V.
1. , , = () ', '',, (n), ..
F(; ; '; '';; (n)) = 0.
, = () .
2. , .
3. n- (; b) = j(), (; b) n- , , ..
F(x; j(); j'();; j(n)()) º 0.
.
1. :
F(x; ; ') = 0. (4.1)
(4.1) ',
' = f(x; ) (4.2)
- , .
j(; ) d + g(x; ) d = 0.
2. = j(; C), C C.
(; ; ) = 0, , .
3. , - = 0, .. = j(; 0).
, , .
, , .
4. , , .
= () ' = f(x; y), (0) = 0 (), 0, 0 - .
1. j() d = g(x) dx .
.
|
|
F(y) = G(x) + C.
2. ' = j() g(x) .
.
1. '
.
2. d
d = j() g(x) d.
3. , . . , , , d, , , d.
j(). d = g(x) d .
4. ;
F() = G(x) + C .
5. (), (0) = 0.
F(y) = G(x) + C0.
1. ' + 1 + 2= 0, (1) = 1.
. . , .
1. + 1 + 2= 0 ½ × d.
2. d + (1 + 2) d = 0.
3. , ( ¹ 0)
dy = - .
4.
,
*, ¹ 0.
,
x = .
5. ,
= 1 =1.
1 = e-1, C = e.
= .
1. , .
' + () = Q(x), (4.3)
() Q(x) .
Q(x) º 0, (4.3) .
= .
( ), , (4.3)
= ,
() .
1.
' + 2 = 2 .
. . ' + 2 = 0, . . , , 3.
+ 2 = 0 ½ × d
d + 2 d = 0.
,
+ 2 d = 0.
:
;
ln|| + 2+ ln|1| = 0, ln|1| = -2.
= , .
= () ,
() .
.
+ 2 () = 2 .
():
.
() = 2+ C.
,
= (2+ C) ,
C .
|
|
( ) (4.3), (4.3)
= u v, (4.4)
u v .
(4.4),
' = .(4.5)
' (4.3),
,
. (4.6)
(4.4) u v, , u, , u = 0. u ,
. (4.7)
, u = u(x). (4.6).
U(x) = Q(x).(4.8)
, v = v(x; C), C (4.8). = u v u v ,
= u(x) v(x; C).
, .
2.
( 1) ' + = 2(2 1), (2) = 4.
. . , x ( 1) ( ¹ 0; ¹ 1) (4.3).
' + . (4.9)
= u v. ' = u' v + u v'.
' (4.9),
u' v + u v' + ,
v . (4.10)
u , , . .
. (4.11)
(4.10)
. (4.12)
(4.11) :
,
.
,
,
. (4.13)
u(x) (4.12), v.
,
,
,
v(x; C) = 2 + C. (4.14)
= u v u v (4.13) (4.14),
(2 + ).
, = 4 = 2.
4 = (4 - 2 + ), . . = 0.
,
= 2.
* , , C ln|C|.