Методические указания по выполнению контрольной работы № 2 по математике для студентов инженерно-технических специальностей заочной формы обучения

F( x, y, yc) 0

(4.1)

ɢɥɢ, ɟɫɥɢ ɪɚɡɪɟɲɢɬɶ ɟɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ yc, ɜ ɧɨɪɦɚɥɶɧɨɣ ɮɨɪɦɟ

yc f ( x, y).

(4.2)

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

Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɧɚɡɵɜɚɟɬɫɹ

ɮɭɧɤɰɢɹ

y M(x,ɋ) , ɤɨɬɨɪɚɹ ɩɪɢ ɥɸɛɨɦ ɞɨɩɭɫɬɢɦɨɦ ɡɧɚɱɟɧɢɢ ɩɨɫɬɨɹɧɧɨɣ ɋ

ɹɜɥɹɟɬɫɹ

ɜ ɜɢɞɟ y M(x,C0 ) C0 R .

Ɍɟɨɪɟɦɚ Ʉɨɲɢ. ȿɫɥɢ ɮɭɧɤɰɢɹ

f ( x, y) ɨɩɪɟɞɟɥɟɧɚ, ɧɟɩɪɟɪɵɜɧɚ ɢ ɢɦɟɟɬ

ɧɟɩɪɟɪɵɜɧɭɸ ɱɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ

wf (x, y)

ɜ ɨɛɥɚɫɬɢ D,

ɫɨɞɟɪɠɚɳɟɣ ɬɨɱɤɭ

wy

Ɇ(x0 , y0 ) , ɬɨɝɞɚ ɧɚɣɞɟɬɫɹ ɢɧɬɟɪɜɚɥ

(x0 G;

x0 G) ,

ɧɚ ɤɨɬɨɪɨɦ ɫɭɳɟɫɬɜɭɟɬ

ɟɞɢɧɫɬɜɟɧɧɨɟ

ɪɟɲɟɧɢɟ

y M(x)

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ (4.2),

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ y(x0 )

y0 .

ɉɚɪɭ ɱɢɫɟɥ ( x0 , y0 ) ɧɚɡɵɜɚɸɬ ɧɚɱɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ. Ɋɟɲɟɧɢɹ, ɤɨɬɨɪɵɟ

ɩɨɥɭɱɚɸɬɫɹ

ɢɡ ɨɛɳɟɝɨ

ɪɟɲɟɧɢɹ

y M(x,ɋ) ɩɪɢ

ɨɩɪɟɞɟɥɟɧɧɨɦ ɡɧɚɱɟɧɢɢ

ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ

P(x)dx Q( y)dy 0

(4.3)

ɨɛɳɢɦ ɢɧɬɟɝɪɚɥɨɦ ɛɭɞɟɬ ³P(x)dx ³Q( y)dy ɋ, ɝɞɟ ɋ

– ɩɪɨɢɡɜɨɥɶɧɚɹ

ɩɨɫɬɨɹɧɧɚɹ.

ɍɪɚɜɧɟɧɢɟ ɜɢɞɚ

M1(x) M2 ( y)dx N1(x)N2 ( y)dy 0

(4.4)

ɢɥɢ

dy

f1(x) f2 ( y),

(4.5)

dx

ɨɛɪɚɡɨɦ:

ɟɫɥɢ

N1( x) z 0, M2 ( y) z 0 , ɬɨ ɪɚɡɞɟɥɢɦ ɨɛɟ ɱɢɫɬɢ ɭɪɚɜɧɟɧɢɹ (4.4)

ɧɚ

N1(x) M2 ( y) . ȿɫɥɢ f2 ( y) z 0 , ɬɨ ɭɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (4.5) ɧɚ dx

ɢ

ɪɚɡɞɟɥɢɦ

ɧɚ

f2 ( y) . ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ ɫ ɪɚɡɞɟɥɟɧɧɵɦɢ

ɩɟɪɟɦɟɧɧɵɦɢ ɜɢɞɚ:

M1 (x)

dx

N2 ( y)

dy

0;

N1 (x)

M 2 ( y)

f1 (x)dx

dy

.

f2 ( y)

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 .

1 y

2

x(1 x

2

)

1 y

2

x(1

x

2

)

Ɋɚɡɥɨɠɢɦ ɩɨɞɵɧɬɟɝɪɚɥɶɧɭɸ ɞɪɨɛɶ ɧɚ ɩɪɨɫɬɟɣɲɢɟ:

1

A

Bx D

, A 1, B 1, D 0.

x(1 x2 )

1 x2

x

Ɉɬɫɸɞɚ

1

ln(1

y2 )

ln | x |

1

ln(1 x2 ) ln | C |;

2

2

ln | (1 x2 )(1 y2 ) |

2ln | Cx | .

(1 x2 )(1 y2 )

ɋ2 x2

– ɨɛɳɢɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ.

ȼɵɪɚɡɢɜ ɢɡ ɧɟɝɨ y ,

ɢɦɟɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ

y r

C2 x2

1 .

1

x

2

4.2. Ɉɞɧɨɪɨɞɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ 1 ɩɨɪɹɞɤɚ

Ɏɭɧɤɰɢɹ

f (x, y)

ɧɚɡɵɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɨɣ ɮɭɧɤɰɢɟɣ n–ɝɨ ɢɡɦɟɪɟɧɢɹ

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

f (tx, ty)

tn f (x, y) .

(4.6)

ɇɚɩɪɢɦɟɪ: f (x, y)

x3 3x2 y

–

ɨɞɧɨɪɨɞɧɚɹ ɮɭɧɤɰɢɹ

ɬɪɟɬɶɟɝɨ

ɢɡɦɟɪɟɧɢɹ

ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɟɦɟɧɧɵɯ x ɢ y, ɬɚɤ ɤɚɤ

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

ɮɭɧɤɰɢɢ M (x, y) ɢ N(x, y)

– ɨɞɧɨɪɨɞɧɵɟ ɮɭɧɤɰɢɢ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɢɡɦɟɪɟɧɢɹ.

ɉɪɢ ɩɨɦɨɳɢ ɩɨɞɫɬɚɧɨɜɤɢ

y

ux , ɝɞɟ u(x) –

ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɨɞɧɨɪɨɞɧɨɟ

ɭɪɚɜɧɟɧɢɟ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɫ ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ.

ɉɪɢɦɟɪ 4.2. Ɋɟɲɢɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ

yc

y2

2 .

x2

Ɋɟɲɟɧɢɟ. ɗɬɨ ɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɬɚɤ ɤɚɤ

f (x, y)

y2

2

– ɨɞɧɨɪɨɞɧɚɹ

x2

ɮɭɧɤɰɢɹ ɧɭɥɟɜɨɝɨ ɢɡɦɟɪɟɧɢɹ. ɉɨɥɨɠɢɦ y

ux,

y

c

c

u x u .

c

u

u

2

2,

c

2

u 2 .

Ɍɨɝɞɚ u x

u x u

du

x

u2

u 2,

du

dx

–

ɭɪɚɜɧɟɧɢɟ

ɫ

ɪɚɡɞɟɥɟɧɧɵɦɢ

u2 u 2

x

dx

ɩɟɪɟɦɟɧɧɵɦɢ. ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɢɦ

³

du

³

dx

,

1

u 2

ln | x | ln | C |,

ln

1

9

3

u 1

(u

)

2

x

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

( y2 3x2 )dy 2xydx 0 ,

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɨɦɭ ɭɫɥɨɜɢɸ y

x

0

1.

Ɋɟɲɟɧɢɟ. M (x, y)

2xy,

N (x, y)

y2 3x2

– ɨɞɧɨɪɨɞɧɵɟ ɮɭɧɤɰɢɢ ɜɬɨɪɨɝɨ

ɢɡɦɟɪɟɧɢɹ. ɉɨɞɫɬɚɧɨɜɤɚ y

ux, y

c

c

u x u ɩɪɢɜɨɞɢɬ ɭɪɚɜɧɟɧɢɟ ɤ ɜɢɞɭ

(u2 3)du dx

.

u(1 u2 )

x

ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɚɟɦ

³

(u2 3)du

³

dx

;

u2 3

A

B

D

;

u(1 u)(1 u)

x

u(1 u)(1 u)

u

1

u

1

u

A

3,

B

1;

D

1;

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

ln | x | ln | C |;

1 u2

1

y2

Cx

;

x2

Cx.

u3

y3

x3

x2 y2

Cy3 –

ɨɛɳɢɣ

ɢɧɬɟɝɪɚɥ

ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ɇɚɣɞɟɦ ɱɚɫɬɧɵɣ

ɢɧɬɟɝɪɚɥ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ ɭɫɥɨɜɢɸ

y

x 0

1; 0 1 C; C 1;

y3

y2 x2

– ɱɚɫɬɧɵɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ.

ɡɚɩɢɫɚɬɶ ɫɨɨɬɧɨɲɟɧɢɟɦ

yc P(x) y Q(x) ,

(4.7)

ɝɞɟ u(x) ɢ v(x) – ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ.

Ɍɨɝɞɚ

dy

v

du

u

dv

ɢ ɭɪɚɜɧɟɧɢɟ (4.7) ɩɪɢɦɟɬ ɜɢɞ

dx

dx

dx

v

du

§dv

·

u¨

P(x)v ¸

Q(x) .

(4.8)

dx

©dx

¹

dv

P(x)v

0 .

dx

ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɟ

v v(x)

ɜ ɭɪɚɜɧɟɧɢɟ (4.8), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɫ

ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ ɩɟɪɟɦɟɧɧɵɦɢ

v

du

Q(x) .

dx

ɇɚɣɞɹ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ u u(x,C) , ɩɨɥɭɱɢɦ ɨɛɳɟɟ

ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4.3)

y u(x,C)v(x) .

ɉɪɢɦɟɪ 4.4. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ

y

c

y ctg x

1

sin x .

ɉɨɥɚɝɚɟɦ y

u(x)v(x) ,

ɬɨɝɞɚ y

c

c

c

u v v u ɢ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ

ɜɢɞ

c

c

uv ctg x

1

;

sin x

u v v u

(4.9)

1

c

c

v ctg x)

sin x .

u v u(v

dv

vctg x, dv

ctg xdx;

dx

v

ln | v |

ln | sin x | v

sin x.

ɉɨɞɫɬɚɜɥɹɹ v ɜ ɭɪɚɜɧɟɧɢɟ (9), ɩɨɥɭɱɚɟɦ

c

sin x

1

du

1

sin x ;

dx

sin2 x ;

u

du

dx

u

ctg x C.

sin2 x

Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɚɤɨɜɨ

y

uv

( ctg x C)sin x

cos x C sin x .

4.4. ɍɪɚɜɧɟɧɢɹ Ȼɟɪɧɭɥɥɢ

ɍɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ ɢɦɟɟɬ ɜɢɞ

yc P(x) y

Q(x) ym , ɝɞɟ m z 0, m z1.

Ɍɚɤɨɟ ɭɪɚɜɧɟɧɢɟ ɦɨɠɧɨ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɫ ɩɨɦɨɳɶɸ ɩɨɞɫɬɚɧɨɜɤɢ y uv

ɢɥɢ ɫɜɟɫɬɢ ɤ ɥɢɧɟɣɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɫ ɩɨɦɨɳɶɸ ɡɚɦɟɧɵ z y1 m .

ɉɪɢɦɟɪ 4.5. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ yc

y

x2

.

x

y

ɉɨɥɚɝɚɹ y

uv , ɩɪɢɜɨɞɢɦ ɭɪɚɜɧɟɧɢɟ ɤ ɜɢɞɭ

§du

u

·

§dv

x2 ·

v¨

¸ ¨

u

¸

0 .

(4.10)

© dx

x ¹

¨

¸

©dx

uv ¹

ɍɪɚɜɧɟɧɢɟ

du

u

0 ɢɦɟɟɬ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ u x .

x

dx

ɉɨɞɫɬɚɜɥɹɹ u ɜ ɭɪɚɜɧɟɧɢɟ (4.10), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ

dv

x

x2

0,

dv

1

.

dx

xv

dx

v

ȿɝɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ v

r 2x C . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ:

y x(r

2x C ) .

Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ n–ɝɨ ɩɨɪɹɞɤɚ ɢɦɟɟɬ ɜɢɞ

c

cc

(n)

) 0

F(x, y, y , y

,..., y

ɢɥɢ, ɟɫɥɢ ɨɧɨ ɪɚɡɪɟɲɟɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ

y

(n)

, ɬɨ y

(n)

c

(n 1)

) . Ɂɚɞɚɱɚ

f (x, y, y ,..., y

ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɹ

y

M(x) ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ,

ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ ɧɚɱɚɥɶɧɵɦ

ɭɫɥɨɜɢɹɦ

y

x x0 y0 , yc

x

x0

y0c ,..., y(n 1)

x

x0

y0(n 1) ,

(k 1) ɜɤɥɸɱɢɬɟɥɶɧɨ:

F(x, y(k ) , y(k 1) ,..., y(n) ) 0 .

ɉɨɪɹɞɨɤ ɬɚɤɨɝɨ

ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ

ɩɨɧɢɡɢɬɶ

ɧɚ

k ɟɞɢɧɢɰ

ɡɚɦɟɧɨɣ

y(k ) (x)

ɪ(x) . ɍɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ

c

(n k )

) 0 .

F(x, p, p ,..., p

ɂɡ

ɩɨɫɥɟɞɧɟɝɨ

ɭɪɚɜɧɟɧɢɹ,

ɟɫɥɢ

ɷɬɨ

ɜɨɡɦɨɠɧɨ,

ɨɩɪɟɞɟɥɹɟɦ

p f (x,C ,C

2

,...,C

n k

) ,

ɚ ɡɚɬɟɦ ɧɚɯɨɞɢɦ y ɢɡ ɭɪɚɜɧɟɧɢɹ

y(k )

f (x,C ,C

2

,...,C

n k

)

1

1

k–ɤɪɚɬɧɵɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ.

3. ɍɪɚɜɧɟɧɢɟ ɧɟ ɫɨɞɟɪɠɢɬ ɧɟɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ:

c

cc

(n)

)

0.