математика

ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ z( y)

ɩɨ y:

yc

z;

ycc

dz

dz

dy

dz

z;

yccc

d

2 z

z2

§ dz ·2

z

ɢ ɬ. ɞ.

¨

¸

dx

dy

dx

dy

dy2

¨

¸

©dy ¹

ɉɨɞɫɬɚɜɢɜ

ɷɬɢ

ɜɵɪɚɠɟɧɢɹ

ɜ

ɭɪɚɜɧɟɧɢɟ ɜɦɟɫɬɨ

c

cc

(n)

,

ɩɨɥɭɱɢɦ

y , y ,..., y

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (n 1) –ɝɨ ɩɨɪɹɞɤɚ.

ɍ

Ɂɚɦɟɱɚɧɢɟ. ɉɪɢ

ɪɟɲɟɧɢɢ

ɡɚɞɚɱɢ Ʉɨɲɢ:

ɧɚɱɚɥɶɧɵɟ

ɭɫɥɨɜɢɹ

ɥɭɱɲɟ

Ɍ

ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜ ɩɪɨɰɟɫɫɟ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ

ɭɪɚɜɧɟɧɢɹ.

ɇ

ɉɪɢɦɟɪ 4.8. Ɋɟɲɢɬɶ ɡɚɞɚɱɭ Ʉɨɲɢ

Ȼ

yy

cc

y

4

c

2

,

y(0)

c

( y )

1, y (0) 0 .

Ɋɟɲɟɧɢɟ.

Ⱦɚɧɧɨɟ

ɭɪɚɜɧɟɧɢɟ

ɧɟ

ɫɨɞɟɪɠɢɬ

ɧɟɡɚɜɢɫɢɦɭɸ

ɩɟɪɟɦɟɧɧɭɸ,

ɩɨɷɬɨɦɭ ɩɨɥɚɝɚɟɦ y

c

z( y) . Ɍɨɝɞɚ

y

cc

z

dz

ɣ

, ɢ ɭɪɚɜɧɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ

dyɢ

yz

dz

z2 y4 .

ɪ

dy

ɬɨɝɞɚ ɦɵɨɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɉɭɫɬɶ

yz z 0,

z z( y)

ɬ

dz

z

y3

ɢ

dy

y

z

.

ɡ

y2 C1 . ɂɡ ɭɫɥɨɜɢɹ yc

z

ɩɪɢ y

1 ɢɦɟɟɦ

Ɋɟɲɚɹ ɟɝɨ, ɧɚɯɨɞɢɦ z

ry

0

ɨ

z

ry

y2 1

ɢɥɢ

dy

ry

y2 1 .

ɂɧɬɟɝɪɢɪɭɹ ɷɬɨ

C

1 ,

ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

1

ɩ

dx

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ

ɭɪɚɜɧɟɧɢɟ

ɫ

ɪɚɡɞɟɥɹɸɳɢɦɢɫɹ

ɩɟɪɟɦɟɧɧɵɦɢ,

ɢɦɟɟɦ

ɟ

1

arccos

1

r x

C2 . ɉɨɥɚɝɚɹ

y

1 ɢ

x

0 ,

ɩɨɥɭɱɢɦ C2

0 ,

ɨɬɤɭɞɚ

cos x ɢɥɢ

y

Ɋ

y

sec x .

y

ek1x (C C

2

x C

3

x2 ... C

r

xr 1 ) .

1

4. ɉɚɪɚ ɤɨɦɩɥɟɤɫɧɨ–ɫɨɩɪɹɠɟɧɧɵɯ ɤɨɪɧɟɣ k1,2

DrEi

ɭɪɚɜɧɟɧɢɹ (5.2) ɢɦɟɟɬ

ɤɪɚɬɧɨɫɬɶ r.

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ r ɩɚɪ ɱɥɟɧɨɜ

C ek1x

... C

ek2 r x ɜ

1

2r

ɮɨɪɦɭɥɟ (5.3) ɡɚɦɟɧɹɸɬɫɹ ɫɥɚɝɚɟɦɵɦ

ɍ

4

2

xr 1 ) cosEx (C

Ɍ

eDx [(C C

2

x ... C

r

r 1

C

r 2

x ... C

2n

xr 1 )sinEx] .

1

ɉɪɢɦɟɪ

5.1.

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

y

IV

5y

cc

4 y

0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

k

5k

4

0

ɢɦɟɟɬ

ɤɨɪɧɢ

k1,2

r1,

k3,4

ɇ

r2 . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ

ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ

Ȼ

y C e x

C

2

e x C

3

e2 x C

4

e 2 x .

1

ɉɪɢɦɟɪ

5.2.

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

ycc 2 yc 5y

0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ k 2 2k 5

0 ɢɦɟɟɬ ɤɨɪɧɢ k

1 r 2i . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ

1,2

ɣ

ex (C cos 2x C

ɪ

y

2

sin 2x) .

ɢ

1

ɨ

ɉɪɢɦɟɪ

5.3.

y

cc

2 y

c

y

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

0 .

ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

k 2

2k 1

0

ɬ

2–ɤɪɚɬɧɵɣ

ɤɨɪɟɧɶ

k

1, ɩɨɷɬɨɦɭ

ɨɛɳɟɟ

ɢɦɟɟɬ

ɢ

1,2

ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ

y

e x (C

C x) .

1

ɡ

2

ɨ

ɭɪɚɜɧɟɧɢɟ y

IV

8 y

ccc

16 y

c

0 . ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɉɪɢɦɟɪ 5.4.

Ɋɟɲɢɬɶ

ɭɪɚɜɧɟɧɢɟ

k

5

8k 3

16k

0

ɢɦɟɟɬ ɤɨɪɧɢ

k1

0 ,

k2,3

2i,

k4,5

2i .

Ɉɛɳɟɟ

ɩ

ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɬɚɤɨɜɨ

ɟ

Ɋ

y

C1 C2 cos 2x C3 sin 2x C4 x cos 2x C5 x sin 2x .

y(n)

a1 y(n 1)

... an 1 yc an y

f ( x) ,

(5.4)

ɝɞɟ ai

Ɍ

R,

f ( x) – ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ.

ɇ

ɍ

ɉɭɫɬɶ

y

C1 y1 C2 y2 ... Cn yn

–

Ȼ

(5.5)

– ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (5.1), ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɭɪɚɜɧɟɧɢɸ

(5.4).

Ɇɟɬɨɞ ɜɚɪɢɚɰɢɢ

ɩɪɨɢɡɜɨɥɶɧɵɯ

ɩɨɫɬɨɹɧɧɵɯ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɨɛɳɟɟ

ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (5.4) ɢɳɟɬɫɹ ɜ ɜɢɞɟ

ɣ

ɢ

y

C1 (x) y1 C2 ( x) y2 ... Cn ( x) yn ,

ɪ

ɝɞɟ C1(x),..., Cn (x) – ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ. ɗɬɢ ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ

ɫɢɫɬɟɦɵ

­

C1c(x) y1 C2c (x) y2

Cnc

(x) yn

0;

°

ɬ

°

C

c

(x) y

c

C

c

(x) y

c

c

c

0;

°

1

1

2

2

n

n

®

ɢ

°

(n 1)

(n 1)

(n 1)

... Cnc

f (x),

¯C1c

(x) y1

C2c (x) y2

(x) yn

i

dCi (x)

ɡ

i

c

ɝɞɟ C

ɨ

dx

y

cc

p x y

c

q x y

f (x)

ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ ɢɦɟɟɬ ɜɢɞ

ɟ

(x) y

0;

­Cc

(x) y Cc

2

ɩ1 1

2

®

(x) y1c C2c (x) y2c

f (x).

Ɋ

¯C1c

ɉɪɢɦɟɪ 5.5. Ɋɟɲɢɬɶ ɭɪɚɜɧɟɧɢɟ ycc

yc

1

.

1 e x