4. (+) dxxdy= 0.
. . ,
P (λx, λy) =λx+λy=λ (x+y) = λP (x,y), Q (λx, λy) = λx=λ (x) =λQ (x, y).
=u, u . : dy=du+ud. dy ,
(+u) d (du+ud) = 0, d + ud 2 duud= 0; d 2 du= 0 ddu= 0.
, . ,
, .
u , =x ln (Cx). .
, (9.3.2):
. . ..
5. : .
. . (10.6):
= u, : . , . , :
.
. .
,
6. = 1 = 1.
.
2 dy = (xy + y 2 ) dx (*)
=u. dy= ud+ du. dy (*),
2 (ud + du) = (.u+u 2 2 ) d;
x 2 (udx+xdu) =x 2 (u+u 2 ) dx;
udx+xdu= udx+ u 2 dx; ..
xdu= u 2 dx.
. ,
.
u = , .
=1, = 1,1=ln1 + C, =1. ,
,
.
7.
.
. . , x=u+α, y=v+β, (u+β+ 2 ) du ( 2 u+ 2 α+v+β+ 6 ) dv= 0, . .
(u+ (β+ 2 )) du ( 2 u+v+ ( 2 α+β+ 6 )) dv= 0.
α β , ,
α= 2, β= 2. (10.5): , .. .
8. .
. P (x)= ctg x, Q (x) = sin x. .
I.
, .. . , ( =0 ), ,
, ,
.
.
.
:
.
. .
, y= (x+C) sin x.
II.
|
|
.
,
.
u (x) , , .. ,
.
u= 1 × sin x.
1=1, u= sin x.
, , .. .
, y= (x+C) sin x, .
9.
. . , (10.6):
, .. :