(. .) 1- F(x, , ¢) = 0 (1)
¢ = f(x, ), (1¢)
d = f(x,)dx. (1¢¢)
(1) - (1¢¢).
.. (1) - (1¢¢) = 
, (2)
- , :
) .. ;
) 
. (*)
, = 0, 
(*).
= 0 (*) . ..
.. (, , ) =0, .
.. .
: m1(x) * m2(y) dx + n1(x) * n2(y)dx = 0, (m2(y) ¹ 0 n1(x) ¹ 0). (3)
: 
. 
(3)
2) ..
: ¢ = f(x,y), (4)
f(x,y) - , f(tx,ty) = f(x,y) t. (4) (4¢):
. (4¢)
= u * x : y¢ = u¢x + u.
; 
-
(4).
.. 1- .
:
¢ + () * = Q (). (5)
y = u * V, u = u(x), V = V(x); y¢ = u¢V + uV¢.
u¢V + uV¢ + P(x) * uV = Q(x). (5¢)
V , V¢ + P(x) * V = 0. - .. .
, (5¢) :
u¢V = Q (x), .. (V !): 
, : 
.
, 
. (5).
4) .
: y¢ + P(x) * y = Q(x) * yn (6)
(n ¹ 0 n ¹ 1). - , (5).
1
1. 
; 2. 
;
3. 
; 4. 
.
1. 
.
.
; 
.
.
2. = t * x, ¢ = t¢x + t :
, 
,
.
3. 
- , 
,
= U * V, y¢ = U¢V + UV¢,
- . . . 
;
- .
,
- ( ).
, ( ).
2-
:
F(x, y, y¢, y¢¢) = 0 (7)
y¢¢ = f (x, y, y¢). (7¢)
. (8)
(9) (7) (7¢) (1,2 - ), 1 2 (7), (8).
|
|
|
2-
:
¢¢ + 1¢ + 2 = 0. (13)
:
2 + 1 + 2 = 0. (13¢)
1, 2 - . 3 :
) , ;
) , : 1 = 2 = (2- );
) , 1,2 = i, i = 
).
(13) . 1.
1
| 1, 2 | (13) ( )
|
| () |  .
|
| () |  .
|
| () |  .
|
2
1. ¢¢ - 4112 ¢ + 3111 = 0; 2. ¢¢ + 222 ¢ + 222 = 0;
3. ¢¢ + 21¢ = 0; 4. ¢¢ + 332 = 0.
1. : 
.
, 1 = 311, 2 = 11 ( , ). 1- . 1. 
- .
2. : 
.
. 1 ...
3. : 
.
.
4. : 
.
.
. . (..), .
5. ¢¢ - 312 ¢ + 212 = 0, (1) = 1, ¢(1) = 2;
6. ¢¢ - 222 = 0, (0) = 2, ¢(0) = 3.
5. :
, 1, 2 ...
6. :
, 1, 2 ...
