2
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82-08
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2005 .
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: - 4 -
1. dy/dx = - (1 - y2)1/2 - 4 -
2. dy/dx = (y+(y2-4x2)1/2) / x - 5 -
3. dy/dx = (4-y2)1/2 / y - 6 -
4. dy/dx = 1-y4 - 7 -
5. dy/dx = 1/2×(y2-1) - 8 -
6. x=y-0.1x - 9 -
y=-x-0.1y - 9 -
7. x=y-x(x2+y2) - 10 -
y=-x-y(x2+y2) - 10 -
8. dy/dx = xy(1+2y)/(1-y2) - 11 -
. - 12 -
, .
, .
, . , .
:
1. dy/dx = - (1 - y2)1/2
(H) : . .. Y 1 , f(x,y). Ÿ , (H).
?
, Yn → 1 , . .
Sn , S (n+1) .
, , , , . , , , Y = 1, , - .
, , , .
.
2. dy/dx = (y+(y2-4x2)1/2) / x
. Y=0.
: .
. . . .
|
|
. -, , .
Y → 0, .
.
H > 0 , . , H < 0. .
: : . , .
: .
(. .), . Y=0, 0, , , . . , .
, x=0, y>0. .
3. dy/dx = (4-y2)1/2 / y
, Y=2 , Y = 0 .
, > 0 (- . ).
. - . , (- . ).
H < 0, , .
, .
, Y0 Y=0, . , , . , , , . .
4. dy/dx = 1-y4
- . , . . 0.00202, . . , Y = 1, , Y = 1, ( ). , . 0.00202 .
Y = 1, , ;).
- :
, dy/dx = f (x,y) Y=a¹0. . Y(x0) = Y0¹a, f(xn,yn)0 n¥. h×f(xn,yn)0, .., , , yn ; . Yn , , , , . , h .
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5. dy/dx = 1/2×(y2-1)
X = -1,62566841
Y = 0,94285715
= 0,05
. , , . , . , . .
, , - .
.
6. x=y-0.1x
y=-x-0.1y
, . - , , , , .
h>0,198 , -, .
, , . h=0.198 , , , h .
. n Jn. n+1 Jn+1.
yn+1 + Jn+1 =yn + jn + hf(xn, yn+jn) +o(h)
jn, h , .
h, | 1+h f(xn,yn)/y | . ,
( +h Df(xn,yn)/Dy) ( , - ) .
: Ÿ :
| | >1 h> 0.2/1.01 0.2
7. x=y-x(x2+y2)
y=-x-y(x2+y2)
, , . , , .
, /2. .
8. dy/dx = xy(1+2y)/(1-y2)
5 , , , -, . , - .
1) Y<-1, X>0, H>0
2) Y<-1, X<0, H<0
3) -1<Y<0, X
h>0
h<0
4) Y>1, X ≥ 0.60962558, h<0
5) Y>1, X ≤ 0.60962558, h<0
, , , .
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y=1, , y 1. .. .
, .
,
: .
.
:
- , .
- , .
- .
- , , , .
- , - , , , .