23.
23.1. , .
, :
. (23.1)
,
. (23.1) . (23.2)
, (23.2) :
. (23.3)
L :
1) L [ cX ] = cL [ X ];
2) L [ X1 + X2 ] = L [ X1 ] + L [ X2 ],
(23.3) ( F = 0) : 1 2 (23.3), .
1, 2,, :
23.2. () 1, 2,, ,
, , α1,α2,, α , ,
α11 + α22 ++ α ≡ 0 (23.4)
. (23.4) αi = 0, .
. (23.4)
, (23.5)
, (23.4). , , , W = 0 1, 2,, [ a,b ]. :
23.1. .
(23.6)
: , (23.7)
αi . (23.7) (23.6) ekt, :
. (23.8)
, , , :
, (23.9)
k, .
, , (23.8), (23.6). . , β1, β2,, β ,
, , i. , , . , (23.7) , : , (23.10)
ci .
.
. :
k1 = 1, k2 =5. k = 1 : ,
. , . k = 5 ,
. . , : .
|
|
(23.6)
, γ ks.
.
. :
k1 = k2 = 3. x = (c1 + c2 t) e3t, y = (c3 + c4 t) e3t. 3 4 1 2. e3t te3t: (3 c1 + c2 + 3 c2t) e3t = (2 c1 + c3) e3t + (2 c2 + c4) te3t, c3 = c1 + c2,
c4 = c2. , : x = (c1 + c2 t) e3t, y = (c1+ 2 + c2t) e3t.
. (23.1) , , . .
.
. : . : , = 3, = 1. , : x = c1et + 2 c2e4t + 3 e5t, y = -c1et + c2e4t + e5t.
24.
. . . (), . . . .
, , , , . , , , , .
(24.1)
yi (t0) = yi0.
24.1. φi (t) (ǐ = 1,2,, n) ,
, yi (t) , , (24.2)
( ).
yi (t) (24.2) , φi (t) .
φi (t) ,
(24.3)
, .
. (24.3) .
.
(24.4)
. (24.5)
(24.4) (24.5) . , t. (24.5)
(24.6)
.
|
|
, (24.7)
.
.
24.2. (24.6) , ; . , .
. .
(24.1) , , : - , . (24.1) :
, (24.8)
.
= 0, = 0 :
, . (24.9)
:
.
:
1) k1 k2 . (24.9) : . :
) k1 < 0 k2 < 0, , , , t = t0 δ , t , ε , . .
) k1 > 0, k2 > 0, t t. , , t , .
) k1 > 0, k2 < 0 ,
t ε . .
2) k1,2 = p qi. (24.9)
, - 1, 2. :
) p < 0, q ≠ 0. , . , . , . .
) p > 0, q ≠ 0. , , .
) = 0. , , . , δ, , δ , ε (x ² (t) + y ² (t) < ε²).
3) : k1 = k2.
) k1 = k2 < 0. , . .
) k1 = k2 > 0. - .