2009) (. , 2009) , (2009) , (. , 2008, 2010) () (2008, 2009)

: tan(x), cot(x), arctan(x).

. .

1-1+, 2-1, 3-5, 4-3 1-2+ [0, 6], 3-3 [pi, pi]

( ) 4-2 [pi, pi], 6-1 [1, 1] ( ) 5-2 [pi, pi]

1-3+, 2-2, 3-1, 4-1, 5-4, 6-2 2-3+, 2-4+, 8-2+ 1-4+, 3-4, 4-4, 5-5, 6-3 3-2+, 4-5, 6-4 1-5+, 5-6, 6-5, 8-1+ I 2-6+ II 1-6+, 3-6, 5-7

. .

1, 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6 {1}, 2, 3, 4, 5, 6, 7 1, 2, 3, 4, 5, {6} 1, 2

1 1.

n=1

(1)n xn4

4n 3() R .() x = 1, x = 3/2, x = 1/2 (

, ).

2. f : R R T = 6, , f(x) = x/3 1 x (0, 6]. f(x) . 3. A(1,1, 1)

f(x, y) = x2 + y2 2x+ 2y 7,

F (x, y, z) = f(x, y) + e2x2+3y2+4z2 .

:() f .

() g = gradF (A) F (A)

g

4. .

y + 4y =6

cos(2x)+ (2x 3)e2x

5. T , -.

T : z = 6 x2 y2, z2 = x2 + y2 (z 0) 6. ,

C

xdy ydx

C

x2

16+y2

9= 1,

.

10 .

II 1 5

1. n=1

(1)n xn4

4n 3() R .() x = 1, x = 3/2, x = 1/2 (

, ).

. . :

( , )n=0

un = u1 + u2 + u3 + u4 + u5 + u6 + . . .

( )n=1

un = u1 + u2 u3 + u4 u5 u6 + . . .

n=1

|un| = |u1|+ |u2|+ |u3|+ |u4|+ |u5|+ |u6|+ . . .

( )n=1

(1)n+1un = u1 u2 + u3 u4 + u5 u6 + . . .

(an , x )n=0

anxn = a0 + a1x+ a2x

2 + a3x3 + . . .+ anx

n + . . .

.- R . R

, (,). R , e (R,R), x = R x = R.

R = limn

anan+1 R = limn 1n|an|

: , , . ( x = R x = R).

II 1 6

n=1

un,

n=1

vn, un 0, vn 0; un vn =n=1

un n=1

vn

vn ,

un .

un ,

vn .

limn

nun =

< 1 = > 1 = = 1 =

limn

un+1un

=

< 1 = > 1 = = 1 =

limn

[n

(unun+1

1)]

=

< 1 = > 1 = = 1 =

. f(x). f(x) x k, :

I =

k

f(x)dx

(), . (), .

n=1

|un| , n=1

un .

n=1

|un| n=1

un , .

n=1

|un| n=1

un , .

II 1 7

|un+1| |un| limn

|un| = 0, n=1

un .

/ , - ,

n=1

|un|.

() : (R,R)

R = limn

anan+1 R = limn 1n|an|

R : (,). , x = R x = R .

,

:n=0

|anxn|. :

n=0

(1)nx3n3n!

, R =

. .

n=1

(1)n xn4

4n 3 =n=1

(1)n4n 3x

n4

an =(1)n4n 3

.

R = limn

anan+1 = limn

(1)n4n 3 4(n+ 1) 3(1)n+1 = limn

4n+ 4 34n 3 =

= limn

4n+ 14n 3 = limn

n(4 + 1/n)n(4 3/n) = 4 + 04 0 = 1

1 . . R = 1.

II 1 8

x = 1. .

n=1

(1)n 1n4

4n 3 =n=1

(1)n4n 3

. . un+1 un. (1)n+14(n+ 1) 3 , (1)n4n 3

= 1|4n+ 1| < 1|4n 3|4n + 1 > 4n 3, . , . .

limn

(1)n4n 3 = limn 1|4n 3| = limn 1|n(4 3/n)| = 0

. , |un|.

n=1

(1)n4n 3 =

n=1

1

4n 3

( n=1

1

n, .

.) , .

limn

un+1un

= limn

1

4(n+ 1) 34n 3

1= lim

n4n 34n+ 1

= limn

n(4 3/n)n(4 + 1/n)

= 1

, ( ).

I =

1

dx

4x 3 = limN N1

dx

4x 3 = limN1

4

N1

d(4x)

4x 3 =

= limN

1

4ln |4x 3||N1 = lim

N1

4(ln |4N 3| ln 1) = 1

4limN

ln |4N 3| =

, , . , x = 1 , :

n=1

|un| =n=1

1

4n 3 =

n=1

un =n=1

(1)n4n 3 =

II 1 9

x = 1/2.

n=1

(1)n (1/2)n4

4n 3 =n=1

(1)n2n4(4n 3) = 16

n=1

(1)n2n(4n 3)

. . (1)n+12n(4(n+ 1) 3) , (1)n2n(4n 3)

= 1|2n(4n+ 1)| < 1|2n(4n 3)|limn

(1)n2n(4n 3) = limn 1|2n(4n 3)| =

[1

]

= 0

n=1

un . n=1

|un|.

16n=1

(1)n2n(4n 3) = 16

n=1

1

|2n(4n 3)|

.

limn

un+1un

= limn

2n(4n 3)2n+1(4(n+ 1) 3) = limn

4n 32(4n+ 1)

=

=1

2limn

n(4 3/n)n(4 + 1/n)

=1

2

(4 04 + 0

)=

1

2< 1

n=1

|un| . , n=1

un .

x = 3/2.n=1

(1)n (3/2)n4

4n 3 =n=1

(1)n(1)n(1)43n42n4(4n 3) =

16

81

n=1

3n

2n(4n 3)

. .

limn

un+1un

= limn

3n+1

2n+1(4(n+ 1) 3)2n(4n 3)

3n= lim

n3(4n 3)2(4n+ 1)

=

=3

2limn

n(4 3/n)n(4 + 1/n)

=3

2

(4 04 + 0

)=

3

2> 1

.: : R = 1, x = 1: , x = 1/2:

, x = 3/2: .

II 1 10

2. f : R R T = 6, , f(x) = x/3 1 x (0, 6]. f(x) .. .

[pi, pi]. f(x) [pi, pi]. :

an =1

pi

pipif(x) cos(nx)dx, bn =

1

pi

pipif(x) sin(nx)dx, n = 0, 1, 2, . . .

:

f(x) =a02

+n=1

[an cos(nx) + bn sin(nx)].

, : f(x+2pi) = f(x) ( ). :

a0 =1

pi

pipif(x) cos(0)dx =

1

pi

pipif(x)dx, b0 =

1

pi

pipif(x) sin(0)dx = 0.

( ). f(x) [0, pi]. - f(x) [pi, pi] : f(x) =f(x). , F (x) = f(x) x [pi, pi]:

an =1

pi

pipiF (x) cos(nx)dx =

2

pi

pi0

F (x) cos(nx)dx, bn = 0,

an: F (x) cos(nx) ( ), bn: F (x) sin(nx) ( ). :

f(x) =a02

+n=1

an cos(nx), a0 =1

pi

pipiF (x)dx =

2

pi

pi0

F (x)dx.

: F (x+ 2pi) = F (x).

( ). f(x) [0, pi]. - f(x) [pi, pi] : f(x) =f(x). , F (x) = f(x) x [pi, pi]:

an = 0, bn =1

pi

pipiF (x) sin(nx)dx =

2

pi

pi0

F (x) sin(nx)dx,

an: F (x) cos(nx) ( ), bn: F (x)

II 1 11

sin(nx) ( ). -:

f(x) =n=1

bn sin(nx).

: F (x+ 2pi) = F (x).

. f(x) [a, b]. : l = (b a)/2. :

an =1

l

ba

f(x) cos(npix

l

)dx, bn =

1

l

ba

f(x) sin(npix

l

)dx.

:

f(x) =a02

+n=1

[an cos

(npixl

)+ bn sin

(npixl

)].

: f(x+ 2l) = f(x). :

a0 =1

l

ba

f(x)dx, b0 = 0.

( ). f(x) [0, c]. f(x) [c, c] : f(x) = f(x). , F (x) = f(x) x [c, c]:

an =1

c

ccF (x) cos

(npixc

)dx =

2

c

c0

F (x) cos(npix

c

)dx, bn = 0,

- . :

f(x) =a02

+n=1

an cos(npix

c

), a0 =

1

c

ccF (x)dx =

2

c

c0

F (x)dx.

: F (x+ 2c) = F (x).

( ). f(x) [0, c]. f(x) [c, c] : f(x) = f(x). , F (x) = f(x) x [c, c]:

an = 0, bn =1

c

ccF (x) sin

(npixc

)dx =

2

c

c0

F (x) sin(npix

c

)dx,

- . :

f(x) =n=1

bn sin(npix

c

).

: F (x+ 2c) = F (x).

II 1 12

-: [pi, pi] [c, c]; - [a, b].

. .

-4

-3

-2

-1

0

1

2

3

4

-1 0 1 2 3 4 5 6 7-10

-5

0

5

10

-5 0 5 10 15 20

: l = (6 0)/2 = 3. : f(x+ 6) = f(x). a0:

a0 =1

3

60

(x3 1)dx =

1

3

(x2

6

60

x|60)

=1

3

(62

6 0 (6 0)

)= 0.

an:

an =1

3

60

(x3 1)

cos(npix

3

)dx =

1

3

[ 60

x

3cos(npix

3

)dx

60

cos(npix

3

)dx

] : - :

I1 =

60

x

3cos(npix

3

)dx =

1

npi

60

x cos(npix

3

)d(npix

3

)=

1

npi

60

xd sin(npix

3

)=

=1

npi

[x sin

(npix3

)60 60

sin(npix

3

)dx

]=

=1

npi

[6 sin(2npi) 0 sin(0) 3

npi

60

sin(npix

3

)d(npix

3

)]=

=1

npi

[0 +

3

npicos(npix

3

)]=

3

n2pi2cos(npix

3

)60

=

=3

n2pi2[cos(2npi) cos(0)] = 3

n2pi2(1 1) = 0,

II 1 13

sin(2npi) = 0, cos(2npi) = (1)2n = 1. :cos(npi) = (1)n, sin(npi) = 0.

:

I2 =

60

cos(npix

3

)dx =

3

npisin(npix

3

)60

=3

npi[sin(2npi) sin(0)] = 0.

, ( I2), ( I1, ). an = 0. bn:

bn =1

3

60

(x3 1)

sin(npix

3

)dx =

1

3

[ 60

x

3sin(npix

3

)dx

60

sin(npix

3

)dx

].

, ( I1 ). . :

I1 =

60

x

3sin(npix

3

)dx = 1

npi

60

xd cos(npix

3

)=

= 1npi

[x cos

(npix3

)60 60

cos(npix

3

)dx

]=

= 1npi

[6 cos(2npi) 0 cos(0) 0] = 1npi

(6 0) = 6npi

.

, I2 -. bn :

bn =1

3

[ 6npi 0]

= 2npi

f(x) :

f(x) =n=1

[ 2npi

]sin(npix

3

)= 2

pi

n=1

1

nsin(npix

3

).

, , .

3. A(1,1, 1) f(x, y) = x2 + y2 2x+ 2y 7,

F (x, y, z) = f(x, y) + e2x2+3y2+4z2 .

:() f .

() g = gradF (A) F (A)

g

II 1 14

. y = f(x) x :

y = 2x =dy

dx= f x

f(x, y), x - y , y x :

f x =f

x= 2x+ 0 2 + 0 0 = 2x 2

x. y( y):

f y =f

y= 0 + 2y 0 + 2 0 = 2y + 2

df f(x, y):

df =f

xdx+

f

ydy = (2x 2)dx+ (2y + 2)dy

.

f x = 2x 2 = 0, f y = 2y + 2 = 0 =M(1,1)

.

f xx = 2, fyy = 2, f

xy = 0, f

yx = 0, f

xy = f

yx

, ( - M(1,1), x y [ ]):

1 = fxx = 2 > 0

2 =

f xx f xyf yx f yy = 2 00 2

= 4 > 0 2 > 0, ( - (- )). 1 > 0 , 1 < 0 .

: M(1,1)

II 1 15

-10 -5 0 5 10-10-5

0 5

10-50

0 50

100 150 200 250

M(1,-1)

- f(x, y, z) :

gradf =f

x

i +

f

y

j +

f

z

k =

(f

x,f

y,f

z

) :

F (x, y, z) = x2 + y2 2x+ 2y 7 + e2x2+3y2+4z2

. A(1,1, 1).

F x =F

x= 2x 2 + e2x2+3y2+4z2(4x), F x(A) = 2 2 + 4e9 = 4e9

F y =F

y= 2y + 2 + e2x

2+3y2+4z2(6y), F y(A) = 2 + 2 6e9 = 6e9

F z =F

z= 0 + e2x

2+3y2+4z2(8z), F z(A) = 8e9

F (x, y, z) A(1,1, 1) :g = gradF = 4e9i 6e9j + 8e9k = (4e9,6e9, 8e9) = e9(4,6, 8)

. f(x, y, z), gradf a (a1, a2, a3). () a 0 - ( ):

a 0 =a|a | =

(a1|a | ,

a2|a | ,

a3|a |

), |a | =

a21 + a

22 + a

23

f(x, y, z) a :f

a= a 0.gradf

II 1 16

:

f

g= g 0.gradf = g 0.g

. g .

< g, g >= |g||g| cos(](g, g)) = |g||g| cos(0) = |g|2

g 0 :

g 0 =g|g |

:

f

g= g 0.g =

g|g | .g = < g, g >|g | =

|g |2|g | = |

g |

, , g.

|g | =

(4e9)2 + (6e9)2 + (8e9)2 = e9

42 + 62 + 82 = e9

116 = 2e9

29

: f(x, y, z) M(1,1), g = gradF = e9(4,6, 8),F

g= 2e9

29.

4. .

y + 4y =6

cos(2x)+ (2x 3)e2x

. ( ): ( ), , . : , .

an, y = y(x) - n- :

a0y(n) + a1y

(n1) + . . .+ an1y + any = 0

. , :

a0kn + a1k

n1 + . . .+ an1 + an = 0

II 1 17

k1, k2, . . . , kn. :

y1 = ek1x, y2 = e

k2x, . . . yn = eknx

:

y(x) = c1ek1x + c2e

k2x + . . .+ cneknx

.

-

y1 = ek1x, y2 = e

k2x, . . . yn = eknx

y(x) = c1ek1x + c2e

k2x + . . .+ cneknx

x k e

y1 = ekx, y2 = xe

kx, y3 = x2ekx

y(x) = c1ekx + c2xe

kx + c3x2ekx

- ( , )

k1,2 = i : y1 = ex cos(x), y2 = ex sin(x)

k3,4 = i : y3 = ex cos(x), y4 = ex sin(x)y(x) = ex[c1 cos(x) + c2 sin(x)] + e

x[c3 cos(x) + c4 sin(x)]

x k e ( )

k1,2 = i : y1 = ex cos(x), y2 = ex sin(x)k3,4 = i : y3 = xex cos(x), y4 = xex sin(x)k5,6 = i : y5 = x2ex cos(x), y6 = x2ex sin(x)

y(x) = ex[c1 cos(x) + c2 sin(x)] + xex[c3 cos(x) + c4 sin(x)]+

+ x2ex[c5 cos(x) + c6 sin(x)]

II 1 18

. ( .)

y + 4y = 0

:

k2 + 4 = 0

k1,2 = 2i, : k1,2 = 02i. :

y1 = e0x cos(2x), y2 = e

0x sin(2x)

:

y(x) = c1 cos(2x) + c2 sin(2x)

. , . : - ( ). ? .

: sin / cos sin / cos - . : cos(2x) cos(2x), e3x e3x ( ).

: . ( , . .)

.

y + 4y =6

cos(2x)

:

y(x) = c1 cos(2x) + c2 sin(2x)

, . ( (x)). .

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

- x. .

II 1 19

, - . .

C 1(x) cos(2x) + C2(x) sin(2x) = 0

C 1(x)[cos(2x)] + C 2(x)[sin(2x)]

=6

cos(2x)

C1(x), C2(x) ( ) ( ). - . : , ( : ..).

, , - - , .

, .

C 1(x) cos(2x) + C2(x) sin(2x) = 0

2C 1(x) sin(2x) + 2C 2(x) cos(2x) =6

cos(2x)

.

C 1(x) =C 2(x) sin(2x)

cos(2x)

.

2C2(x) sin(2x)

cos(2x)sin(2x) + 2C 2(x) cos(2x) =

6

cos(2x)

C 2(x)sin2(2x)

cos(2x)+ C 2(x) cos(2x) =

3

cos(2x)

.

C 2(x)sin2(2x)

cos(2x)+ C 2(x)

cos2(2x)

cos(2x)=

3

cos(2x)

C 2(x)[sin2(2x) + cos2(2x)] = 3

C 2(x) = 3

, - .

C 2(x) = 3 : C1(x) =

3 sin(2x)cos(2x)

II 1 20

: C 1(x) = 3 tan(2x), C 2(x) = 3. .

C1(x) = 3

sin(2x)

cos(2x)dx =

3

2

d(cos(2x))

cos(2x)=

3

2ln | cos(2x)|

C2(x) = 3

dx = 3x

1(x) :

1(x) =3

2cos(2x) ln | cos(2x)|+ 3x sin(2x)

.

y + 4y = (2x 3)e2x

:

y(x) = c1 cos(2x) + c2 sin(2x)

, . , . . :

xex[Pm(x) cos(x) +Qn(x) sin(x)]

l = i , - . , l = i0 = :

xexPm(x)

, :

xex[Pm(x) cos(0x) +Qn(x) sin(0x)] = xex[Pm(x).1 + 0] = x

exPm(x)

x . k 6= l, : = 0 x = x0 = 1. k = l , = 1 x = x1 = x. : = 2 x = x2. .

Pm(x) m. 2x 3 : m = 1. P2(x) = Ax+B, A B . x3 + 1: P3(x) = Ax3 +Bx2 +Cx+D. ( .) Qn(x).

2x 3, x3 + 1,

- : .

P3(x) = Ax3 +Bx2 + Cx+D, Q3(x) = Ex

3 + Fx2 +Gx+H

II 1 21

!

.

y + 4y = (2x 3)e2x, k1,2 = 0 2i l = +i = 2+i0 = 2 , ( . . ); l 6= k; = 0; 2x 3 m = 1.

2(x) = x0e2x(Ax+B)

- : y, y, y.

2(x) = Axe2x +Be2x

2(x) = 2Axe2x + Ae2x + 2Be2x

2(x) = 4Axe2x + 2Ae2x + 2Ae2x + 4Be2x

e2x .

2(x) = e2x(Ax+B)

2(x) = e2x(2Ax+ A+ 2B)

2(x) = e2x(4Ax+ 2A+ 2A+ 4B)

.

e2x(4Ax+ 4A+ 4B) + 4e2x(Ax+B) = e2x(2x 3)4Ax+ 4A+ 4B + 4Ax+ 4B = 2x 3

x.

8Ax+ 4A+ 8B = 2x 3 8A = 24A+ 8B = 3A = 1/4, B = 1/2.

2(x) = e2x

(1

4x 1

2

) 2(x) -

( 1(x)). ( , .)

:

y(x) = c1 cos(2x) + c2 sin(2x) +3

2cos(2x) ln | cos(2x)|+ 3x sin(2x) + e2x

(1

4x 1

2

) .

II 1 22

5. T , -.

T : z = 6 x2 y2, z2 = x2 + y2 (z 0). : z = 6 x2 y2 ( ), z2 = x2 + y2 ( ). z 0 , Oxy. . .

-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2

-1 0 1 2

3 4 5

-20-15-10-5 0 5

10

-3 -2 -1 0 1 2 3-3-2 -1

0 1 2 3

0 1 2 3 4 5 6 7

-.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.5

0 0.5 1 1.5

2 0 1 2 3 4 5 6

(, ) (, ).

, - z, z. ( , . - .)

-, , -. : z = 6x2 y2. : z = x2 + y2, Oxy, z 0, . ( Oxy, ).

II 1 23

z : x2 + y2 z 6 x2 y2

x y . z. . : x2 = z2 y2. :

z = 6 (z2 y2) y2

z2 + z 6 = 0, z1 = 3, z2 = 2 , : x2+y2 =(3)2 x2 + y2 = 22. z = 2, Oxy. :

D :

x = r cos()y = r sin()r [0, 2], [0, 2pi]

x2+y2 = 22, Oxy , [0, 2pi]. : r [0, 2]. = r. :

G :

D :

x = r cos()y = r sin()r [0, 2], [0, 2pi]

x2 + y2 z 6 x2 y2

:G

dxdydz =

Gdxdydz

z:Gdxdydz =

D

[ 6x2y2x2+y2

dz

]dxdy =

D

[6 x2 y2

x2 + y2

]dxdy

x y r r:D

[6 x2 y2

x2 + y2

]dxdy =

=

D

[6 (r cos())2 (r sin())2

(r cos())2 + (r sin())2

]rdrd =

=

D

[6 r2

r2]rdrd =

D

[6 r2 r] rdrd

II 1 24

sin2()+cos2() = 1. , ( ).

D[6r r2 r3]drd =

2pi0

[ 20

(6r r2 r3)dr]d =

=

2pi0

d

20

(6r r2 r3)dr = |2pi0(

3r2|20 r3

3

20

r4

4

20

)=

= 2pi

(3.4 8

3 16

4

)=

32pi

3

32pi/3. .

6. ,

C

xdy ydx

C

x2

16+y2

9= 1,

.

. . F

( ):

F (x, y, z) = P (x, y, z)

i +Q(x, y, z)

j +R(x, y, z)

k

:C

P (x, y, z)dx+Q(x, y, z)dy +R(x, y, z)dz

:

C :

x = x(t)y = y(t), t [, ]z = z(t)

:C

P (x, y, z)dx+Q(x, y, z)dy +R(x, y, z)dz =

=

{P [x(t), y(t), z(t)]

dx

dt+Q[x(t), y(t), z(t)]

dy

dt+R[x(t), y(t), z(t)]

dz

dt

}dt

II 1 25

- x y, z. : , . :

C

P (x, y)dx+Q(x, y)dy =

D

(Q

x Py

)dxdy

, . .

: C

xdy ydx =C

(y)dx+ xdy, P (x, y) = y, Q(x, y) = x

:D

(Q

x Py

)dxdy =

D

(1 (1))dxdy = 2

D

dxdy

, :

x2

16+y2

9= 1

: x = 4r cos()y = 3r sin() r [0, 1]. , : [0, 2pi].

D :

x = 4r cos()y = 3r sin()r [0, 1], [0, 2pi]

= 4.3.r = 12r ( = r ).

2

D

dxdy = 2

D

12rdrd = 24

2pi0

d

10

rdr =

= 24|2pi0r2

2

10

= 24.2pi.1

2= 24pi

: pi.a.b.r2 = pi.4.3.1 = 12pi. , . 24pi.

2 1.

n=0

xn+2

n+ 2

R =? x = R x = R. 2.

z = f(x, y) = x3 + y3 12xy + 3 3.

y sin(x) = (y2 + 1) arctan(y), y(pi

2

)= 1

4.y y = 4x+ 2 sin(4x)

5. y = 4 x2y = x2 2x 6.

(C)

(2z 3x2 3y2)dl

(C) :

x = 3 cos(2t)y = 3 sin(2t), t [pi, 2pi]z = 4t

10 .

II 2 27

1. n=0

xn+2

n+ 2

R =? x = R x = R.. .

an =1

n+ 2

R = limn

anan+1 = limn

(n+ 1) + 2n+ 2 = limn n+ 3n+ 2 = limn n(1 + 3/n)n(1 + 2/n) = 1

R = 1. x = 1.n=0

1n+2

n+ 2=n=0

1

n+ 2

. . x = 1.

n=0

(1)n+2n+ 2

. .(1)n+3n+ 3 , (1)n+2n+ 2

= 1n+ 3 < 1n+ 2limn

(1)n+2n+ 2 = limn 1n+ 2 =

[1

]

= 0

x = 1. , / ( ).

: : R = 1, x = 1: , x = 1: . : () : [1, 1). 2.

z = f(x, y) = x3 + y3 12xy + 3. . .

f x = 3x2 + 0 12y + 0 = 3x2 12y

f y = 0 + 3y2 12x+ 0 = 3y2 12x

: 3x2 12y = 03y2 12x = 0

II 2 28

x .

x =3y2

12= 3

(3y2

12

)2 12y = 0

: y = 0 y = 4. M0(0, 0) M1(4, 4).

.

f xx = 6x, fyy = 6y, f

xy = 12, f yx = 12

. M0(0, 0).

1 = 6x = 6.0 = 0

2 =

6x 1212 6y = 0 1212 0

= 0 122 = 144 < 0 M0(0, 0) .

M1(4, 4).

1 = 6.4 = 24 > 0

2 =

6.4 1212 6.4 = 242 122 = 3.122 = 432 > 0

2 (1 > 0). ( , .)

-10 -5 0 5 10-10-5

0 5

10-3500-3000-2500-2000-1500-1000-500 0 500 1000

1500 2000 M0(0,0) M1(4,4)

: M1(4, 4), M0(0, 0).

3.y sin(x) = (y2 + 1) arctan(y), y

(pi2

)= 1

II 2 29

. .

y =dy

dx

x, y.

dy

dxsin(x) = (y2 + 1) arctan(y)

sin(x), x 6= kpi.dy

dx=

(y2 + 1) arctan(y)

sin(x)

(y2 + 1) arctan(y), y 6= kpi/2.dy

(y2 + 1) arctan(y)dx=

1

sin(x)

dx.

dy

(y2 + 1) arctan(y)=

dx

sin(x)

, ( , ).

dy

(y2 + 1) arctan(y)=

dx

sin(x)+ C

arctan(y) 1/(1 + y2).d(arctan(y))

arctan(y)=

dx

sin(x)+ C

, ? :

tan(x

2

)= t, sin(x) =

2t

1 + t2

dx.

x = 2 arctan(t), dx =2dt

1 + t2

:dx

sin(x)=

1 + t2

2t

2dt

1 + t2=

dt

t= ln |t| = ln

tan(x2

)

II 2 30

:ln | arctan(y)| = ln

tan(x2

)+ C y(pi/2) = 1, y 1 x pi/2:

ln | arctan(1)| = lntan(pi

4

)+ Cln(pi

4

)= ln 1 + C = ln

(pi4

)= 0 + C = C = ln

(pi4

) :

ln | arctan(y)| = lntan(x

2

)+ ln(pi4

)= ln | arctan(y)| = ln

pi4

tan(x

2

)arctan(y) =

pi

4tan(x

2

)= y = tan

(pi4

tan(x

2

)) :

y = tan(pi

4tan(x

2

))

4.y y = 4x+ 2 sin(4x)

.

y + P (x)y = Q(x),

y = y(x), P (x) Q(x) x. :

y(x) = eP (x)dx

(C +

eP (x)dxQ(x)dx

) (, ):

y(x) = exp

(P (x)dx

)[C +

exp

(P (x)dx

)Q(x)dx

] y, y y, y y, yIV , .. , . - :y = u(x). : y = (y) = (u(x)) = u. :

u u = 4x+ 2 sin(4x)

II 2 31

, P (x) = 1, Q(x) = 4x+ 2 sin(4x). :

u(x) = e(1)dx

(C +

e(1)dx[4x+ 2 sin(4x)]dx

)u(x) = ex

(C +

ex[4x+ 2 sin(4x)]dx

)u(x) = ex

(C + 4

xexdx+ 2

ex sin(4x)dx

) . .

I1 =

xexdx =

xd(ex) =

(xex

exdx

)=

= (xex +

d(ex)

)= (xex + ex) = ex(x+ 1)

( , ).

I2 =

ex sin(4x)dx =

sin(4x)d(ex) =

= (ex sin(4x)

exd(sin(4x))

)=

= (ex sin(4x) 4

ex cos(4x)dx

)=

= (ex sin(4x) + 4

cos(4x)d(ex)

)=

= (ex sin(4x) + 4

(ex cos(4x)

exd cos(4x)

))=

= (ex sin(4x) + 4ex cos(4x) + 16

ex sin(4x)dx

)I2 = ex sin(4x) 4ex cos(4x) 16I2

17I2 = ex sin(4x) 4ex cos(4x)I2 = e

x

17(sin(4x) + 4 cos(4x))

II 2 32

u(x).

u(x) = ex (C + 4I1 + 2I2)

u(x) = ex(C + 4[ex(x+ 1)] + 2

[ex

17(sin(4x) + 4 cos(4x))

])u(x) = ex

(C 4ex(x+ 1) 2e

x

17(sin(4x) + 4 cos(4x))

)u(x) = Cex 4x 4 2

17sin(4x) 8

17cos(4x)

y = u(x). y, . . .

y =d2y

dx2=

d

dx

(dy

dx

)=

d

dxy =

dy

dx

:dy

dx= Cex 4x 4 2

17sin(4x) 8

17cos(4x)

y x ( ). dx.

dy =[Cex 4x 4 2

17sin(4x) 8

17cos(4x)

]dx

, .dy =

[Cex 4x 4 2

17sin(4x) 8

17cos(4x)

]dx+ C1

y ( ).

y = Cexdx 4

xdx 4

dx 2

17

sin(4x)dx 8

17

cos(4x)dx+ C1

y = Cex 4x2

2 4x+ 2

17

1

4cos(4x) 8

17

1

4sin(4x) + C1

y = Cex 2x2 4x+ 134

cos(4x) 217

sin(4x) + C1

. :

y =d1y

dx1=

d

dxy =

dy

dx

.

dy

dx= Cex 2x2 4x+ 1

34cos(4x) 2

17sin(4x) + C1

II 2 33

dx

dy =

[Cex 2x2 4x+ 1

34cos(4x) 2

17sin(4x) + C1

]dx

, .dy =

[Cex 2x2 4x+ 1

34cos(4x) 2

17sin(4x) + C1

]dx+ C2

y = C

exdx2

x2dx4

xdx+

1

34

cos(4x)dx 2

17

sin(4x)dx+C1

dx+C2

y = Cex 2x3

3 4x

2

2+

1

34

1

4sin(4x) +

2

17

1

4cos(4x) + C1x+ C2

y = Cex 23x3 2x2 + 1

136sin(4x) +

1

34cos(4x) + C1x+ C2

.

6. (C)

(2z 3x2 3y2)dl

(C) :

x = 3 cos(2t)y = 3 sin(2t), t [pi, 2pi]z = 4t

. : t [5, 5] t [pi, 2pi].

-3 -2 -1 0 1 2 3-3-2 -1

0 1 2 3

-20-15-10-5

0 5 10 15 20

-3 -2 -1 0 1 2 3-3-2 -1

0 1 2 3

12 14 16 18 20 22 24 26

. f(x, y, z) . C, - :

C :

x = x(t)y = y(t), t [, ]z = z(t)

II 2 34

:

C

f(x, y, z)dl =

f [x(t), y(t), z(t)]

(dx

dt

)2+

(dy

dt

)2+

(dz

dt

)2dt

, . z, .

t:

dx

dt=

d

dt[3 cos(2t)] = 6 sin(2t), dy

dt=

d

dt[3 sin(2t)] = 6 cos(2t),

dz

dt=

d

dt[4t] = 4

:C

(2z 3x2 3y2)dl =

=

2pipi

[2.4t 3(3 cos(2t))2 3(3 sin(2t))2]

(6 sin(2t))2 + (6 cos(2t))2 + 42 dt =

=

2pipi

[8t 27 cos2(2t) 27 sin2(2t)]

36 sin2(2t) + 36 cos2(2t) + 16 dt =

=

2pipi

(8t 27)36 + 16 dt =

52

2pipi

(8t 27)dt = 2

13(4t2|2pipi 27t|2pipi ) =

= 2

13[4(4pi2 pi2) 27(2pi pi)] = 2

13(12pi2 27pi) = 6

13pi(4pi 9).

.

5. y = 4 x2y = x2 2x. y, .

x2 2x = 4 x2

2x2 2x 4 = 0x2 x 2 = 0

2 1. x = 2 y = 0, x = 1 y = 3.: x = 2, y = 0; x = 1, y = 3.

3 1. z = x3 + y3 15xy. 2. , -

D

(1 y

2

x2

)dxdy,

D x2 + y2 pi2. 3. :

f(x) =

{x pi < x < 0,0 0 < x < pi.

4. .

y + y 2y = cos(x) 3 sin(x)

5. .

n=1

xn

n2 + 1

6. K

ydx+ 2xdy,

K , , -

x

3+y

2= 1, x

3 y

2= 1.

10 .

II 3 36

5. .

n=1

xn

n2 + 1

. .

an =1

n2 + 1

R = limn

anan+1 = limn

(n+ 1)2 + 1n2 + 1 = limn n2 + 2n+ 2n2 + 1 =

= limn

n2(1 + 2/n+ 2/n2)

n2(1 + 1/n2)=

1 + 0 + 0

1 + 0= 1

R = 1. x = 1.

n=1

1n

n2 + 1=n=1

1

n2 + 1

. ( ).

1

dx

x2 + 1= lim

N

N1

dx

x2 + 1= lim

Narctan(x)|N1 =

= limN

(arctan(N) arctan(1)) = pi2 pi

4=pi

4

, . x = 1.

n=1

(1)nn2 + 1

. . (1)n+1(n+ 1)2 + 1 , (1)nn2 + 1

= 1n2 + 2n+ 2 < 1n2 + 1limn

(1)nn2 + 1 = limn 1n2 + 1 =

[1

]

= 0

.: : [1, 1].

II 3 37

3. :

f(x) =

{x pi < x < 0,0 0 < x < pi.

. .

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4-10

-5

0

5

10

-10 -5 0 5 10 15

: f(x+ 2pi) = f(x). , . a0:

a0 =1

pi

0pixdx = 1

pi

x2

2

0pi

= 1pi

[0 pi

2

2

]=pi

2.

an:

an =1

pi

0pix cos(nx)dx = 1

npi

0pixd sin(nx) =

= 1npi

[x sin(nx)|0pi

0pi

sin(nx)dx

]=

= 1npi

[0 sin(0) (pi) sin(npi) 1

n

0pi

sin(nx)d(nx)

]=

= 1npi

[0 0 + 1

ncos(nx)|0pi

]= 1

n2pi[cos(0) cos(npi)] =

= 1n2pi

[1 (1)n] = (1)n 1

n2pi,

II 3 38

sin(npi) = sin(npi) = 0, cos(npi) = cos(npi) = (1)n. bn:

bn =1

pi

0pix sin(nx)dx = 1

npi

0pixd cos(nx) =

=1

npi

[x cos(nx)|0pi

0pi

cos(nx)dx

]=

=1

npi

[0 cos(0) (pi) cos(npi) 1

nsin(nx)|0pi

]=

=1

npi

[0 + pi(1)n 1

n(sin(0) sin(npi))

]=

(1)nn

,

. :

f(x) =pi

4+n=1

[(1)n 1

n2picos(nx) +

(1)nn

sin(nx)

].

1. z = x3 + y3 15xy.. 2, 2. . .

f x = 3x2 + 0 15y = 3x2 15y

f y = 0 + 3y2 15x = 3y2 15x

:M0(0, 0) M1(5, 5).

.

f xx = 6x, fyy = 6y, f

xy = 15, f yx = 15

. M0(0, 0).

1 = 6x = 6.0 = 0

2 =

6x 1515 6y = 0 1515 0

= 0 152 = 225 < 0 M0(0, 0) .

M1(5, 5).

1 = 6.5 = 30 > 0

2 =

6.5 1515 6.5 = 302 152 = 3.152 = 675 > 0

2 (1 > 0). ( , .)

II 3 39

-10 -5 0 5 10-10-5

0 5

10-4000-3000-2000-1000

0 1000 2000

M0(0,0) M1(5,5)

: M1(5, 5), M0(0, 0).

4. .

y + y 2y = cos(x) 3 sin(x). :

y + y 2y = 0 :

k2 + k 2 = 0 k1 = 1 k2 = 2. :

y(x) = c1ex + c2e

2x

:

y + y 2y = e0x[1 cos(x) + (3) sin(x)] , sin / cos .

l = i = 0 1i = i, :k 6= l, : = 0. : 1 3, : A B.

( . 19):

xex[Pm(x) cos(x) +Qn(x) sin(x)]

:

(x) = x0e0x[A cos(1x) +B sin(1x)] = A cos(x) +B sin(x)

II 3 40

.

(x) = A sin(x) +B cos(x)

(x) = A cos(x)B sin(x) :

A cos(x)B sin(x) A sin(x) +B cos(x) 2[A cos(x) +B sin(x)] = cos(x) 3 sin(x)

:

(B 3A) cos(x) + (A 3B) sin(x) = 1 cos(x) + (3) sin(x) B 3A = 1A 3B = 3A = 0, B = 1.

(x) = sin(x)

:

y(x) = c1ex + c2e

2x + sin(x)

2. , -

D

(1 y

2

x2

)dxdy,

D x2 + y2 pi2.. . x2 + y2 = pi2.

: x = r cos()y = r sin() r , . , [0, 2pi] ( ). , - . (: (x1)2+(y1)2 = 1 , [0, pi/2].) r.

(r cos())2 + (r sin())2 = pi2, r2 = pi2, r = pi

II 3 41

, r [0, pi]. D, D.

D :

r [0, pi] [0, 2pi] , . x y r .

=

xr yrx y = cos() sin()r sin() r cos()

= r cos2() + r sin2() = r( , .) -, , r. , = r. :

D

(1 y

2

x2

)dxdy =

D

(1 r

2 sin2()

r2 cos2()

)rdrd

:

1 r2 sin2()

r2 cos2()= 1 1 cos

2()

cos2()= 1 1

cos2()+ 1 = 2 1

cos2()

:D

(2 1

cos2()

)rdrd =

D

(2r r

cos2()

)drd

- : - . . . , . .

D

(2r r

cos2()

)drd =

2pi0

[ pi0

(2r r

cos2()

)dr

]d

. . 2pi

0

[ pi0

2rdr

]d =

2pi0

d

pi0

2rdr = |2pi0 r2|pi0 = 2pipi2 = 2pi3

e . (- , ). 2pi

0

[ pi0

( r

cos2()

)dr

]d =

pi0

rdr

2pi0

d

cos2()=

= r2

2

pi0

tan()|2pi0 = pi2

2(tan(2pi) tan(0)) = 0

2pi3. .

II 3 42

6. K

ydx+ 2xdy,

K , , -

x

3+y

2= 1, x

3 y

2= 1.

. .

-5-4-3-2-1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

A(3,0)

B(0,2)

C(-3,0)

D(0,-2)

. ( ), . , x: y = y(x) y: x = x(y).

x t, x (y dy ).

y , x dx .

x y ( ):

x

3+y

2= 1 : y = 2 2

3x x = 3 3

2y

x

3 y

2= 1 : y = 2 + 2

3x x = 3 + 3

2y

x

3+y

2= 1 : y = 2 2

3x x = 3 3

2y

x

3 y

2= 1 : y = 2 + 2

3x x = 3 +

3

2y

II 3 43

( ). - x, y = y(x) . dy x.

AB : x [3, 0], y = 2 23x, dy = 2

3dx

BC : x [0,3], y = 2 + 23x, dy =

2

3dx

CD : x [3, 0], y = 2 23x, dy = 2

3dx

DA : x [0, 3], y = 2 + 23x, dy =

2

3dx

( , - ):

K

ydx+ 2xdy =

AB

+

BC

+

CD

+

DA

:AB

ydx+ 2xdy =

03

(2 2

3x

)dx+ 2x

(2

3

)dx =

03

(2 6

3x

)dx =

=

03

(2 2x) dx = 2x|03 x2|03 = 2(0 3) (0 9) = 6 + 9 = 3BC

ydx+ 2xdy =

30

(2 +

2

3x

)dx+ 2x

(2

3

)dx =

30

(2 +

6

3x

)dx =

=

30

(2 + 2x) dx = 2x|30 x2|30 = 2(3 0) (9 0) = 6 + 9 = 3CD

ydx+ 2xdy =

03

(2 2

3x

)dx+ 2x

(2

3

)dx =

03

(2 6

3x

)dx =

=

03

(2 2x) dx = 2x|03 x2|03 = 2(0 + 3) (0 9) = 6 + 9 = 3DA

ydx+ 2xdy =

30

(2 + 2

3x

)dx+ 2x

(2

3

)dx =

30

(2 + 6

3x

)dx =

=

30

(2 + 2x) dx = 2x|30 + x2|30 = 2(3 0) + (9 0) = 6 + 9 = 3

12. ( ).

y, x = x(y) . dx

II 3 44

y.

AD : x = 3 +3

2y, y [0,2], dx = 3

2dy

DC : x = 3 32y, y [2, 0], dx = 3

2dy

CB : x = 3 + 32y, y [0, 2], dx = 3

2dy

BA : x = 3 32y, y [2, 0], dx = 3

2dy

( , ):

K

ydx+ 2xdy = K

ydx+ 2xdy = AD

DC

CB

BA

:AD

ydx+ 2xdy =

20

y

(3

2

)dy + 2

(3 +

3

2y

)dy =

20

(6 +

9

2y

)dy =

= 6y|20 +9

2

y2

2|20 = 6(2 0) +

9

4(4 0) = 12 + 9 = 3

DC

ydx+ 2xdy =

02y

(3

2

)dy + 2

(3 3

2y

)dy =

02

(6 9

2y

)dy =

= 6y|02 9

2

y2

2|02 = 6(0 + 2)

9

4(0 4) = 12 + 9 = 3

CB

ydx+ 2xdy =

20

y

(3

2

)dy + 2

(3 + 3

2y

)dy =

20

(6 + 9

2y

)dy =

= 6y|20 +9

2

y2

2|20 = 6(2 0) +

9

4(4 0) = 12 + 9 = 3

BA

ydx+ 2xdy =

02

y

(3

2

)dy + 2

(3 3

2y

)dy =

02

(6 9

2y

)dy =

= 6y|02 9

2

y2

2|02 = 6(0 2)

9

4(0 4) = 12 + 9 = 3

:K

ydx+ 2xdy = 12

, , , , :

K

ydx+ 2xdy = K

ydx+ 2xdy = (12) = 12

: 12.

4 1. z = x2y3(6 x y) M(1, 2).

2. f(x) = pi2 x2 [pi, pi]. 3. R - x = R x = R.

n=1

(1)n+1xn

n2

4. .

y + 4y + 4y = xe2x

5.

J =

D

(x2 + y2)dxdy,

D , y = x, y = x + 3, y = 3,y = 9.

6. C

2xydx+ x2dy,

C y2 = x A(0, 0) B(1, 1)

10 .

II 4 46

3. R - x = R x = R.

n=1

(1)n+1xn

n2

. .

an(1)n+1n2

R = limn

anan+1 = limn

(1)n(n+ 1)2n2(1)n+1 = limn n2 + 2n+ 2n2 =

= limn

n2(1 + 2/n+ 2/n2)

n2.1=

1 + 0 + 0

1= 1

R = 1. x = 1.

n=1

(1)n+1 1n

n2=n=1

(1)n+1n2

. . (1)n+2(n+ 1)2 , (1)n+1n2

= 1n2 + 2n+ 1 < 1n2limn

(1)n+1n2 = limn 1n2 =

[1

]

= 0

. x = 1.

n=1

(1)n+1 (1)n

n2=n=1

1n2

. ( ).

1

dxx2

= limN

N1

x2dx = limN

x1|N1 = limN

1

x

N1

=

= limN

(1

N 1)

= 0 1 = 1

, .: : [1, 1].

II 4 47

2. f(x) = pi2 x2 [pi, pi].. .

0

2

4

6

8

10

-6 -4 -2 0 2 4 6

0

2

4

6

8

10

-10 -5 0 5 10 15

: f(x+ 2pi) = f(x). , : f(x) = f(x):

f(x) = pi2 (x)2 = pi2 x2 = f(x).

( ). bn = 0. - a0 an.

a0 an ( bn ):

an =1

pi

pipif(x) cos(nx)dx =

2

pi

pi0

f(x) cos(nx)dx.

a0:

a0 =2

pi

pi0

(pi2 x2)dx = 2pi

[pi2x|pi0

x3

3

pi0

]=

=2

pi

[pi2pi 0

(pi3

3 0)]

=2

pi

[pi3 pi

3

3

]=

4

3pi2.

an:

an =2

pi

pi0

(pi2 x2) cos(nx)dx = 2pi

[ pi0

pi2 cos(nx)dx pi0

x2 cos(nx)dx

].

II 4 48

:

I1 =

pi0

pi2 cos(nx)dx =pi2

nsin(nx)|pi0 =

pi2

n[sin(npi) sin(0)] = 0,

sin(npi) = 0. :

I2 =

pi0

x2 cos(nx)dx =1

n

pi0

x2d sin(nx) =

=1

n

[x2 sin(nx)|pi0

pi0

sin(nx)d(x2)

]=

=1

n

[pi2 sin(npi) 0 sin(0)

pi0

2x sin(nx)dx

]=

=1

n

[0 0 + 2

n

pi0

xd cos(nx)

]=

=2

n2

[x cos(nx)|pi0

pi0

cos(nx)dx

]=

=2

n2

[pi cos(npi) 0 cos(0) 1

nsin(nx)|pi0

]=

=2

n2

[pi(1)n 1

n(sin(npi) sin(0))

]=

2pi(1)nn2

,

, cos(npi) = (1)n. an:

an =2

pi

[0 2pi(1)

n

n2

]=

4(1)n(1)n2

=4(1)n+1

n2.

:

f(x) =2

3pi2 + 4

n=1

(1)n+1n2

cos(nx).

1. z = x2y3(6 x y) M(1, 2).

. : df =f

xdx+

f

ydy.

z = f(x, y) = 6x2y3 x3y3 x2y4

f x =f

x= 12xy3 3x2y3 2xy4

f y =f

y= 18x2y2 3x3y2 4x2y3

II 4 49

.

f x(M) = 12.8 3.8 2.16 = 8(12 3 4) = 8.5 = 40f y(M) = 18.4 3.4 4.8 = 4(18 3 8) = 4.7 = 28

df = 40dx+ 28dy

-10 -5 0 5 10-10-5

0 5

10-3e+006-2.5e+006-2e+006-1.5e+006-1e+006-500000

0 500000 1e+006 M(1,2)

4. .

y + 4y + 4y = xe2x

. :

y + 4y + 4y = 0

:

k2 + 4k + 4 = 0 = (k + 2)2 = 0 k1,2 = 2. :

y(x) = c1e2x + c2xe2x

:

y + 4y + 4y = xe2x

- . ( , -; . 17. : x2e2x(Ax+B)).

. :

(x) = C1(x)e2x + C2(x)xe2x

II 4 50

. C 1(x)e2x + C 2(x)xe2x = 0C 1(x)[2e2x] + C 2(x)[2xe2x + e2x] = xe2x C 1(x) = C 2(x)x2C 1(x) 2C 2(x)x+ C 2(x) = x C 1(x) .

2[C 2(x)x] 2C 2(x)x+ C 2(x) = xC 2(x) = x

: C 1(x) = x2, C 2(x) = x. .

C1(x) = x2dx = x

3

3, C2(x) =

xdx =

x2

2

(x):

(x) = x3

3e2x +

x2

2xe2x =

(2x

3

6+

3x3

6

)e2x =

x3

6e2x

:

y(x) = c1e2x + c2xe2x +

x3

6e2x

5.

J =

D

(x2 + y2)dxdy,

D , y = x, y = x + 3, y = 3,y = 9.

. .

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

y = 3

y = 9

y = xy = x + 3

II 4 51

y ( Oy), : y [3, 9]x [y 3, y] x ( Ox), : x [0, 3]y [3, x+ 3] ,

x [3, 6]y [x, x+ 3] , x [6, 9]y [x, 9]

.

D :

y [3, 9]x [y 3, y] . , - x y. x , y . : 9

3

[ yy3

(x2 + y2)dx

]dy

x y : 9

3

[ yy3

(x2 + y2)dx

]dy =

93

[x3

3

yy3

+ y2x|yy3]dy

x, y: 93

[1

3[y3 (y 3)3] + y2[y (y 3)]

]dy

, . 93

[1

3[y3 (y 3)3] + y2[y (y 3)]

]dy =

93

1

3[y3 (y 3)3] + 3y2dy =

=

93

1

3[y3 (y3 3y23 + 3y32 33)] + 3y2dy =

=

93

3y2 9y + 9 + 3y2dy = 93

(6y2 9y + 9)dy =

= 2y3|93 9y2

2

93

+ 9y|93 = 2(93 33)9

2(92 32) + 9(9 3) =

= 2(3333 33) 92

(3232 32) + 9.6 = 2.26.33 92

8.32 + 54 =

= 2.26.27 9.4.9 + 54 = 1404 324 + 54 = 1134

II 4 52

1134. - .

I1 :

x [0, 3]y [3, x+ 3] , I2 : x [3, 6]y [x, x+ 3] , I3 :

x [6, 9]y [x, 9] y, x. , y x .

I1 =

30

[ x+33

(x2 + y2)dy

]dx =

30

[x2y|x+33 +

y3

3

x+33

]dx =

=

30

[x2(x+ 3 3) + 1

3[(x+ 3)3 33]

]dx =

=

30

[x3 +

1

3(x3 + 9x2 + 27x+ 27 27)

]dx =

=

30

[x3 +

x3

3+ 3x2 + 9x

]dx =

30

[4x3

3+ 3x2 + 9x

]dx =

=x4

3

30

+ x3|30 +9x2

2

30

=34

3+ 33 +

9.32

2= 2.33 +

34

2= 94, 5

I2 =

63

[ x+3x

(x2 + y2)dy

]dx =

63

[x2y|x+3x +

y3

3

x+3x

]dx =

=

63

[x2(x+ 3 x) + 1

3[(x+ 3)3 x3]

]dx =

=

63

[3x2 +

1

3(x3 + 9x2 + 27x+ 27 x3)

]dx =

=

63

[3x2 + 3x2 + 9x+ 9

]dx =

63

[6x2 + 9x+ 9

]dx =

= 2x3|63 +9x2

2

63

+ 9x|63 = 2(63 33) +9

2(62 32) + 9(6 3) =

= 2(2333 33) + 92

(2232 32) + 9.3 = 2.7.33 + 92

3.32 + 27 =

= 15.27 +9

227 = 526, 5

II 4 53

I3 =

96

[ 9x

(x2 + y2)dy

]dx =

96

[x2y|9x +

y3

3

9x

]dx =

=

96

[x2(9 x) + 1

3(93 x3)

]dx =

96

[9x2 x3 + 35 x

3

3

]dx =

=

96

[9x2 4x

3

3+ 35

]dx = 3x3|96

x4

3

96

+ 35x|96 =

= 3(93 63) 13

(94 64) + 35.3 = 3(3333 2333) 13

(3434 2434) + 36 =

= 3.19.33 13

65.34 + 36 = 33(3.19 65 + 33) = 33.19 = 513 : 94, 5 + 526, 5 + 513 = 1134. : 1134. 6.

C

2xydx+ x2dy,

C y2 = x A(0, 0) B(1, 1). y2 = x.

-5-4-3-2-1 0 1 2 3 4 5

0 5 10 15 20

. t. : y = t, x = t2. t: A(0, 0) B(1, 1), t [0, 1]. x y t:

x = t2, dx = 2tdt, y = t, dy = dt

:C

2xydx+ x2dy =

10

2t2 t 2tdt+ t4dt =

10

4t4 + t4dt =

10

5t4dt = t5|10 = 1

5 1. - .

2. :

f(x) =

pi

4 x, 0 < x pi

2,

pi4,pi

2< x < pi.

3. , (x, y) , T (x, y) = ex cos(y) + ey cos(x). T (x, y) O(0, 0) .

4. z(x, y) = x3+y315xy .

5. y + y = x2 + 1.

6. G

xyzdxdydz,

G ,

G : {4z2 x2 + y2, x 0, y 0, 0 z 1}.

7. C

(x+ y)dx (x y)dy,

C x2 + y2 2y = 0.: 1,2,6: 10, 3,7: 5, 4: 8, 5: 12.

II 5 55

2. :

f(x) =

pi

4 x, 0 < x pi

2,

pi4,

pi

2< x < pi.

. .

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

, . Oy.

, ; , . f(x) = f(x). ( ):

f(x) =

pi

4, pi < x < pi

2,

pi4 x, pi

2 x < 0.

:

f(x) = pi4

= f(x),

f(x) = pi4 (x) = pi

4+ x =

(pi4 x)

= f(x). . -. -:

f(x) :[pi,pi

2

] pi

4,[pi

2, 0] pi

4 x,

f(x) :[0,pi

2

] pi

4 x,

[pi2, pi] pi

4.

II 5 56

F (x) :

F (x) =

pi/4, x [pi,pi/2]pi/4 x, x [pi/2, 0]pi/4 x, x [0, pi/2]pi/4, x [pi/2, pi]

.

: F (x+ 2pi) = F (x).

-4

-2

0

2

4

-10 -5 0 5 10 15

, a0 = 0 an = 0. bn. bn ( a0 an ):

bn =1

pi

pipiF (x) sin(nx)dx =

2

pi

pi0

F (x) sin(nx)dx =2

pi

pi0

f(x) sin(nx)dx,

f(x) [0, pi], f(x) [pi, 0]., f(x),

f(x). , , f(x) . , .

bn:

bn =2

pi

[ pi/20

(pi4 x)

sin(nx)dx+

pipi/2

(pi

4

)sin(nx)dx

]=

=2

pi

[pi

4

pi/20

sin(nx)dx pi/20

x sin(nx)dx pi4

pipi/2

sin(nx)dx

]=

=1

2

[ pi/20

sin(nx)dx pipi/2

sin(nx)dx

] 2pi

pi/20

x sin(nx)dx.

II 5 57

, :

I1 =

pi/20

sin(nx)dx =1

n

pi/20

sin(nx)d(nx) =

= 1n

cos(nx)|pi/20 = 1

n

[cos(npi

2

) cos(0)

]=

1

n

[1 cos

(npi2

)].

:

I2 =

pipi/2

sin(nx)dx =1

n

pipi/2

sin(nx)d(nx) =

= 1n

cos(nx)|pipi/2 = 1

n

[cos(npi) cos

(npi2

)]=

1

n

[cos(npi

2

) (1)n

].

, I1 I2:

I1 I2 = 1n

[1 cos

(npi2

)] 1n

[cos(npi

2

) (1)n

]=

1

n

[1 + (1)n 2 cos

(npi2

)].

bn :

bn =1

2n

[1 + (1)n 2 cos

(npi2

)] 2pi

pi/20

x sin(nx)dx.

:

I3 =

pi/20

x sin(nx)dx =1

n

pi/20

x sin(nx)d(nx) = 1n

pi/20

xd cos(nx) =

= 1n

[x cos(nx)|pi/20

pi/20

cos(nx)dx

]=

= 1n

[pi

2cos(npi

2

) 0 cos(0) 1

n

pi/20

cos(nx)d(nx)

]=

= 1n

[pi

2cos(npi

2

) 1n

sin(nx)|pi/20]

=

= 1n

[pi

2cos(npi

2

) 1n

(sin(npi

2

) sin(0)

)]=

1

n

[1

nsin(npi

2

) pi

2cos(npi

2

)].

bn, 1/n :

bn =1

2n

[1 + (1)n 2 cos

(npi2

)] 2pin

[1

nsin(npi

2

) pi

2cos(npi

2

)]=

=1

n

{1

2

[1 + (1)n 2 cos

(npi2

)] 2pi

[1

nsin(npi

2

) pi

2cos(npi

2

)]}=

=1

n

[1

2+

(1)n2 cos

(npi2

) 2pin

sin(npi

2

)+ cos

(npi2

)]=

=1

n

[1

2+

(1)n2 2pin

sin(npi

2

)].

II 5 58

:

F (x) =n=1

1

n

[1

2+

(1)n2 2pin

sin(npi

2

)]sin(nx).

, . n : n = 2k.: (1)2k = 1, sin(2kpi/2) = sin(kpi) = 0. :

F (x) =k=1

1

2k

[1

2+

1

2 2pin

0

]sin(2kx) =

k=1

1

2k[1 0] sin(2kx) =

k=1

sin(2kx)

2k.

n : n = 2k + 1. : (1)2k+1 = 1, sin[(2k + 1)pi/2] =(1)k. :

F (x) =k=1

1

2k + 1

[1

2 1

2 2pi(2k + 1)

(1)k]

sin[(2k + 1)x] =

=k=1

2(1)k(1)pi(2k + 1)2

sin[(2k + 1)x] =2

pi

k=1

(1)k+1(2k + 1)2

sin[(2k + 1)x].

: sin(npi) = 0, cos(npi) = (1)n,

cos(npi) : n = 2k = cos(2kpi) = (1)2k = 1,

cos(npi) : n = 2k + 1 = cos[(2k + 1)pi] = (1)2k+1 = 1.

cos(npi

2

): n = 2k = cos

(2kpi

2

)= cos(kpi) = (1)k,

cos(npi

2

): n = 2k + 1 = cos

[(2k + 1)pi

2

]= 0,

sin(npi

2

): n = 2k = sin

(2kpi

2

)= sin(kpi) = 0,

sin(npi

2

): n = 2k + 1 = sin

[(2k + 1)pi

2

]= (1)k.

4. z(x, y) = x3+y315xy .

. , 3, 1.

5. y + y = x2 + 1.

II 5 59

. :

y + y = 0

:

k2 + 1 = 0

k1,2 = 0i. :

y(x) = e0x[c1 cos(1x) + c2 sin(1x)] = c1 cos(x) + c2 sin(x)

:

y + y = x2 + 1

sin / cos, . . :

y + y = e0x(x2 + 1)

l = i = 0 i0, k = 0 i, l 6= k, : = 0. : m = 2, : Ax2 +Bx+ C.

( . 19):

xexPm(x)

:

(x) = x0e0x(Ax2 +Bx+ C) = Ax2 +Bx+ C

:(x) = 2Ax+B

(x) = 2A

:

2A+ Ax2 +Bx+ C = x2 + 1

Ax2 +Bx+ 2A+ C = x2 + 1

: A = 1B = 02A+ C = 1

A = 1, C = 1.(x) = x2 1

:

y(x) = c1 cos(x) + c2 sin(x) + x2 1

II 5 60

6. G

xyzdxdydz,

G ,

G : {4z2 x2 + y2, x 0, y 0, 0 z 1}.

. : 4z2 = x2 + y2 , x = 0, y = 0, z = 0 z = 1 . , z = 1, . .

-10 -8 -6 -4 -2 0 2 4 6 8 10-10-8-6-4

-2 0 2 4

6 8 10

-6-4-2 0 2 4 6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.5

0 0.5 1 1.5

2 0

0.2 0.4 0.6 0.8

1

-.

0 0.5 1 1.5 2 0 0.5

1 1.5

2 0

0.2 0.4 0.6 0.8

1

( ) ( ). z = 1.

II 5 61

( -, ). z ( ) z = 1. z:

z = x2 + y2

4

z = 0, . z :

x2 + y2

2 z 1

x y. z . :

D :

x = r cos()y = r sin() z = 1, - : 4 = x2 + y2. : r [0, 2].

. , . ( x y), . : [0, pi/2].

= r. :

G :

D :

x = r cos()y = r sin()r [0, 2], [0, pi/2]

x2 + y2

2 z 1

z: G

xyzdxdydz =

D

[ 1x2+y2

2

xyzdz

]dxdy

:

(2z) = (2z1/2) = 2

1

2z1/21 = z1/2 =

1z

z:

D

[xy 2z1

x2+y2

2

]dxdy =

D

2xy1

x2 + y2

2

dxdy ==

D

[2xy

(1

4x2 + y2

2

)]dxdy

II 5 62

x y r r:D

[2r cos()r sin()

(1

4r2 cos2() + r2 sin2()

2

)]rdrd

sin2() + cos2() = 1 r :D

[2r3 cos() sin()

(1

4r22

)]drd

r ( ), : pi/20

2 cos() sin()d

20

r3(

1r2

)dr

sin(2) = 2 sin() cos(): pi/20

sin(2)d

20

(r3 r

3r

2

)dr =

=1

2

pi/20

sin(2)d(2)

20

(r3 r

7/2

2

)dr =

= 12

cos(2)pi/20

(r4

4

20

12

2

9r9/220

)=

= 12

(cos(pi) cos(0))(

16

4 0 2

9

2(29/2 0)

)=

= 12

(1 1)(

4

2

929/2

)=

= 12

(2)(

4 210/2

9

)= 4 2

5

9=

36

9 32

9=

4

9

: 4/9.

7. C

(x+ y)dx (x y)dy,

C x2 + y2 2y = 0.. x2 + y2 = 2y.

II 5 63

-1

0

1

2

3

-2 -1 0 1 2

(0, 1) 1.

x2 + y2 2y + 1 1 = 0 = x2 + (y 1)2 = 1

( ): x = r cos()y = r sin() , : [0, pi]. :

r2 cos2() + r2 sin2() = 2r sin()

r2 = 2r sin()

r = 2 sin()

= r. :

D :

x = r cos()y = r sin()r [0, 2 sin()], [0, pi]

, :

C

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

D

(

x(y x)

y(x+ y)

)dxdy =

=

D

(1 1)dxdy = 2

D

dxdy

II 5 64

, 2.

2

D

dxdy = 2

Drdrd =

pi0

[ 2 sin()0

2rdr

]d =

= pi0

r2|2 sin()0 d = pi0

4 sin2()d

: sin2() = [1 cos2()]/2.

pi0

4 sin2()d = pi0

41 cos2()

2d =

pi0

[2 + 2 cos(2)]d == 2|pi0 + sin(2)|pi0 = 2(pi 0) + (sin(2pi) sin(0)) = 2pi

pi.r = pi.1 = pi. 2 : 2pi. 3. , (x, y) , T (x, y) = ex cos(y) + ey cos(x). T (x, y) O(0, 0) .

. O(0, 0).

T x = ex cos(y) ey sin(x), T x(0, 0) = 1

T y = ex sin(y) + ey cos(x), T y(0, 0) = 1

.

gradT = T xi + T y

j = 1

i + 1

j

.

|gradT | = 1 + 1 =

2

6 1. :

f(x) = |x|, x [1, 1].

2. ?

z = xy(4 x y)

3.y + 2y + y = 3ex

x+ 1 + xex

4. ?

D : {y = x2 + 4, y = 4x, x = 0}

5. G

x2 + y2dxdydz, G : {x2 + y2 = 2x, z = 0, z = x2 + y2}

6.

F () =

ba

f(x, )dx, D = {(x, y) : a x b, c a}.

, f(x, ) C(D), F () C[c, a]. 10 .

II 6 66

1. :

f(x) = |x|, x [1, 1].

. .

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3 4 5

: l = [1 (1)]/2 = 1. : f(x+ 2) = f(x). : f(x) = | x| = |x| = f(x).

( ). bn = 0. a0 an.

a0 an ( bn ):

an =1

pi

pipif(x) cos(nx)dx =

2

pi

pi0

f(x) cos(nx)dx.

f(x) = |x|:

an =1

1

11|x| cos

(npix1

)dx = 2

10

x cos(npix)dx,

x [0, 1] |x| = x (x > 0). a0:

a0 =1

1

11|x|dx = 2

10

xdx = x2|10 = 1 0 = 1.

II 6 67

an:

an = 2

10

x cos(npix)dx =2

npi

10

xd sin(npix) =

=2

npi

[x sin(npix)|10

10

sin(npix)dx

]=

=2

npi

[1 sin(npi) 0 sin(0) + 1

npicos(npix)|10

]=

=2

n2pi2[cos(npi) cos(0)] = 2(1)

n 1n2pi2

,

sin(npi) = 0, cos(npi) = (1)n. :

f(x) =1

2+

2

pi2

n=1

(1)n 1n2

cos(npix).

2. ?

z = xy(4 x y)

. .

z = f(x, y) = 4xy x2y xy2

f x = 4y 2xy y2, f y = 4x x2 2xy .

y(4 2x y) = 0 = y = 0, y = 4 2xx(4 x 2y) = 0 = x = 0, x = 4 2y

M0(0, 0) M1(

4

3,4

3

). .

f xx = 2y, f yy = 2x, f xy = 4 2x 2y, f yx = 4 2x 2y

.

1 = 2y = 02 =

2y 4 2x 2y4 2x 2y 2x = 0 44 0

= 0 16 < 0

II 6 68

2 , M0(0, 0) .

1 = 243

= 83< 0

2 =

2y 4 2x 2y4 2x 2y 2x = 8/3 4/34/3 8/3

= 649 169 > 0 2 , M1

(4

3,4

3

) (1 < 0).

-10 -5 0 5 10-10-5

0 5

10-2000-1500-1000-500

0 500 1000 1500 2000 2500

M0M1

: M1(

4

3,4

3

), M0(0, 0).

3.y + 2y + y = 3ex

x+ 1 + xex

. :

y + 2y + y = 0

:

k2 + 2k + 1 = 0 = (k + 1)2 = 0 k1,2 = 1. :

y(x) = c1ex + c2xex

:

y + 2y + y = ex(3x+ 1 + x)

, ( . 17). ( , , x:

x+ 1.)

(x) = C1(x)ex + C2(x)xex

II 6 69

. C 1(x)ex + C 2(x)xex = 0C 1(x)(ex) + C 2(x)[xex + ex] = ex(3x+ 1 + x) C 1(x) = C 2(x)xC 1(x) C 2(x)x+ C 2(x) = 3x+ 1 + x C 1(x) .

[C 2(x)x] C 2(x)x+ C 2(x) = 3x+ 1 + x

C 2(x) = 3x+ 1 + x

: C 2(x) = 3x+ 1 + x, C 1(x) = 3x

x+ 1 x2.

.

C2(x) =

[3x+ 1 + x]dx = 3

(x+ 1)1/2dx+

xdx =

= 32

3(x+ 1)3/2 +

x2

2= 2(x+ 1)3/2 +

x2

2

C1(x) x+ 1 .

C1(x) = 3x(x+ 1)1/2dx

x2dx =

= 323

xd(x+ 1)3/2 x

3

3=

= 2(x(x+ 1)3/2

(x+ 1)3/2dx

) x

3

3=

= 2(x(x+ 1)3/2 2

5(x+ 1)5/2

) x

3

3=

= 2x(x+ 1)3/2 + 45

(x+ 1)5/2 x3

3

(x):

(x) =

[2x(x+ 1)3/2 + 4

5(x+ 1)5/2 x

3

3

]ex +

[2(x+ 1)3/2 +

x2

2

]xex =

=

[4

5(x+ 1)5/2 x

3

3+x3

2

]ex =

[4

5(x+ 1)5/2 +

x3

6

]ex

:

y(x) = c1ex + c2xex +

[4

5(x+ 1)5/2 +

x3

6

]ex

II 6 70

4. ?

D : {y = x2 + 4, y = 4x, x = 0}. .

0

2

4

6

8

10

-1 -0.5 0 0.5 1 1.5 2 2.5 3

y = x2 + 4

y = 4xx = 0

:

D :

x [0, 2]y [4x, x2 + 4] :

D

dxdy =

Ddxdy

:Ddxdy =

20

[ x2+44x

dy

]dx =

20

y|x2+44x dx = 20

(x2 + 4 4x)dx =

=x3

3

20

+ 4x|20 2x2|20 =8

3+ 8 8 = 8

3

8/3.

5. G

x2 + y2dxdydz, G : {x2 + y2 = 2x, z = 0, z = x2 + y2}

. : z = x2 + y2 , x2 + y2 = 2x . :

x2 + y2 = 2x 1 + 1 = x2 2x+ 1 + y2 = 1 = (x 1)2 + y2 = 1 (1, 0) . , . .

II 6 71

-6 -4 -2 0 2 4 6-6-4 -2

0 2 4 6

-5 0 5

10 15 20 25

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.5

0 0.5 1 1.5

2 0 1 2 3 4 5

.

0 0.5 1 1.5 2-1-0.5

0 0.5

1 0 0.5 1 1.5 2 2.5 3 3.5 4

: ( ), z = 0 ( ) ( ).

( ). z z = 0 :

0 z x2 + y2

:

D :

x = r cos()y = r sin() x2 + y2 = 2x ( (x 1)2 + y2 = 1):

r2 cos2() + r2 sin2() = 2r cos()

r2 = 2r cos()

r = 2 cos()

: r [0, 2 cos()].

II 6 72

x2 + y2 = 2x (1, 0) . : [pi/2, pi/2].

= r. :

G :

D :

x = r cos()y = r sin()r [0, 2 cos()], [pi/2, pi/2]

0 z x2 + y2 z:

G

x2 + y2dxdydz =

D

[ x2+y20

x2 + y2dz

]dxdy =

=

D

x2 + y2z|x2+y20 dxdy =

D

x2 + y2(x2 + y2 0)dxdy =

=

D

x2 + y2(x2 + y2)dxdy =

D

(x2 + y2)3/2dxdy

x y r r:D

(x2 + y2)3/2dxdy =

D

[r2 cos2() + r2 sin2()]3/2rdrd =

=

D

[r2]3/2rdrd =

Dr4drd =

=

pi/2pi/2

[ 2 cos()0

r4dr

]d =

pi/2pi/2

r5

5

2 cos()0

d =

=1

5

pi/2pi/2

[25 cos5() 0]d = 325

pi/2pi/2

cos5()d

. :

cos2() =1 + cos(2)

2, cos3() =

1

4[3 cos() + cos(3)]

cos(1) cos(2) =1

2[cos(1 2) + cos(1 + 2)]

cos5() :

[cos3()]2 =1

42[3 cos() + cos(3)]2 =

=1

16[9 cos2() + 6 cos() cos(3) + cos(3)] =

=1

16

[9

1 + cos(2)

2+ 6

1

2(cos() + cos(4)) +

1 + cos(6)

2

]=

=1

32[9 + 9 cos(2) + 6 cos(2) + 6 cos(4) + 1 + cos(6)] =

=1

32[10 + 15 cos(2) + 6 cos(4) + cos(6)]

II 6 73

:

32

5

pi/2pi/2

1

32[10 + 15 cos(2) + 6 cos(4) + cos(6)]d =

=1

5

[10|pi/2pi/2 +

15

2sin(2)|pi/2pi/2 +

6

4sin(4)|pi/2pi/2 +

1

6sin(6)|pi/2pi/2

]=

=1

5

[10(pi

2(pi

2

))+ 0 + 0 + 0

]= 2

2pi

2= 2pi

pi, pi ( ). 2pi.

7

1. , z =y2

3x+ xy.

x2z

x xyz

y+ y2 = 0

2. .

n=0

2n

n!xn

3. .

z = x3 + y3 3xy

4. .

y = y tan(x) + sin2(x)

5. .

y + 2y + 2y = 6x2 + xex

6. , y = 2xx2, y = x. 7. x (0, pi).

f(x) =pi

2 x

2

: 4 7 5 , 10.

II 7 75

1. . .

n=1

3nxn

2n 1

2. x (0, pi).

f(x) =pi

4 x

2

3. z

xz

y, z =

y2

x ln(y).

3. z = x2 + xy + y2 3x 6y. 4. y + y = cos(x) + xe2x.

5. , 2z = 2 + x2 + y2 z = 3, .

6. .

y = y cos(x) sin(2x)

: 3 6 5 , 10.

II 7 76

1. .

n=1

xn+4

6n+ 11

2. F : R R T = 8, F (x) = 1 x/2 x [0, 4]. F (x) .

3. .

f(x, y) = x2 y2 + 2x 4y + 6

4. .

y + 9y = 4sin(3x)

+ (3x 4)e2x

5. D

2xy2dxdy,

D x = 0 (x 0) x2 + y2 = 16. 6.

C

(2x2 + 2y2)dl

C, x = 3 cos(2t), y = 3 sin(2t), t [0, pi]. 10 .

II 7 77

1. .

n=1

xn

5nn2

2. .

z = ex2x(4x+ xy2)

3. .

f(x) =

{0, x [1, 0]x, x (0, 1)

4. y sin(x) = y ln(y), - y

(pi2

)= 1.

5. .

y y = 2x+ 4 cos(2x)

6. , .

y = 4 x2, y = x2 2x

10 .

II 7 78

1. .

f(x, y) = 2x3 + xy2 216x

2. .

y y = 4xex + 1ex + 2

3. .

(y x)dx+ (y + x)dy = 0

4. V

ydxdydz,

V , y =x2 + z2 -

y = 2.

5. .

n=1

(1)n 1n ln(n)

6. L

3xydx+ 2x2dy,

L , y = x y = x22x, .

10 .

8 1. , :

x2 + y2 + z2 = 9, x2 + y2 + z2 = 6z.

. : x2+y2+z2 = 9 , x2+y2+z2 =6z (0, 0, 3), :

x2 + y2 + z2 = 6z 9 + 9,x2 + y2 + z2 6z + 9 = 9,x2 + y2 + (z 3)2 = 9.

3. :

-3 -2 -1 0 1 2 3-3-2

-1 0

1 2

3-3-2-1 0 1 2 3 4 5 6

-, :

-3 -2 -1 0 1 2 3-3-2 -1

0 1 2 3

0 0.5

1 1.5

2 2.5

3

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5 2

2.5 3

( ), ( ).

: , - .

II 8 80

. :x = r sin() cos()y = r sin() sin()z = r cos()

.

:

=

xr y

r z

r

x y z

x y z

=

sin() cos() sin() sin() cos()r cos() cos() r cos() sin() r sin()r sin() sin() r sin() cos() 0

== cos()(1)4[r2 cos() sin() cos2() + r2 cos() sin() sin2()] r sin()(1)5[r sin2() cos2() + r sin2() sin2()] == r2 cos2() sin() + r2 sin3() = r2 sin().

= || = r2 sin(). . ( .)

. G:

G :

x2 + y2 + z2 = 9x2 + y2 + z2 = 6z . r, . :

r2 sin2() cos2() + r2 sin2() sin2() + r2 cos2() = 9,

r2 sin2() + r2 cos2() = 9,

r2 = 9 = r = 3 = r = 3. . :

r2 sin2() cos2() + r2 sin2() sin2() + r2 cos2() = 6r cos(),

r2 sin2() + r2 cos2() = 6r cos(),

r2 = 6r cos() = r = 6 cos(). , :

3 = 6 cos() = 1 = 2 cos() = cos() = 12

= = pi3.

z Oxy, 0 pi/2. = pi/3 . Ozy:

II 8 81

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

: z y (- Oxy), = pi/3 = 60. .

Oxy, 0 2pi ( ). z Oxy ( -), Oxy z. :

x = r cos() cos(), y = r cos() sin(), z = r sin(), = r2 cos().

, ( ). sin() = 1/2, =pi/6 = 30.

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

y ( Oxy) z. , ,

II 8 82

. . -

, . : http://en.wikipedia.org/wiki/ISO_31-11#Coordinate_systems,http://en.wikipedia.org/wiki/Spherical_coordinate_system#Cartesian_coordinates.

. :

G1 :

r [0, 3] [0, pi/3] [0, 2pi]

, G2 :

r [0, 6 cos()] [pi/3, pi/2] [0, 2pi]

.

0 3 0 6 cos(). Oxy, , 0 2pi.

: V = VG1 + VG2 . :

x = r sin() cos(), y = r sin() sin(), z = r cos(), = r2 sin().

:

VG1 =

G1

dxdydz =

G1

r2 sin()drdd =

2pi0

d

30

r2dr

pi/30

sin()d =

= |2pi0r3

3

30

[ cos()]|pi/30 = 2pi27

3[cos(pi/3) cos(0)] = 18pi[1/2 1] = 9pi.

:

VG2 =

G2

dxdydz =

G2

r2 sin()drdd =

=

2pi0

d

pi/2pi/3

[ 6 cos()0

r2 sin()dr

]d = |2pi0

pi/2pi/3

r3

3

6 cos()0

sin()d =

= 2pi

pi/2pi/3

63 cos3()

3sin()d = 36pi

pi/2pi/3

4 cos3() sin()d =

= 36pi pi/2pi/3

4 cos3()d[cos()] = 36pi cos4()|pi/2pi/3 =

= 36pi[cos4(pi/2) cos4(pi/3)] = 36pi[0 1/16] = 9pi4.

:

V = VG1 + VG2 = 9pi +9pi

4=

45pi

4.

. ( - ) ( , Oxy)

II 8 83

( , Oxy), .

:x = r cos()y = r sin()z = z

.

() :

r2 cos2() + r2 sin2() + z2 = 9,

r2 + z2 = 9.

:

r2 cos2() + r2 sin2() + z2 = 6z,

r2 + z2 = 6z.

:

9 = 6z = z = 32.

z = 3/2 :

r2 +9

4= 9, r2 +

9

4= 6

3

2.

( ), :

r2 +9

4= 9 = r2 = 27

4= r = 3

3

2= r = 3

3

2.

. r = 3

3/2 (0, 0, 1.5),

:

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5 2

2.5 3

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5 2

2.5 3

II 8 84

Oxy:

D : x2 + y2 =

(3

3

2

)2.

r :

D :

0 r 33/20 2pi . z

:

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5 2

2.5 3

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

Ozy. z. () :

x2 + y2 + z2 = 9,

z2 = 9 x2 y2,z =

9 x2 y2.

z =

9 x2 y2 ( - ), z = 9 x2 y2 .

V1 () Oxy, :

II 8 85

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5 2

2.5 3

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

Ozy. :

x2 + y2 + z2 = 6z,

z2 6z + 9 = 9 x2 y2,(z 3)2 = 9 x2 y2,z = 3

9 x2 y2.

z = 3 +

9 x2 y2 , z = 3 9 x2 y2 , .

V2 Oxy, :

-2 -1 0 1 2 -2-1

0 1

2 0

0.5 1

1.5

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

II 8 86

Ozy. z :

3

9 x2 y2 z

9 x2 y2.

:

G :

D :

0 r 33/20 2pi39 x2 y2 z 9 x2 y2 .

:x = r cos(), y = r sin(), z = z, = r.

:

V =

G

dxdydz =

D

[ 9x2y23

9x2y2dz

]dxdy =

D

z

9x2y2

3

9x2y2dxdy =

=

D

9 x2 y2dxdy

D

[3

9 x2 y2]dxdy = V1 V2.

V1 :

V1 =

D

9 x2 y2dxdy =

Dr

9 r2 cos() r2 sin()drd =

=

Dr

9 r2drd = 2pi0

d

33/20

r

9 r2dr =

= |2pi01

2

33/20

2r

9 r2dr = 2pi12

33/20

9 r2d(r2) =

= pi 33/20

(9 r2)1/2d(9 r2) = pi (9 r2)3/2

3/2

33/2

0

=

= 2pi3

(9 r2)3/233/20

= 2pi3

[(9 27

4

)3/2 (9 0)3/2

]=

= 2pi3

[(9

4

)3/2 27

]= 2pi

3

[27

8 27

]=

= 2pi3

[189

8

]=

63pi

4.

: 0 3:

V = pir2h = pi

(3

3

2

)23 = pi

27

43 =

81pi

4.

II 8 87

V1 0 () 1.5 ( ):

V = pir2h = pi

(3

3

2

)21.5 = pi

27

4

3

2=

81pi

8,

, h = 1.5:

V = pih2

(R 1

3h

)= pi

(3

2

)2(3 1

3

3

2

)=

9pi

4

(3 1

2

)=

9pi

4

5

2=

45pi

8.

:

V1 = V + V =81pi

8+

45pi

8=

126pi

8=

63pi

4.

. V2 Oxy:

V2 =

D

[3

9 x2 y2]dxdy =

Dr[3

9 r2 cos() r2 sin()]drd =

=

Dr[3

9 r2]drd =

2pi0

d

33/20

r[3

9 r2]dr =

= |2pi0[ 33/2

0

3rdr 33/20

r

9 r2dr]

=

= 2pi

33/20

3rdr 2pi 33/20

r

9 r2dr.

:

I2 = 2pi

33/20

r

9 r2dr = 63pi4.

:

I1 = 2pi

33/20

3rdr = 3pi

33/20

2rdr = 3pir2|33/2

0 = 3pi27

4=

81pi

4.

V2 :V2 = I1 I2 = 81pi

4 63pi

4=

18pi

4.

: I1 ( -), I2 V1, -: , 1.5 3 .

II 8 88

, .

:

V = V1 V2 = 63pi4 18pi

4=

45pi

4.

. , h = 1.5:

V = pih2

(R 1

3h

)= pi

(3

2

)2(3 1

3

3

2

)=

9pi

4

(3 1

2

)=

9pi

4

5

2=

45pi

8.

:

V = 245pi

8=

45pi

4.

2. :

2(y)2 = (y 1)y, y(1) = 2, y(1) = 1.. y x:

y = y(x).

p, y:

p = p(y) = p[y(x)].

y p:

yx =dy

dx yx = p.

y p:

yx =d2y

dx2 yx = px.

p x:

px = px[y(x)] = p

yyx.

:

px =dp

dx=dp

dy

dy

dx= pyy

x.

yx = p:pyy

x = p

yp.

II 8 89

p y, py = p:

yx = pyyx = p

yp = p

p.

:y = p, y = pp.

:2p2 = (y 1)pp,2p = (y 1)p,

2p = (y 1)dpdy.

dy, y 1, y 6= 1:2pdy

y 1 = dp.

2p, p 6= 0:dy

y 1 =dp

2p.

, ( -):

dy

y 1 =dp

2p+ C,

d(y 1)y 1 =

1

2

dp

p+ C,

ln |y 1| = 12

ln |p|+ C. . :

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

y(1) = 1 x = 1, y = 1. x, x . -, p = y. :

ln |y 1| = 12

ln |y|+ C,

ln |2 1| = 12

ln | 1|+ C,

ln 1 =1

2ln 1 + C,

0 = 0 + C = C = 0.

II 8 90

:

ln |y 1| = 12

ln |p|,

ln |y 1| = ln |p|1/2. e ():

eln |y1| = eln |p|1/2

,

|y 1| = |p|1/2. :

|y 1|2 = (|p|)2.

, . , :

(y 1)2 = |p|.

y y -:

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

: a, a :

|a| =

a, a > 0

a, a < 00, a = 0

.

: |5| = 5, | 3| = (3) = 3, |0| = 0. : y = 1, |y| = y. :

(y 1)2 = y.

:

(2 1)2 = (1) = 1 = 1.

|y| = y, (y 1)2 = y, :

(2 1)2 = 1 = 1 = 1. .

:

(y 1)2 = y,

II 8 91

(y 1)2 = dydx.

dx, (y 1)2, y 6= 1:

dx = dy(y 1)2 .

: dx =

dy

(y 1)2 + C1,

x = d(y 1)(y 1)2 + C1.

t1:(1

t

)= (t1) = 1t2 = 1

t2.

, t = y 1:

x =1

y 1 + C2.

:y(1) = 2 x = 1, y = 2,

y(1) = 1 x = 1, y = 1. :

1 =1

2 1 + C2,1 = 1 + C2 = C2 = 0.

:x =

1

y 1 .

y, x 6= 0, y 6= 1:

y 1 = 1x,

y =1

x+ 1.

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