обыкновенные диф ур-я 1-порядка

1

ϕ(x+ 21 x2y)

= 1 · ψ y22 +y .

A, ((B

y

ϕ(U1) = 1 # ! &/

/ A9B !

&

"

y1 = ψ(y) 5 & &10 !"

& ψ ! U2 ! U2 = y J A, 9*B

&

A, (9B & μ =

1

/ &

A, (9B

μ = 1

y

y

!&/ & ! 2 ! 2

(1 + xy)dx +

2 x2 + 1 + y dy = 0.

1

1

= &

2

x2

ϕ

1

U

x

y

y

ϕ y

y

ϕ

y

y

y.

=

+

2

+

+

) = 1 + y

) =

+ ln

( );

(

(

/6 xex + x2 − x

= 0

/

y dx +

y3 + ln x

dy

= 0

x

y

y

/ (y3 + cos x)dx + (3xy2 + ey)dy = 0

7 xdx + ydy =

xdy

ydx

2−

2

(

2xy + x

y +

x +y

+ y

)dy = 0

3

dx + (x

2

y3

2

2

) 4 /.

x3

2

+x

2

= C / 4y ln x+y

4

2

x

x

+

3 +xy

= C /6 y = x e

−xe

( x x2

*

2

2

−

/

x

2

3

=

Cx / xy3 +sin x+ey

= C 7 x2 +y2

2 arctg y

= C yex

x2 +

y2

C

y +

= C

2x + ln(x + y ) = C

(x − C)y = 2x

−y + 1 =

2

xy(arctg x + C)

x = f (y, y ).

A+ :B

D ! p = dy

x = f (y, p) 7 " M !

dx !&/

!

∂f

∂f

dx =

dy +

dp.

∂p

∂y

! & dx = dy

p 7 &!" !&/ & &

dy

=

∂f

dy +

∂f

dp,

x = f (y, p),

A+ @B

p

∂y

∂p

! &

A+ :B /

y = ϕ(p, C),

x = f (ϕ(p, C), p).

A+ *B

= !1/ p !&/ 0 !

Φ(x, y, C) = 0.

A+ 9)B

D ! & A+ 9B !" y

y = f (x, y ),

A+ 99B

& ! /

pdx =

∂f

dx +

∂f

dp,

y = f (x, p).

A+ 9(B

∂x

∂p

99 ? &

y = y 2

− xy +

x2

.

A+ 9;B

2

! 7 y = p

y = p2

− xp +

x2

.

A+ 9,B

2

= !1/ p

y = p2 − xp + x2/2,

A+ 9+B

p = x + C.

0

y = x2/2 + Cx + C2.

A+ 9.B

?

∂y=

∂

2 +

+

y = x

2 +

+

A+ 9:B

∂C = ∂C

x2/2 + Cx + C2

.

0 = x + 2C.

y x2/

Cx C

2,

y = x

2/

Cx C2,

C

/4 ?

= !1/

!&/ & 1

2

5 & 2 & A+ 9;B 0

! !" &

y = x2/4 !

( 2p − x = 0 p = x/2. $1

y = x /4

y = p2 − xp + x2/2,

A+ 9@B

p = x/2.

2

, % 2 ,4

. y 2 + xy = y2 + xy

y 2 + x y

+ 1 = 0

y

6 y = xy 2 − 2y 3

−

x = y

y 2 + 1

.7yy 2

2xy + y = 0

. 2xy

− y = y ln yy

)

4 . y = Cex G

y = Ce−x + x −1 (x2C2 + 1 −2Cy)(x2 + C2 −

2Cy) = 0

x

y

= 0

2p3

3p2

+C

y = x 2

G !

2

−

2

6

y = xp2

−

2p3,

−

G

=

x = (1− p)2

$ y = 0

3

−

−

.7 y2

+ C2 =

y

2(p2+1)3/2

(p2

+ 1)1/2 + C,

.

2CxG ! x

x = p

p2 + 1

C ln C

y

= 0

y = 2Cx

−

2

−

.

2

G !

2

+ 9 # !" &

y = xϕ(y ) + ψ(y )

A+ 9*B

y = xϕ(p)+ψ(p) pdx = ϕ(p)dx+ xϕp

(p) dp

dx

xϕ (p) + ψp(p)

(p) + ψp

=

p

dp

p − ϕ(p)

y = xy + ψ(y ).

A+ ()B

9( ? &

y = 2y x + 1/y .

A+ (9B

x =

1

(ln p + C),

p2

A+ ((B

y = 2px + 1/p.

C !

y & A+ (9B & / !&/ &

!

! !" & A+ (9B

!

, % 2 ,4

.* 2y = x

y + y

2

.(

4

= 3(xy

./y 3

y)

y = y +

1

−

y

−

. y = xy + y 2

=

C + x2 G

! y = x .*

) 4

.( y

.. y = xy

+ 1 + y

2

C

±

x = ln |p| − arcsin p + C, ./ C3 = 3(Cx

y) G ! 9y2 = 4x3

.

−

y = p +

1 − p2

=

− 4 ..

y = Cx + √

+ C2 G ! y =

x2

G

. y

Cx

1 + C2

! x2 + y2 = 1

"

/

+ ; # !" &

F (y ) = 0,

F (y, y ) = 0, F (x, y ) = 0

A+ (;B

1

!

A+ (;B & " 0"1

! &

D ! & F

(y ) = 0, 0 ! ' F

y−C

= 0.

x

C & & F (x, y ) = 0

x = ϕ(t),

A+ (,B

y = ψ(t).

= ( ) = ( ) t(

t

!&/ /

$1 dy

ψ t dx

ψ t ϕ

t)dt y =

ψ(t)ϕ (t)dt + C J

x = ϕ(t),

(t)dt + C.

A+ (+B

y = ψ(t)ϕt

1

t

dy = sin tdx = − sin2 tdt y =

sin 2t −

+ C.

4

2

C !&/ /

x

=

1

t

A+ ;(B

cos t,

√ 2

2

2 !

2

y =

4 sin 2t −

+ C.

2

± y

(

a

a

± x

/ ! ! /

9, C " &

y ctg x =

.

a2 − y 2

! # ! & y

= a sin t = 2 &

ctg x = ctg t $1

x = t,

A+ ;;B

y = a sin t.

x = ω(t),

A+ ;,B

y = tx = tω(t).

D ! P Q R '

y y

! &

y = ty

& &