n -
, (14)
1, 2,..., n .
(7), :
i () = i = cons t (i = 1, 2,..., n).
(14) :
1) , . 2.5;
2) (10):
= 0 + *,
0 (11);
* (14).
0, , , . 2.3.
* (14) ( ). , (14)
f (x) = a x × (Pn (x)cosb x + Qm (x)sinb x), (15)
Pn (x), Qm (x) n m ; a, b .
* (14)
*= s × a x × ( cosb x + sinb x), (16)
s ab i ( ab i , s = 0); k = max (n, m); , k .
f (x) (14), (15) *, (16), . 2.
2.
n -
| f (x) (14) | * (14) | |
l (12) | l (12) | ||
1 | f (x) = Pn (x) n | l = 0 Þ * = Qn (x) n | l = 0 (12) s Þ Þ * = s × Qn (x) s + n |
2 | f (x) = a x × Pn (x) | l = a Þ Þ * = a x × Qn (x) | l =a s Þ Þ * = a x × s × Qn (x) |
3 | f (x) = Pn (x) × cosb x + + Qm (x)sinb x | l = b i Þ Þ * = P'k (x) cosb x + Q'k (x) sin b x | l = b i s Þ Þ * = s × ( cos b x + Q'k (x) sin b x) |
4 | f (x) = a x × (Pn (x) × × cosb x + Qm (x) sinb x) | l = a b i Þ Þ * = a x × (P'k (x) cosb x + + Q'k (x) sin b x) | l = a b i s Þ Þ * = s × a x (P'k (x) cos b x+ + Q'k (x) sin b x) |
1.
.
. . (10):
|
|
= 0 + *,
0 ; * .
1. 0
.
.
: k 1,2 = 1 (. 2.3, 1); k 3,4 = i (. 2.3, 3; a = 0, b = 1). ,
0 = 1 x + C 2 + 3cos x + C 4sin x.
2. * .
f (x) = 10 cos x
(15); . 2 3, Pn (x)= P 0(x)=10,
b =1, Qn (x) = Q 0(x) = 0.
i , . 2 ( ) * :
* = × ( 1cos x + 2sin x).
1, 2. :
*' = ( × ( 1 cos x + 2sin x))' = 1 cos x + 2sin x + × ( 1sin x + 2 cos x);
*'' = 2 2cos x 2 1sin x × ( 1 cos x + 2sin x);
*''' = 3 1cos x 3 2sin x × ( 1 sin x + 2cos x);
* IV = 4 1sin x 4 2 cos x + × ( 1cos x + 2sin x).
IV IV ‑ y = 10 cos x:
4 1sin x 4 2cos x + x ( 1cos x + 2sin x) 1cos x 2sin x = 10 cos x.
:
4 1sin x 4 2cos x = 10 cos x.
sin x cos x:
4 1 = 0; 4 2 = 10.
1 = 0; 2 = 5/2. * :
* = × ( 1cos x + 2sin x) = 5/2 sin x.
3. ,
= 0 + * = 1 x + C 2 - + 3cos x + C 4sin x ‑ 5/2 sin x.
'' + p ' + qy = f (x),
p, q ; f (x) 0.
, . 2.4, .. (14) n = 2.
2.
'' ‑ 3 ' + 2 y = x 2 + 3 x.
. . (10):
= 0 + *,
0 '' 3 ' + 2 y = 0;
* .
1. 0
'' ‑ 3 ' + 2 y = 0.
k 2‑ 3 k + 2 = 0.
k 1= 1, k 2= 2 (. 2.3, 1). ,
|
|
0 = 1 x + C 2 2.
2. * .
f (x) = x 2 + 3 x 2- (15). . 2 1, Pn (x)= P 2(x)= x 2 + 3 x. 0 , . 2 ( ) * :
* = Q 2(x) = 1 x 2 + 2 x + 3.
1, 2, 3. :
*' = ( 1 x 2 + 2 x + 3 = 2 1 x + 2; *'' = 2 1.
'', ' :
2 1 6 1 x 3 2 + 2 1 x 2+ 2 2 x + 2 3= x 2+ 3 x.
:
2 1 x 2+ (2 2 6 1) x + (2 1 3 2 + 2 3) = x 2+ 3 x.
x:
, 1 = 1/2; 2 = 3; 3 = 4. * :
* = x 2 + 3 x + 4.
3. ,
= 0 + * = 1 x + C 2 2 + x 2 + 3 x + 4.
3.
'' ‑ 2 ' + y = x ∙ ex.
. . (10):
= 0 + *,
0 '' ‑ 2 ' + y = 0;
* .
1. 0
'' ‑ 2 ' + y = 0.
k 2 2 k + 1 = (k 1)2 = 0.
k 1 = k 2= 1 2 (. 2.3, 2). ,
0 = x ( 1+ C 2 ).
2. * .
f (x) = x ∙ x (15). . 2 2, Pn (x)= P 1(x)= , a = 1. a = 1 2, . 2 ( ) * :
* = x × 2 × Q 1(x) = x × 2 × ( 1 x + 2) = x × ( 1 3+ 2 2).
1, 2. :
*' = ( x × ( 1 3+ 2 2) = x × ( 1 3+ ( 2+ 3 1) 2+ 2 2 x);
*'' = x × ( 1 3+ ( 2+ 3 1) 2+ 2 2 x + 3 1 2+ 2( 2+ 3 1) + 2 2).
'', ' :
x × ( 1 3+ ( 2+ 3 1) 2+ 2 2 x + 3 1 2+ 2( 2+ 3 1) + 2 2) 2 x × ( 1 3+ ( 2+ 3 1) 2+ 2 2 x +
+ x × ( 1 3+ 2 2) = ∙ .
x
6 1 x + 2 2= x.
x: 6 1 = 1; 2 2 = 0. 1 = 1/6; 2 = 0. * :
* = x × 3.
3. ,
= 0 + * = x ( 1+ C 2 ) + x × 3 = x ( 1+ C 2 + 3).
4. (), (t), , L, R C. i t, .
|
|
. , :
(t) = UL + UR + UC,
UL = ; UR = R i; UC = .
, : , , UC = , UC = + . q 0= 0. , - (t) = + Ri + . t, :
+ R + .
: 1) (t) = = onst; 2) (t) = E sinw t.
. . :
+ + .
.
³ 0, . . .
< 0,
,
, .
, .
,
; .
i t,
.
t = 0 i ,
0 = 1, ,
1 = 0, .
,
.
. (t) = E sinw t, E × w × cosw t.
+ + E × w × cosw t.
(10):
,
i 0 - ( , ); - . .
f (t) = E × w × cosw t (15);
. 2 3, Pn (t)= P 0 (t) == E × w × cosw t, b =w, Qn (t) = Q 0 (t) = 0. w i , . 2 (- ) :
= 1 cosw t + 2 sinw t.
1, 2. :
= (1 cosw t + 2 sinw t = - 1 wsinw t + 2 wcosw t;
= - 1 w2cosw t 0 -2 w2sinw t.
, i :
- 1 w2cosw t - 2 w2sinw t + (- 1 wsinw t + 2 wcosw t) + (1 cosw t + 2 sinw t) =
= E × w × cosw t.
, sinw t cosw t, :
1 = ; 2 = .
, :
= cosw t + sinw t.
,
+ cos w t + sin w t,
, . , .
;
t:
.