4.
n -
,(7)
1(), 2(),..., n (), f (x) (, b).
(7) f (x) ¹ 0, , .
(7) f (x) = 0,
(8)
, .
, , .
, .
.
. 1, 2,..., n (8),
= 1 1 + 2 2 +... + n n, (9)
1, 2 ,..., n , (8).
(8) :
1) 1, 2 ,..., n ;
2) 1, 2,..., n (9) .
, 1(), 2(),..., n () (, b), a1, a2,..., a n, ,
a1 1 + a2 2 +... + a n n = 0.
.
䠠
a1 1 + a2 2 = 0, ,
.
. 1) 1 = , 2 = 3 ( + 1) ; 2) 1 = 2 ,
2 = 2 ‑ 1 ; 3) 1 = 3 cos 2 x, 2 = 4 cos 2 ; 4) 1 = , 2 = .
(8) 1, 2,..., n .
, (8) 1, 2,..., n , .. (, b), , " Î (, b) () :
.
(7)
, (10)
0 (8);
* (7).
1, 2,..., n (8), (7) , . 2.5.
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, (11)
1, 2,..., n .
(8), :
i () = i =const (i = 1, 2,..., n).
(11)
, (12)
(11) k, . (12) n - k, n , .
(11) (9) (12) :
1) k (9) × kx;
1) m (9) ( 1 + C 2 x + C 3 x 2 +... + Cmxm - 1) × kx;
3) k 1,2 = i (9) a x × ( 1cosb x + C 2sinb x);
4) k 1,2 = a b i m (9)
a x (( 1 + C 2 x +... + Cmxm 1)cosb x + ( 1 + C 2 x +... + C m xm 1)sinb x).
1. .
. .
.
k 1 = 0, k 2 = 1, k 3 = 2 ( 1). , 0× x = 1, 1× x = x, x ,
= 1 + C 2 x + C 3 2.
2. .
. .
.
: k 1= 0 ( 1), k 2,3 = 1 2 ( 2). , 0× x = 1, 1× x = x, x × x ,
= 1 + C 2 x + C 3 2.
3. .
. .
.
: k 1,2 = i, k 3,4 = 2 i ( 3; a = 0, b = 1 b = 2 ). , 0× x × cos x = cos x, 0× x × sin x = sin x, 0× x × cos 2 x = cos 2 x, 0× x × sin 2 x = sin 2 x ,
= 1 cos x + C 2sin x + 3 cos 2 x + C 4sin 2 x.
4. .
. .
|
|
.
. : k 1,2 = 2 i 2 ( 4; a = 0, b = 2); k 3,4= i ( 3; a = 0, b = ). ,
= ( 1+ C 2 x)cos 2 x + (3 + C 4 x)sin 2 x + 5cos x + C 6sin x.
n - (11) n = 2, .. .
, (13)
p, q .
1.
, . 2.3, .
1. (13).
2.
.
3. D = p 2 4 q.
) D > 0, k 1, k 2,
= 1 × + C 2 × .
) D = 0, k 1 = k 2= k,
= 1 × kx + C 2 × × kx.
) D < 0, k 1,2 = ab i,
a x × ( 1cosb x + C 2sinb x).
. 1.
1.
2-
D > 0 | D = 0 | D < 0 | |
k 1¹ k 2Î R | k 1= k 2= k Î R | k 1,2 = a b i Î C | |
1 + C 2 | 1 kx + C 2 kx | a x ( 1cos b x + C 2sin b x) |
5. .
. .
.
(D < 0) k 1,2 = 1 3 i ( 3; a = 1, b = 3), , x cos 3 x, x sin 3 x ,
= x ( 1cos 3 x + C 2sin 3 x).
6. , : (0) = 1, .
. .
.
(D > 0) k 1= 0, k 2= 3 ( 1). ,
= 1+ C 2 3.
: ' = 3 C 2 -3. , 1 C 2:
1= 5/3, C 2 = 2/3. , ,
.
7. .
, l, , .. . , w 0 = const, a. w , w = w () , -, , (. 11).
. l, , Q. dx, (. 11).
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. 11
w. D t , + D .
, D t ,
‑ = =
.
, , :
= .
w 0 = const, w w 0 = u,
, .
, .
u = 1 a + C 2 - a ,
w = w0 + 1 a + C 2 - a ,
1 2 () .
, , w 1 w 2(w 2 < w 1). : w (0) = w 1; w (l) = w 2. ,
, 1 2, , .