: y,
y¢ = f(x),
dy = f(x)dx. , ,
y = ò f(x)dx
, , . : y, ,
F(x, y, y¢, y¢¢,..., y(n)) = 0. (4.1)
, x, y n , .
, , , ( ) . , , .
:
y¢ - x2y + x3 = 0 - ,
y¢¢ + 4y¢ + cos x = 0 - ,
x y(5) + yy¢¢¢ = 1 - . .
, , , . - . y , , , , .
n - n 1, 2,..., cn
y = j(x, 1, 2,..., cn).
, x, y n , , y -
(x, y, 1, 2,..., cn) = 0,
(4.1).
, . . , , , . () , 1, 2,..., cn .
. , , . , n - - n .
( ) n - , n :
y(xo) = yo, y¢(xo) = yo¢,..., y(n-1)(xo) = yo(n-1),
n 1, 2,..., cn. 1-
F(x, y, y¢) = 0,
, y¢:
y¢ = f(x, y).
46. y¢ = 3x.
. ,
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y = ò 3x dx, y = 3x2/2 + C,
- . , , ,
y = 3x2/2 (= 0),
y = 3x2/2 + 5 ( = 5)
..
47. , 100 r . Yo , Yx - x . ,
Yx+1 = (1+r)Yx,
x = 0, 1, 2, 3,.... ( ),
Yx+1/2 = (1 + r/2)Yx,
x = 0, 1/2, 1, 3/2,.... , n x 0, 1/n, 2/n, 3/n,...,
Yx+1/n = (1 + r/n)Yx,
.
1/n = h, :
.
n ( n¥, h0) :
,
x 1- . , Yx - , x - , r - . :
Yx = e r x+C, Yx = P e r x, P eC.
Y(0) = Yo, P: Yo = Peo, , Yo = P. :
Yx =Yo e r x.
. 1- , Y .
48. Y , :
,
, , Y ( q). D:
dD/dt = qY.
Y D t.
Y = Yo D = Do t = 0. ,
, Y= Yo e k t. Y , dD/dt = qYo e k t.
D = (q/ k) Yo e k t +, = const, . , Do = (q/ k)Yo + . , ,
D = Do+(q/ k)Yo (e k t -1),
, k, .
n -
y(n) = f(x).
n .
49. y¢¢¢ = cos x.
. ,
y¢¢ = ò cos x dx = sin x + C1,
y¢ = ò (sin x + C1)dx = - cos x + C1x + 2,
y = ò (- cos x + C1x +C2)dx = - sin x + C1x2/2 +C2x+C3.
,
y = - sin x + C1x2/2 +C2x+C3.
, . (9.1) , :
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o(x)y(n)(x) + 1(x)y(n- 1)(x) +... + n - 1(x)y¢(x) + n(x)y(x) = f(x), (9.2)
o(x), 1(x),..., n(x), f(x) - . f(x) º 0, (9.2) , - . (9.2) - y(x) :
o(x)y(n)(x) + 1(x)y(n- 1)(x) +... + n - 1(x)y¢(x) + n(x)y(x) = 0. (9.3)
o(x), 1(x),..., n(x) , (9.2) :
oy(n)(x) + 1y(n- 1)(x) +... + n - 1y¢(x) + ny(x) = f(x) (9.4)
n .
(9.4) :
oy(n)(x) + 1y(n- 1)(x) +... + n - 1y¢(x) + ny(x) = 0. (4.5)
o = 1 (4.5)
y(n)(x) + 1y(n- 1)(x) +... + n - 1y¢(x) + ny(x) = 0. (4.6)
(9.6) y = e kx, k - . : y¢ = k e kx, y¢¢ = k2 e kx,..., y(n) = kn e kx. (9.6), :
e kx (kn + 1kn-1 +... + n-1k + n) = 0.
.. e kx ¹ 0,
kn + 1kn-1 +... + n-1k + n = 0. (4.7)
(4.7) k. (4.6). n - , , n , . k1, k2,..., kn - (4.7), - (4.7),
y = .
:
y¢¢ + y¢ +qy = 0. (4.8)
k2 + k + q=0 (4.9)
D = 2 - 4q .
1. D>0, k1 k2 (4.9) , :
y = c1 ex(k1x) + c2 ex(k2x).
2. D = 0, .. k1 k2 , :
y = (c1 + c2x) ex (k1x).
3. D<0, , k1 = a + bi, k2 = a - bi, i - . :
y = (c1 cos bx+c2 sin bx) ex (ax).
50. y¢¢ - y = 0.
. k2 - 1 = 0, k1 = 1, k2 = -1 . :
y = c1 e x + c2 e -x.
51. y¢¢- 4y¢ + 4y = 0.
. :
k2 -4k +4 = 0 (k - 2)2 = 0, .. k1= k2 =2, , :
y = e2x(c1+c2x).
52. y¢¢+9y = 0.
. k: k2+9 = 0, k = 3i, Þ a = 0, b = 3, , :
y = c1 cos 3x + c2 sin 3x.
2- , , , P . ,
D = a +aP, S =b+bP,
l , ( ), :
+ l (b - a) P = l (a - b).
P =`P = (a - b)/(b - a),
. = P -`P
+ l (b - a) = 0. (4.10)
. , k, :
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k2 + l (b - a) = 0.
(a<0, b>0, l>0) l (b - a) . w = . k 1,2 = i w. , (4.10) :
= C cos (wt-e),
C e , , . , `P, :
P = `P+ C cos (wt-e).
. Yx, , x. Yo 100 r . x (x = 0, 1, 2,...). x Yx.
Yx = (1+r)Yx-1.
Yo, , Yx = Yo x = 0. Yx , .. Yx-1; x .
, Y = Y(x), x, , , . , x, x, x+1, x+2,... : x, x-1, x-2,.... Yx,Yx+1,Yx+2,... Yx, Yx-1, Yx-2,.... Yx :
D Yx = Yx+1 - Yx,
D Yx+1 =Yx+2 - Yx+1,
DYx+2 = Yx+3 - Yx+2,
...............
D2Yx= DYx+1 - D Yx,
D2Yx+1= D Yx+2 - DYx+1,
D2Yx+2= D Yx+3 - DYx+2,
...............
D3Yx= D2Yx+1 - D2Yx,
D3Yx+1= D2Yx+2 - D2Yx+1,
...............
, x, Yx DYx, D2Yx, D3Yx,.... :
j(x, Yx, DYx, D2Yx D3Yx, DnYx) = 0, (5.1)
.
, . (5.1) , , , DYx, D2Yx, D3Yx,... Yx, Yx+1, Yx+2,.... (5.1) :
y(x, Yx, Yx+1,...,Yx+n) = 0, (5.2)
x(x, Yx, Yx-1,...,Yx-n) = 0. (5.3)
Yx n - x (x = 0, 1. 2,...), n :
Yx = Y(x, C1, C2,..., Cn).
. - :
D = D(P), S = S(P).
, ,
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D(P) = S(P).
`P ( ), -, `X, - :
`X = D (`P) = S(`P).
. :
Dt = D (Pt) St = S (Pt-1).
, , . : Pt-1 S (Pt-1), Pt , . , Pt - Xt :
Xt = D (Pt) = S (Pt-1).
, Po, P1 X1. , P1, P2 X2 .. Pt ( ):
D (Pt) = S (Pt-1).
, .5, D S - , ( `P `X) Q. Po. Qo S 1. P1, Q1 D (X1), Qo. Q1 S, X2, - Q2 D. P2. , . 5. ( - ) Q1, Q2, Q3,... D. Q. Q. , Pt `P, `P. - (X t).
X
S
(D; S)
x2 Q2
Q
`x
x3 Q3
x1 Q0 Q1
D
O P0 P2 `P P3 P1 P
. 5.
: D = a +aP, S = b +bP. `P `X
`X = a +a`P = b +b`P,
`P = (a - b)/(b - a), `X = (ba - ab)/(b - a). (5.4)
X t = a +aP t = b +bP t-1. (5.5)
, . P t = `P, X t = `X t:
`X = a +a`P = b +b`P. (5.6)
`P `X, (5.4). , - , , (5.5) .
(5.6) (5.5) t = P t -`P, x t = X t -`X.
x t = a t = b t-1. (5.7)
(5.7) (5.5), , ( , ). . c = b/a (5.7), t
t = c t-1. (5.8)
o t = 0 (5.8) :
t = o c t,
P t = `P + (Po - `P) c t.