математика

y*

x cos x x2 sin x .

Ɉɛɳɟɟ

ɪɟɲɟɧɢɟ

ɛɭɞɟɬ

y

y y C cos x C

2

sin x x cos x x2 sin x .

1

ɇɚɯɨɞɢɦ

yc

C1 sin x C2 cos x cos x x sin x 2x sin x x2 cos x .

Ɍɚɤ

ɤɚɤ

c

1, ɬɨ 0

ɍ

y(0) 0, y

(0)

C1, C

C2

1. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, C1

0, C2

0 . ɉɨɞɫɬɚɜɥɹɹ

C1

0, C2 0

Ɍ

ɡɧɚɱɟɧɢɹ

ɜ

ɨɛɳɟɟ

ɪɟɲɟɧɢɟ,

ɩɨɥɭɱɢɦ

ɱɚɫɬɧɨɟ

ɪɟɲɟɧɢɟ

y x cos x x2 sin x .

ɇ

ɉɪɢɦɟɪ 6.4. Ɉɩɪɟɞɟɥɢɬɶ ɜɢɞ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɧɟɨɞɧɨɪɨɞɧɨɝɨ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɤɨɪɧɢ k1

3 2i ,

k2

3 2i

ɟɝɨ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɟɝɨ ɩɪɚɜɚɹ ɱɚɫɬɶ

Ȼ

f (x)

e3x (cos 2x sin 2x) .

ɣ

Ɋɟɲɟɧɢɟ.

ȼ

ɩɪɚɜɨɣ

ɱɚɫɬɢ

D

3, E 2, Pn (x) 1, Qm ( x)

1

– ɦɧɨɝɨɱɥɟɧɵ

ɧɭɥɟɜɨɣ

ɫɬɟɩɟɧɢ,

DrEi

3 r2i

ɢ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ

ɹɜɥɹɸɬɫɹ

ɤɨɪɧɹɦɢ

ɪ

ɭɪɚɜɧɟɧɢɹ. ɉɨɷɬɨɦɭ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ

y

xe

3x

( Acos 2x Bsin 2x) ,

ɬ

ɝɞɟ A ɢ B – ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵɨ .

7. ɋɂɋɌȿɆɕ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ.

ɡ

ɆȿɌɈȾ ɂɋɄɅɘɑȿɇɂəɢ

. ɆȿɌɈȾ ɗɃɅȿɊȺ Ɋȿɒȿɇɂə

ɨ

ɅɂɇȿɃɇɕɏ ɋɂɋɌȿɆ ɋ ɉɈɋɌɈəɇɇɕɆɂ ɄɈɗɎɎɂɐɂȿɇɌȺɆɂ

ɩ

ɟ

7.1 ɇɨɪɦɚɥɶɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɨɛɵɤɧɨɜɟɧɧɵɯ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

Ɋ

ɇɨɪɦɚɥɶɧɚɹ

ɫɢɫɬɟɦɚ

n–ɝɨ

ɩɨɪɹɞɤɚ ɨɛɵɤɧɨɜɟɧɧɵɯ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ

­ dx1

f1(t, x1, x2 ,..., xn );

°

dt

°dx

2

°

f2 (t, x1, x2 ,..., xn );

®

dt

...

°

°dx

n

fn (t, x1, x2 ,..., xn ).

°

¯ dt

ɝɞɟ t

–

ɧɟɡɚɜɢɫɢɦɚɹ

ɩɟɪɟɦɟɧɧɚɹ;

x1, x2 ,..., xn –

ɧɟɢɡɜɟɫɬɧɵɟ

ɮɭɧɤɰɢɢ ɨɬ

t; f1,

f2 ,..., fn

ɍ

– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.

Ɍ

Ɇɟɬɨɞ ɢɫɤɥɸɱɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ

ɩɪɢɜɨɞɢɬɫɹ ɤ

ɨɞɧɨɦɭ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ

ɭɪɚɜɧɟɧɢɸ n–ɝɨ

ɩɨɪɹɞɤɚ

ɫ

ɨɞɧɨɣ

ɇ

ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɟɣ (ɢɥɢ ɤ ɧɟɫɤɨɥɶɤɢɦ ɭɪɚɜɧɟɧɢɹɦ, ɫɭɦɦɚ ɩɨɪɹɞɤɨɜ ɤɨɬɨɪɵɯ

Ȼ

ɪɚɜɧɚ n). Ⱦɥɹ ɷɬɨɝɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɸɬ ɨɞɧɨ ɢɡ ɭɪɚɜɧɟɧɢɣ ɫɢɫɬɟɦɵ

ɢ ɢɫɤɥɸɱɚɸɬ ɜɫɟ ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ, ɤɪɨɦɟ ɨɞɧɨɣ.

ɣ

ɉɪɢɦɟɪ 7.1. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ

dx y

dy y(x 2 y 1)

ɢ

,

ɪ

dt

t

dt

t(x 1)

y(1)

ɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ x(1)

1;

4 .

ɨ

c

Ɋɟɲɟɧɢɟ. Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦɬɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ t:

y

xcc

y t

.

Ɂɚɦɟɧɹɹ

c

ɢ

t2

c

ɡɞɟɫɶ

y

ɟɟ

y

ɡɧɚɱɟɧɢɟɦ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɢ ɩɨɞɫɬɚɜɥɹɹ

x t ,

ɧɚɣɞɟɧɧɨɟ ɢɡ ɩɟɪɜɨɝɨɡɭɪɚɜɧɟɧɢɹ, ɩɨɥɭɱɢɦ ɩɨɫɥɟ ɭɩɪɨɳɟɧɢɹ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ

c 2

2(x )

ɩɨɪɹɞɤɚ

x

cc ɨ

x

1

.

ɩ

ɂɧɬɟɝɪɢɪɭɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɨɧɢɠɚɹ ɩɨɪɹɞɨɤ:

ɟ

dp

dp

2 p

dp

2dx

xc

p;

p p(x); xcc

dx

p;

dx

x 1

;

dx

x 1

;

Ɋ p C (x 1)2 ;

dx

C

(x 1)2

;

1

C t C

2

; x

C1t C2 1

.

1

dt

1

x 1

1

C1t C2

y

C1t

.

(C t C

2

)2

1

Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɛɭɞɟɬ

x

C1t C2 1

,

y

C1t

.

ɍ

C1t C2

(C t

C

2

)2

1

Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɩɨɞɫɬɚɜɢɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ

C1 C2

1

Ɍ

x(1)

1,

y(1)

4 .

ɉɨɥɭɱɢɦ 1

;

4

C1

, ɨɬɤɭɞɚ

C C

2

(C C

2

)

1

1

ɇ

C

1,

C

2

1

.

Ȼ

1

2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɵɦ ɱɚɫɬɧɵɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɛɭɞɟɬ ɩɚɪɚ ɮɭɧɤɰɢɣ:

x

2t 3

, y

4t

.

ɣ

2t 1

(2t 12

ɢ

ɉɪɢɦɟɪ 7.2. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ

dx

dy

ɪ

2 y 5x et

,

x

6 y e 2t .

dt

dt

ɨ

Ɋɟɲɟɧɢɟ.

Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦɬɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ: xcc

2 yc 5xc et . Ɂɚɦɟɧɹɟɦ

c

ɢ

1

c

t

y

y

2 (x

5x e

) .

ɟɟ ɡɧɚɱɟɧɢɟɦ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɩɨɞɫɬɚɜɥɹɟɦ ɡɚɬɟɦ

ɡ

ɉɨɥɭɱɢɦ ɥɢɧɟɣɧɨɟ ɧɟɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ

ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ

ɨ

2t

t

x

cc

11x

c

28x

2e

7e

.

ɩ

ȿɝɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ

ɟ

1

7

Ɋ

x C e 4t

C

2

e 7t

e 2t

et

1

2

40

ɤɚɤ

ɫɭɦɦɚ ɨɛɳɟɝɨ

ɪɟɲɟɧɢɹ x

C e 4t

C

e 7t

ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ

(ɩɨɥɭɱɟɧɨ

2

1

ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ

ɢ

ɱɚɫɬɧɨɝɨ

ɪɟɲɟɧɢɹ x*

1

e

2t

7

e

t

ɧɟɨɞɧɨɪɨɞɧɨɝɨ

5

40

ɭɪɚɜɧɟɧɢɹ).

ɉɨɞɫɬɚɜɥɹɹ x ɢ xc

ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ y, ɩɨɥɭɱɢɦ

ɍ

y

1

(xc 5x e

t

1

C1e

4t

C2 e

7t

3

e

2t

1

e

t

)

.

2

2

10

40

Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ

x C e 4t

C

2

e 7t

1

e 2t

7

et ;