математика

F( x, y, yc) 0

(4.1)

ɢɥɢ, ɟɫɥɢ ɪɚɡɪɟɲɢɬɶ ɟɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ yc, ɜ ɧɨɪɦɚɥɶɧɨɣ ɮɨɪɦɟ

ɍ

yc f ( x, y).

(4.2)

Ɋɟɲɟɧɢɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚ ɢɧɬɟɪɜɚɥɟ (a,ɜɌ) ɧɚɡɵɜɚɟɬɫɹ

ɬɚɤɚɹ ɮɭɧɤɰɢɹ y

M(x), ɤɨɬɨɪɚɹ ɩɪɢ ɩɨɞɫɬɚɧɨɜɤɟ ɜ ɭɪɚɜɧɟɧɢɟ ɜɦɟɫɬɨ ɧɟɢɡɜɟɫɬɧɨɣ

ɮɭɧɤɰɢɢ ɨɛɪɚɳɚɟɬ ɟɝɨ ɜ ɬɨɠɞɟɫɬɜɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɯ (ɚ,ɜ) .

ɇ

Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ

ɭɪɚɜɧɟɧɢɹ

ɩɟɪɜɨɝɨ

ɩɨɪɹɞɤɚȻɧɚɡɵɜɚɟɬɫɹ

ɮɭɧɤɰɢɹ

y M(x,ɋ) ,

ɤɨɬɨɪɚɹ ɩɪɢ ɥɸɛɨɦ ɞɨɩɭɫɬɢɦɨɦ ɡɧɚɱɟɧɢɢ ɩɨɫɬɨɹɧɧɨɣ ɋ

ɹɜɥɹɟɬɫɹ

ɣ

ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ

ɪɟɲɟɧɢɟɦ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɜɫɹɤɨɟ ɪɟɲɟɧɢɟ

y

M(x)

ɜ ɜɢɞɟ y

M(x,C0 ) C0 R .

ɢ

Ɍɟɨɪɟɦɚ Ʉɨɲɢ. ȿɫɥɢ ɮɭɧɤɰɢɹ

f ( x, y)

ɨɩɪɟɞɟɥɟɧɚ,

ɧɟɩɪɟɪɵɜɧɚ ɢ ɢɦɟɟɬ

ɪ

wf (x, y)

ɜ ɨɛɥɚɫɬɢ D, ɫɨɞɟɪɠɚɳɟɣ ɬɨɱɤɭ

ɧɟɩɪɟɪɵɜɧɭɸ ɱɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸɨwy

Ɇ(x0 , y0 ) ,

ɬɨɝɞɚ

ɧɚɣɞɟɬɫɹ

ɬ

(x0 G;

x0 G) ,

ɧɚ

ɤɨɬɨɪɨɦ ɫɭɳɟɫɬɜɭɟɬ

ɢɧɬɟɪɜɚɥ

ɟɞɢɧɫɬɜɟɧɧɨɟ

ɢ

M(x)

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ (4.2),

ɪɟɲɟɧɢɟ

y

ɡ

y0 .

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ y(x0 )

ɉɚɪɭ

ɨɱɢɫɟɥ ( x0 , y0 ) ɧɚɡɵɜɚɸɬ ɧɚɱɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ. Ɋɟɲɟɧɢɹ, ɤɨɬɨɪɵɟ

ɩɨɥɭɱɚɸɬɫɹɩɢɡ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ y M(x,ɋ) ɩɪɢ

ɨɩɪɟɞɟɥɟɧɧɨɦ

ɡɧɚɱɟɧɢɢ

ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɨɫɬɨɹɧɧɨɣ ɋ, ɧɚɡɵɜɚɸɬɫɹ ɱɚɫɬɧɵɦɢ.

ɟ

Ɂɚɞɚɱɚ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɧɚɱɚɥɶɧɨɦɭ

ɭɫɥɨɜɢɸ

y

y0 ɩɪɢ x x0 ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ Ʉɨɲɢ.

Ɋ

ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ

ɍ

P(x)dx Q( y)dy

0

(4.3)

ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫ ɪɚɡɞɟɥɟɧɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ. ȿɝɨ

ɨɛɳɢɦ

ɢɧɬɟɝɪɚɥɨɦ

ɛɭɞɟɬ

³P(x)dx ³Q( y)dy

ɋ, ɝɞɟ ɋ

–

ɩɪɨɢɡɜɨɥɶɧɚɹ

ɇ

ɩɨɫɬɨɹɧɧɚɹ.

Ȼ

Ɍ

ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ

M1(x) M2 ( y)dx N1(x)N2 ( y)dy

0

(4.4)

ɢɥɢ

dy

ɣ

f1(x) f2 ( y),

ɢ

(4.5)

dx

ɪ

ɚ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɹ, ɤɨɬɨɪɵɟ ɫ ɩɨɦɨɳɶɸ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ

ɩɪɢɜɨɞɹɬɫɹ ɤ ɭɪɚɜɧɟɧɢɹɦ (4.4) ɢɥɢ (4.5) ɧɚɡɵɜɚɸɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢ

ɭɪɚɜɧɟɧɢɹɦɢ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ.

1

ɬ2

Ɋɚɡɞɟɥɟɧɢɟ ɩɟɪɟɦɟɧɧɵɯ ɜɨɭɪɚɜɧɟɧɢɹɯ (4.4) ɢ (4.5) ɜɵɩɨɥɧɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ

ɨɛɪɚɡɨɦ:

ɟɫɥɢ

ɢ

N ( x) z

0, M ( y) z 0 , ɬɨ ɪɚɡɞɟɥɢɦ ɨɛɟ ɱɢɫɬɢ ɭɪɚɜɧɟɧɢɹ (4.4) ɧɚ

ɡ

N1(x) M2 ( y) . ȿɫɥɢ

f2 ( y) z 0 , ɬɨ ɭɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (4.5) ɧɚ dx ɢ

ɨ

ɪɚɡɞɟɥɢɦ

ɧɚ

f2

( y) .

ȼ

ɪɟɡɭɥɶɬɚɬɟ

ɩɨɥɭɱɢɦ

ɭɪɚɜɧɟɧɢɹ

ɫ

ɪɚɡɞɟɥɟɧɧɵɦɢ

ɩɟɪɟɦɟɧɧɵɦɢ ɜɢɞɚ:

ɩ

N2 ( y)

M

1

(x)

dx

dy

0;

ɟ1

Ɋ

N1 (x)

M 2

( y)

f (x)dx

dy

.

ɉɪɢɦɟɪ 4.1. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ

c

1 y2

y

xy(1

x2 ) .

Ɋɟɲɟɧɢɟ. Ɂɚɦɟɧɢɦ

yc

dy

. Ɋɚɡɞɟɥɢɜ ɩɟɪɟɦɟɧɧɵɟ ɢ ɢɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɢɦ

dx

ydy

dx

;

³

ydy

³

dx

C .

ɍ

1 y

2

x(1 x

2

)

1 y

2

x(1

x

2

)

Ɍ

Ɋɚɡɥɨɠɢɦ ɩɨɞɵɧɬɟɝɪɚɥɶɧɭɸ ɞɪɨɛɶ ɧɚ ɩɪɨɫɬɟɣɲɢɟ:

1

A

Bx D

, A 1, B 1, D 0.

x(1 x2 )

x

1 x2

Ȼ

Ɉɬɫɸɞɚ

ɇ

1

ln(1

y2 )

ln | x |

1

ln(1 x2 ) ln | C |;

2

2

ln | (1 x2 )(1 y2 ) |

2ln | Cx | .

ɢ

2

2

2

2

(1 x

)(1

y

)

ɋ

x

ɣ

ȼɵɪɚɡɢɜ ɢɡ ɧɟɝɨ y ,

– ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ.

ɢɦɟɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ

C2 x2

ɨ

y r

1 .

ɪ

1 x2

ɬ

ɢ

4.2. Ɉɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ 1 ɩɨɪɹɞɤɚ

ɡ

ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɨɣ ɮɭɧɤɰɢɟɣ n–ɝɨ ɢɡɦɟɪɟɧɢɹ

Ɏɭɧɤɰɢɹ

f

(x, y)

ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɟɦɟɧɧɵɯ x ɢ y, ɟɫɥɢ ɩɪɢ ɥɸɛɨɦ t ɫɩɪɚɜɟɞɥɢɜɨ ɬɨɠɞɟɫɬɜɨ

ɨ

3

2

f (tx, ty)

tn f (x, y) .

(4.6)

ɇɚɩɪɢɦɟɪ: f (x, y)

x 3x y

–

ɨɞɧɨɪɨɞɧɚɹ ɮɭɧɤɰɢɹ

ɬɪɟɬɶɟɝɨ

ɢɡɦɟɪɟɧɢɹ

ɟ

ɨɬɧɨɫɢɬɟɥɶɧɨɩɩɟɪɟɦɟɧɧɵɯ x ɢ y, ɬɚɤ ɤɚɤ

Ɋ

f (tx, ty)

(tx)3 3(tx)2 ty t3 (x3 3x2 y)

t3 f (x, y) .

Ɏɭɧɤɰɢɹ

M(x, y)

x y

ɹɜɥɹɟɬɫɹ

ɨɞɧɨɪɨɞɧɨɣ

ɮɭɧɤɰɢɟɣ

ɧɭɥɟɜɨɝɨ

x 2 y

ɢɡɦɟɪɟɧɢɹ,

ɬɚɤ

ɤɚɤ

M(tx, ty)

t0M(x, y)

M(x, y) . Ɏɭɧɤɰɢɹ x3 3x2 y x

ɍ

ɨɞɧɨɪɨɞɧɵɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ 1-ɝɨ ɩɨɪɹɞɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɩɟɪɟɦɟɧɧɵɯ x ɢ y, ɟɫɥɢ

f (x, y)

– ɨɞɧɨɪɨɞɧɚɹ ɮɭɧɤɰɢɹ ɧɭɥɟɜɨɝɨ ɢɡɦɟɪɟɧɢɹ.

Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɮɨɪɦɟ

Ɍ

M (x, y)dx N (x, y)dy 0

ɇ

ɧɚɡɵɜɚɟɬɫɹ

ɨɞɧɨɪɨɞɧɵɦ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ

ɭɪɚɜɧɟɧɢɟɦ

1-ɝɨ ɩɨɪɹɞɤɚ, ɟɫɥɢ

ɮɭɧɤɰɢɢ M (x, y) ɢ N(x, y)

Ȼ

– ɨɞɧɨɪɨɞɧɵɟ ɮɭɧɤɰɢɢ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɢɡɦɟɪɟɧɢɹ.

ɉɪɢ ɩɨɦɨɳɢ ɩɨɞɫɬɚɧɨɜɤɢ

y

ux , ɝɞɟ u(x) –

ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɨɞɧɨɪɨɞɧɨɟ

ɣ

ɭɪɚɜɧɟɧɢɟ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ.

ɉɪɢɦɟɪ 4.2. Ɋɟɲɢɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ

yc

y2

2 .

ɢ

x

ɪ

ɨ

y2

Ɋɟɲɟɧɢɟ. ɗɬɨ ɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɬɚɤ ɤɚɤ

f (x, y)

2 – ɨɞɧɨɪɨɞɧɚɹ

x2

ɬ

c

c

ɮɭɧɤɰɢɹ ɧɭɥɟɜɨɝɨ ɢɡɦɟɪɟɧɢɹ. ɉɨɥɨɠɢɦ y

ux,

y

u x u .

ɢ

c

u u

2

2,

c

2

u 2 .

Ɍɨɝɞɚ u x

u x u

du

2

ɡu 2,

du

dx

x

u

–

ɭɪɚɜɧɟɧɢɟ

ɫ

ɪɚɡɞɟɥɟɧɧɵɦɢ

dx

2

x

ɨ

u

u 2

ɩɟɪɟɦɟɧɧɵɦɢ. ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɢɦ

³

ɩdu

³

dx

,

1

u 2

ln | x | ln | C |,

ln

ɟ

1

2

9

x

3

u 1

Ɋ

(u

)

2

4

y

u 2

2

C

3

x

3

,

x

Cx

3

,

y 2x Cx

3

( y

x)

u 1

y

1

x

ɨɬɧɨɫɢɬɟɥɶɧɨ y, ɩɨɥɭɱɢɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ

y

x(2

Cx

3 )

.

1

Cx3

ɉɪɢɦɟɪ 4.3. ɇɚɣɬɢ

ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ

ɍ

( y2 3x2 )dy 2xydx 0 ,

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ y

x 0

1.

Ɍ

Ɋɟɲɟɧɢɟ. M (x, y) 2xy,

N (x, y) y2 3x2

– ɨɞɧɨɪɨɞɧɵɟ ɮɭɧɤɰɢɢ ɜɬɨɪɨɝɨ

ɢɡɦɟɪɟɧɢɹ. ɉɨɞɫɬɚɧɨɜɤɚ y

c

c

ɇ

ux, y

u x u

ɩɪɢɜɨɞɢɬ ɭɪɚɜɧɟɧɢɟ ɤ ɜɢɞɭ

(u2 3)du

dx

.

2

Ȼ

2

u(1 u2 )

x

ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɚɟɦ

ɣ

³

(u 3)du

³dx

;

u 3

A

B

D

;

u(1 u)(1 u)

u(1 u)(1 u)

u

1

u

1

x

u

A

3,

B

1;

D

1;

ɢ

3ln | u | ln |1 u | ln |1 u |

ln | x | ln | C |;

y2

ɨ

1 u2

Cx

;

1 x2

Cx.

ɪ

u3

y3

ɬ

x3

x2 y2

Cy3

ɢ

ɢɧɬɟɝɪɚɥ

ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɇɚɣɞɟɦ ɱɚɫɬɧɵɣ

–

ɨɛɳɢɣ

ɡ

ɢɧɬɟɝɪɚɥ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ ɭɫɥɨɜɢɸ

y

x 0

1; 0 1 C; C 1;

y

3

2

2

yɨx – ɱɚɫɬɧɵɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ.

ɟ

ɩ4.3. Ʌɢɧɟɣɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ 1–ɝɨ ɩɨɪɹɞɤɚ

Ʌɢɧɟɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ 1-ɝɨ ɩɨɪɹɞɤɚ ɜ ɨɛɳɟɦ ɜɢɞɟ ɦɨɠɧɨ

ɡɚɩɢɫɚɬɶ ɫɨɨɬɧɨɲɟɧɢɟɦ

Ɋ yc P(x) y

Q(x) ,

(4.7)

ɝɞɟ P(x),

Q(x) ɡɚɞɚɧɧɵɟ ɧɟɩɪɟɪɵɜɧɵɟ ɮɭɧɤɰɢɢ.

Ɍɨɝɞɚ

dy

v

du

u

dv

ɢ ɭɪɚɜɧɟɧɢɟ (4.7) ɩɪɢɦɟɬ ɜɢɞ

ɍ

dx

dx

dx

v

du

§dv

·

u¨

P(x)v ¸

Q(x) .

(4.8)

dx

©dx

¹

ɇ

Ɏɭɧɤɰɢɸ v(ɯ) ɩɨɞɛɢɪɚɟɦ ɬɚɤ, ɱɬɨɛɵ ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛɤɚɯ ɛɵɥɨɌɪɚɜɧɨ ɧɭɥɸ,

Ȼ

ɬɨ ɟɫɬɶ ɜ ɤɚɱɟɫɬɜɟ v(ɯ) ɜɨɡɶɦɟɦ ɨɞɧɨ ɢɡ ɱɚɫɬɧɵɯ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ

dv

P(x)v

0 .

dx

ɣ

ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɟ v

v(x)

ɜ ɭɪɚɜɧɟɧɢɟ (4.8), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɫ

ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ

ɢ

v

du

Q(x) .

dx