1

pi , ( ) pi pi . , - (...). pi , pi- pi pi (...). 1: :

y(x) + 4y(x) = cosx,dy

dx= x2 + 1

,

2f(x, y)

x2+2f(x, y)

y2= 0, x

f(x, y)

y+ y

f(x, y)

x= 2f(x, y)

.

pi ... pi, pi pi pi , , -, , , pi. pi pi.. , ,pi , , pipi pi . ... pi pi " - , I" ..

1.1: (...) :

F(x, y(x), y(x), ..., y(n)(x)

)= 0, (1.1)

pi n 1 (pi pi), F :I R, I R, Rn+1, y : I R

y(k)(x) =dky(x)

dxk, k = 1, ..., n.

1

1.2: ... pi , pi pi ...

2: ... (1.1) n- , ... y(x) = 2xy2(x) ... dy = ex+ydx pi .

1.3: ... (1.1) pi pi y(x) pi , (1.1) pi ... pi y(n)(x) .

3: ... y(x) + (y(x))3 = x2y(x) pi , ... y(x) + xy(x) 2(y(4)(x))3 = 2x .

1.4: ... (1.1) pi y(x) pi , (1.1) .... ... n- :

an(x)y(n)(x) + an1(x)y(n1)(x) + ...+ a1(x)y(x) + a0(x)y(x) = (x) (1.2)

pi ai(x), i = 0, 1, ..., n I R, an(x) 6= 0 ... (1.2). , (1.2) ... , x (1.2) ... . (x) = 0, (1.2) ..., (x) 6= 0, (1.2) ....

4: ... y(x) + xy(x) 2y(4)(x) = 0 , 4-, , , ... y(x) y(x) +2y(x) = 0 , 2- , , .

1.5: ... (1.1) pi y(x) pi - : y(x0) = y0, ..., y(n1)(x0) = yn1, x0 I, yi, i = 0, 1, ..., n 1, pi- ( ), pi (...). ... (1.1) pi y(x) -pi pi (a, b) I (), pi (...).

2

5: ... y+1 = xy2 y(1) = 2, x 1, ..., .. y(x) + y(x) = 1, x [1, 2], - y(1) = 1, y(2) = 0 ...

1.6: ... (1.1) y = (x), pi pi pi n , (x, (x), (x), ..., (n)(x)

) I , Rn+1 pi (1.1). (1.1) pi (1.1) y = (x, c1, c2, ..., cn) pi ci, i = 1, ..., n - ( ...) pipi y(x0) = y0, ..., y(n1)(x0) = yn1, x0 I, pi pi- ci, i = 1, ..., n, pi, . f(x, (x), c1, ..., cn) = 0, pi ci, i = 1, ..., n , (1.1), (1.1). (1.1) pi pipi pi pi , (1.1) pi pi pipi pi (1.1) , .

6: ... y(x) y(x) = 0. y1(x) = ex, , pi pi -pi pi. y(x) = c1ex + c2ex, c1, c2 , ..., pi -pi pipi pipi, y(0) = 0 y(0) = 2, pi pi. , pi c1 + c2 = 0,c1 c2 = 2 c1 = 1 c2 = 1. y1(x) = ex ..., pipi pi , c1 = 1 c2 = 0.

1.1: f(x, y(x), c1, ..., cn) = 0, - ci, i = 1, ..., n, pi pi xy pi-pi. pi n pi pi f(x, y(x), c1, ..., cn) = 0, pi n pi pi n pi pi pipi , pi - n- , pi pi.

3

1. pi pi ...:

)d2y

dx2+dy

dx 2y = x3 )dy

dx+ y4 = sinx )x2

d2y

dx2 xdy

dx 2y = 0

)x

(d2y

dx2

)4+ 2y = 2x )2x y dy

dx= 0

) ..., 2 , , , -.

) ..., 1 , , pi, 1 .

) ..., 2 , , , -.

) ..., 2 , , pi, 4 .

) ... :

2x y dydx

= 0 dydx

+ y = 2x

, ..., 1 , , , -.

2. ...:

)d2y

dx2=

(1 +

dy

dx

)2, )

(d2y

dx2

)2=

(1 +

dy

dx

)2, )

(1 +

d3y

dx3

)2=

(d2y

dx2

)3){ y

[1 + (y)2]12

}6=(

3y)6

.

) ... pi .

) ... .

) ... .

4

) ... .

3. pi , , pi ... (,, ):

) y(x) = ex2

2 ,dy

dx+xy = 0, ) y(x) = Acosx+Bsinx+ex,

d2y

dx2+y = 2ex,

) x+x2

y= A,

dy

dx=y2 + 2xy

x2,

) pi pi x

y(x) = xex2

2

pi pi ... :

dy

dx+ xy = xex

2

2 + xex2

2 = 0,

pi y(x) = ex2

2 ...

) pi

y(x) = Asinx+Bcosx+ ex

y(x) = AcosxBsinx+ ex

pi pi ... pi :

d2y

dx2+ y = AcosxBsinx+ ex + Acosx+Bsinx+ ex = 2ex,

pi ..., pi y(x) = Acosx+Bsinx+ ex

d2y

dx2+ y = 2ex.

5

) :

x+x2

y= A

y 6=0== (A x)y = x2 Ax6=0==== y = x

2

A x (1)

pipi pi x,

y =2x(A x) x2(1)

(A x)2 y =

2Ax x2(A x)2 (2)

..., (1), pi

y2 + 2xy

x2=

(x2

A x)2

+ 2xx2

A xx2

=x2

(A x)2 +2x

A x =x2 + 2x(A x)

(A x)2

=2Ax x2(A x)2 = y

, (2).

4. ) ... pi lny = Ax2 +B, A,B .

) ... pi: y = Ae3x+Be5x.

) pi x . :

y

y= 2Ax y = 2Axy (1)

pi B. (1) pi x pi A pi (1). :

y = 2A(y + xy)(1)= y = y

xy(y + xy) xyy = yy + x(y)2 (2)

(2) ... pi .

6

) pi x. :

y = 3Ae3x + 5Be5x (1)

(1) pi x, pi :

y = 9Ae3x + 25Be5x (2)

pi (1) (2) pi A,B, :

y = Ae3x +Be5x Ae3x = y Be5x

(1) y = 3(y Be5x) + 5Be5x y = 3y + 2Be5x Be5x = y 3y

(2) y = 9 (y (y 3y)) + 25(y 3y) y = 16y 39y

y 16y + 39y = 0

... y 16y + 39y = 0 ...

7

2 pi -

... pi F (x, y(x), y(x)) = 0. pi y(x) = f(x, y(x)) .

... pi y(x) = f(x, y(x)) pi ... , - ( pi), ... pi . pi pi ... pi - .

2.1 ...

2.1 ... , ... pi pi :

g(y)y(x) = f(x) (2.1)

pi g(y) f(x) pi .

... (2.1) - pi x pi , . :

g(y)y(x)dx =f(x)dx+ c (2.2)

pi dy = ydx y = y(x), -:

g(y)dy =

f(x)dx+ c (2.3)

(2.3) (2.2).

2.1 ... (2.1) pi :

g(y)dy = f(x)dx (2.4) 2.2 (2.1) pi ...

a(x)b(y)y(x) = k(x)h(y) (2.5)

8

, a(x) 6= 0,h(y) 6= 0 (2.5) :b(y)

h(y)y(x) =

k(x)

a(x) pi (2.1),

g(y) =b(y)

h(y) f(x) = k(x)

a(x).

a(x) h(y) , pi - .. (2.5), pi .

1 ...: y = yx, x 6= 0.

... y 6= 0 y

y=

1

x pi

. pipi pi dy

y=

dx

x+ c ln|y| = ln|x|+ c ln

yx

= c y = cx y = 0 ...

2 ...: y xdydx

=y

x, x 6= 0.

...

y xdydx

=y

x y y

x= x

dy

dx y

(x 1x

)= x

dy

dx pi -

, y 6= 0 (x 1x2

)dx =

dy

y

: (x 1x2

)dx =

dy

y+ c ln|x|+ 1

x= ln|y|+ c

y = 0 .

9

3 ...: (x y2x)dx (y x2y)dy = 0

...

(x y2x)dx (y x2y)dy = 0 x(1 y2)dx y(1 x2)dy = 0

1 y2 6= 0 1 x2 6= 0 pi x

1 x2dx =y

1 y2dy, pi pi pi :

x

1 x2dx =

y

1 y2dy + c pi

1

2

1

1 xdx1

2

1

1 + xdx =

1

2

1

1 ydy 1

2

1

1 + ydy + c

12ln|1 x|+ 1

2ln|1 + x| = 1

2ln|1 y|+ 1

2ln|1 + y|+ c

ln1 + x1 x

= ln 1 + y1 y+ c 1 + x1 x = c1 + y1 y

x = 1, y = 1 ...

2.2 .. pi pi ...

(1) ... pi pi x, y ... :

y(x) =P (x, y)

Q(x, y)(2.6)

pi P (x, y), Q(x, y) pi R2, Q(x, y) 6= 0. ... (2.6) pi

M(x, y)dx+N(x, y)dy = 0 (2.7)

10

M(x, y) N(x, y) pi R2.

2.2 ... (2.6) (2.7) - pi x y P (x, y) Q(x, y) pi x, y .(pi (x, y) pi x y, n R : (tx, ty) = tn(x, y)). (x, y) n, t = 1

x, x 6= 0 :

(

1,y

x

)= (tx, ty) = tn(x, y) =

(x, y)

xn

: (x, y) = xn(

1,y

x

)pi ... pi x, y (2.6) :

y(x) =xnP (1, y

x)

xnQ(1, yx) y(x) = P (1,

yx)

Q(1, yx)

(2.8)

pi y(x)x

= z(x), z(x) 6= 0, y(x) = xz(x), pi pi x: y(x) = z(x) + xz(x) (2.8), pi pi ...

z(x) + xz(x) =P (1, z)

Q(1, z) xz(x) = P (1, z)

Q(1, z) z (2.9)

pi pi z(x). (2.9) , - z(x) = y(x)

x pi ... (2.6).

4 ... xydx+ (x2 + y2)dy = 0 (1)

: M(x, y) = xy, N(x, y) = x2 + y2

M(tx, ty) = txty = t2xy = t2M(x, y)

N(tx, ty) = (tx)2 + (ty)2 = t2x2 + t2y2 = t2(x2 + y2) = t2N(x, y)

M(x, y) N(x, y) pi x y 2 , ... (1)

11

pi x y. , yx

= u, x 6= 0, u 6= 0 y = ux,y = ux+ u (1) :

xydx+ (x2 + y2)dy = 0 ux2dx+ (x2 + u2x2)(udx+ xdu) = 0udx+ (1 + u2)(udx+ xdu) = 0 (2u+ u3)dx+ x(1 + u2)du = 0 (2)

... (2) pi :

1 + u2

2u+ u3du+

dx

x= 0

pi :1 + u2

2u+ u3du+

dx

x= c pi :

1

2

du

u+

1

2

u

u2 + 2du =

dx

x+ c

1

2ln|u|+ 1

4lnu2 + 2 = ln|x|+ c ln|u|2 + ln u2 + 2 = ln|x|4 + c

lnu2(u2 + 2)x4 = c u2(u2 + 2)x4 = c

u = yx

..:(yx

)2((yx

)2+ 2

)x4 = c y2(y2 + 2x2) = c

5 ...(

1 + exy

)dx+e

xy

(1 x

y

)dy = 0, y 6= 0 (1)

y(0) = 1.

M(x, y) = 1 + exy N(x, y) = e

xy

(1 x

y

), :

M(tx, ty) = 1 + etxty = 1 + e

xy = M(x, y)

N(tx, ty) = etxty

(1 tx

ty

)= e

xy

(1 x

y

)= N(x, y),

pi x, y 0. ... (1) pi x, y

12

xy= u x = yu dx = udy + ydu, pi :(

1 + exy

)dx+e

xy

(1 x

y

)dy = 0 (1+eu)(udy+ydu)+eu(1u)dy = 0

(u+ eu)dy + y(1 + eu)du = 0

.. pi u(x)

dy

y+

1 + eu

u+ eudu = 0

:dy

y+

1 + eu

u+ eudu = c ln|y|+ ln |u+ eu| = c ln |y (u+ eu)| = c

y(u+ eu) = c

u = xy

pi ..

x+ yexy = c.

pi , pipi : c = 1, pi ... : x+ ye

xy = 1.

(2) ... y(x) = f(a1x+ b1y + c1a2x+ b2y + c2

)

() a1 b1a2 b2

= 0, a1a2 = b1b2 = k pi a1 = ka2, b1 = kb2 ... y(x) = f

(k(a2x+ b2y) + c1a2x+ b2y + c2

) ... -

pi z(x), a2x+ b2y = z y = 1b2

(z a2).

() a1 b1a2 b2

6= 0, {a1x+ b1y + c1 = 0

a2x+ b2y + c2 = 0

(x0, y0) . x = x0 + X y = y0 + Y , pidy

dx=dY

dX ... pi Y (X).

13

6 ... y(x) x 2y + 93x 6y + 19 = 0, 3x 6y+ 19 6= 0.

1 23 6

= 0, x 2y = z 1 2y = z y =

1

2(1 z), pi ... :

1

2(1 z) = z + 9

3z + 19 1 z = 2z + 18

3z + 19 dz

dx=

z + 1

3z + 19

z + 1 6= 0 :3z + 19

z + 1dz = dx pi pi z(x)

pi , (3 +

16

z + 1

)dz = x+ c 3z + 16ln |z + 1| = x+ c

3(x 2y) + 16ln |x 2y + 1| = x+ c x 3y + 8ln |x 2y + 1| = c

z + 1 = 0 x 2y + 1 = 0 y = x+ 12

pipi .

7 ... y(x) = x+ 2y + 2y 2x+ 6 , y 2x+ 6 6= 0.

1 22 1

= 5 6= 0, pi {x+ 2y + 2 = 0

2x+ y + 6 = 0 (2,2) . x = 2 + X y = 2 + Y , pidy

dx=dY

dX ... pi :

dY

dX=

X + 2Y

2X + Y dY

dX=

1 + 2Y

X

2 + YX

pi

: YX

= U Y (X) = U (X)X + U(X) ... pi :

14

U X + U =1 + 2U

2 + U UX =

1 + 4U U22 + U

2 U1 + 4U U2dU =

dX

X,

U2 4U 1 6= 0 pi pi U(X).

pi :2 U

1 + 4U U2dU =dX

X+ c 1

2ln1 + 4U U2 = ln |X|+ c

ln |(1 + 4U U2)X2| = c (1 + 4U U2)x2 = c U=YX===(

1 + 4Y

X Y

2

X2

)X2 = c X2 + 4XY Y 2 = c

.. : X = x 2 Y = y + 2, pi pipi : (x 2)2 + 4(x 2)(y + 2) (y + 2)2 = c x2 y2 12y + 4x 4xy = c. : U2 4U 1 = 0 , :(Y

X

)2 4Y

X 1 = 0 Y 2 4Y X X2 = 1

(y + 2)2 4(y + 2)(x 2) (x 2)2 = 1.

2.3 ... y = f(ax + by + c),

y = f(

1

ax+ by + c

) ... -

z(x) = g(z), z = ax+ by + c.

pipi pi pi pipi, a2 = b2 = 0, c2 6= 0, a1 = b1 = 0, c1 6= 0 .

8 ...: y = (y x)2.

y x = z. pi x: y 1 = z y = z + 1,pi .. :

y = (y x)2 z + 1 = z2 z = z2 1 dzz2 1 = dx, z

2 1 6= 0

15

pi z(x). pi x

dz

z2 1 = x+ c1

2

1

z 1dz 1

2

1

z + 1dz = x+ c

1

2ln |z 1| 1

2ln |z + 1| = x+ c ln

z 1z + 1 = 2x+ c

z 1z + 1

= ce2x z = ce2x + 1

1 ce2x z = y x, pi y = x+

ce2x + 1

1 ce2x z2 1 = 0 z = 1 y = x 1 .

9 ... y = 1x+ y + 1

, x+ y + 1 6= 0.

x+ y + 1 = z y = z 1. :

y =1

x+ y + 1 z 1 = 1

z z = z + 1

z

z+16=0=== z

z + 1dz = dx(

1 1z + 1

)dz = dx pi z(x)

pi x : (1 1

z + 1

)dz = x+ c z + ln|z + 1| = x+ c

y + ln |x+ y + 2| = c, z = x+ y + 1.

y = x 2 .

2.3 ... pi ( pi)

2.3 ... M(x, y)dx+N(x, y)dy = 0, (2.10)

16

pi M(x, y) N(x, y) pi R2, pi x y pi pi, ( pi) pi u(x, y) :

M(x, y)dx+N(x, y)dy = du (2.11)

pi (2.11) pi :

M(x, y) =u(x, y)

x, N(x, y) =

u(x, y)

y(2.12)

u(x, y) = c, (2.10).

2.1: ... (2.10)

M(x, y)

y=N(x, y)

x(2.13)

pi : xx0

M(t, y)dt+

N(x0, y)dy = c (2.14)

pi. () (2.10) ... , pi -, pi u(x, y) (2.11) (2.12). pi (2.12) pi Schwartz pi, :

M(x, y)

y=2u(x, y)

yx=2u(x, y)

xy=N(x, y)

x,

pi pi (2.13).() (2.13). pipi pi - u(x, y), (2.12), pi ... (2.10) . pi pi (2.12) :

u(x, y)

x= M(x, y) u(x, y) =

xx0

M(t, y)dt+ f(y) (2.15)

pi (2.15) pi M(x, y)

17

pi x, y (2.12), :

u(x, y)

y=

y

xx0

M(t, y)dt+f (y) N(x, y) = xx0

M(t, y)

ydt+f (y)

(2.13)===

N(x, y) = xx0

N(t, y)

tdt+f (y) N(x, y) = N(x, y)N(x0, y)+f (y)

f (y) = N(x0, y) f(y) =N(x0, y)dy (2.16)

, pi f(y) u(x, y). (2.15) (2.16) :

u(x, y) =

xx0

M(t, y)dt+

N(x0, y)dy (2.17)

pi , pi pi- (2.11), ... (2.10) .pi pi (2.16) pi u(x, y) = c pi-pi (2.14) (2.10).

2.4 pi (2.12),

yy0

N(x, t)dt+

M(x, y0)dx = c pi

(2.14).

2.5 x0 , pi pi -pipi pi, pi N(x0, y).

10 ... 2xy3 + 3x2y2y = 0.

2xy3 + 3x2y2y = 0 (x2) y3 + x2 (y2) = 0 (x2y3) = 0 x2y3 = c

11 pi a ...

18

(3x2 + axy2) dx+ (2y 3y2 + 4x2y) dy = 0 -.

pipi :

(3x2 + axy2)

y=(2y 3y2 + 4x2y)

x 2axy = 8xy a = 4

a ...

(3x2 + 4xy2) dx+ (2y 3y2 + 4x2y) dy = 0

, pi pi -, x0 = 0 x0

(3t2 + 4ty2)dt+

(2y 3y2)dy = c x3 + 2xy2 + y2 y3 = c.

12 ... x2 + y + (ey + x) y = 0.

... :

(x2 + y) dx+ (ey + x) dy = 0

M(x, y) = x2 + y N(x, y) = ey + x pi

M(x, y)

y= 1 =

N(x, y)

x. ...

, pi x0 = 0 x0

(t2 + y)dt+

eydy = c x

3

3+ xy + ey = c.

2.4 pi Euler ( pi)

2.4 (x, y) pipi Euler ...

19

M(x, y)dx+N(x, y)dy = 0,

...

(x, y)M(x, y)dx+ (x, y)N(x, y)dy = 0 (2.18)

.

(2.18) , :

((x, y)M(x, y))

y=((x, y)N(x, y))

x

(x, y)

yM(x, y) + (x, y)

M(x, y)

y=(x, y)

xN(x, y) + (x, y)

N(x, y)

x

(x, y)

[M(x, y)

y N(x, y)

x

]=(x, y)

xN(x, y) (x, y)

yM(x, y) (2.19)

(2.19) ... pi pi (x, y) pi pi-pi pi pipi. pi , pipi. :

(1) (x, y) = (x), x , pi (2.19):

(x)(x)

=

M(x, y)

y N(x, y)

x

N(x, y)(2.20)

pi (2.20) x, x, f(x). pi (2.20),pi pi pipi, pi :(x) = e

f(x)dx.

:

M(x, y)

y N(x, y)

x

N(x, y)= f(x) (x) = e

f(x)dx. (2.21)

20

pi :

(2)

M(x, y)

y N(x, y)

x

M(x, y) = f(y) (y) = ef(y)dy. (2.22)

(3)

M(x, y)

y N(x, y)

x

N(x, y)M(x, y) = f(xy) (xy) = ef(xy)d(xy).(2.23)

(4)

M(x, y)

y N(x, y)

x

yN(x, y) xM(x, y) = f(xy) (xy) = ef(xy)d(xy). (2.24)

(5) ...M(x, y)dx+N(x, y)dy = 0 pipi - (x, y) = xnym, pi pi n,m pi :

(xnymM(x, y))

y=(xnymN(x, y))

x

2.2: ... M(x, y)dx + N(x, y)dy = 0pi :

1

(x, y)= 0, pi (x, y) pi/ Euler .

pi. (x, y) pi/ Euler .. pi ...

(x, y)M(x, y)dx+ (x, y)N(x, y)dy = 0

, pi , pi u(x, y) :

(x, y)M(x, y)dx+ (x, y)N(x, y)dy = du(x, y).

:

1

(x, y)

[(x, y)M(x, y)dx+ (x, y)N(x, y)dy

]=du(x, y)

(x, y)

21

M(x, y)dx+N(x, y)dy = du(x, y)(x, y)

0 = du(x, y)(x, y)

du(x, y) = 0

1

(x, y)= 0

du(x, y) = 0 u(x, y) = c, .. 1

(x, y)= 0 .

13 ... (x3 + y4) dx xy2 (2y + x2) dy = 0.

M(x, y) = x3 + y4 N(x, y) = 2xy3 x3y2 pi- M(x, y)

y= 4y3 6= 2y3 3x2y2 = N(x, y)

x. ...

. pi/ (x), :

M(x, y)

y N(x, y)

x

N(x, y)=

6y3 + 3x2y2

xy2 (2y + x2) = 3y2 (2y + x2)

xy2 (2y + x2)= 3

x= f(x).

pi/ (x) = e

3xdx =

1

x3 ...(

1 +y4

x3

)dx

(2y3

x2+ y2

)dy = 0

. pi pi x0 = 1, - pi : x1

(1 +

y4

t3

)dt

(2y3 + y2

)dy = c x 1 y

4

2x2 y

3

3= c.

x = 0 .

14 ... 2xydx+ (1 x2 y2) dy = 0.

M(x, y) = 2xy N(x, y) = 1 x2 y2 pi M(x, y)

y= 2x 6= 2x = N(x, y)

x. ... .

pi/ (y), :

22

M(x, y)

y N(x, y)

x

M(x, y) =4x

2xy = 2

y= f(y).

pi/ (y) = e

2ydy =

1

y2 ...

2x

ydx+

1 x2 y2y2

dy = 0

. pi pi x0 = 0, - pi : x0

2t

ydt+

1 y2y2

dy = c t2

y

xt=0

+

1

y2dy

dy = c

x2

y 1y y = c x2 1 y2 = cy

y(x) = 0 ..

15 ... (xy2 + 2x2y3) dx+ (x2y x3y2) dy = 0.

M(x, y) = xy2 + 2x2y3 N(x, y) = x2y x3y2. - : M(x, y)

y= 2xy + 6x2y2 N(x, y)

x= 2xy 3x2y2. pi

M(x, y)

y6= N(x, y)

x

M(x, y)

y N(x, y)

x

yN(x, y) xM(x, y) =9x2y2

3x3y3 = 3

xy= f(xy)

... pi/ (x, y) = e

3xyd(xy) =

1

(xy)3

...(1

x2y+

2

x

)dx+

(1

xy2 1y

)dy = 0

pi x0 = 1 :

23

x1

(1

t2y+

2

t

)dt+

(1

y2 1y

)dy = c 1

xy+

1

y+2ln|x|1

yln|y| = c

1xy

+ ln

x2y = c

... pi, - pi/ (x, y) = xnym. ...

(xn+1ym+2 + 2xn+2ym+3) dx+ (xn+2ym+1 xn+3ym+2) dy = 0

, pi :

(xn+1ym+2 + 2xn+2ym+3)

y=

(xn+2ym+1 xn+3ym+2)x

(m+2)xn+1ym+1+2(m+3)xn+2ym+2 = (n+2)xn+1ym+1(n+3)xn+2ym+2

{m+ 2 = n+ 2

2(m+ 3) = (n+ 3) {m = n

n = 3 n = m = 3

... :(1

x2y+

2

x

)dx+

(1

xy2 1y

)dy = 0

pi pi pi :

1xy

+ ln

x2y = c

16 ...(y2

2+ 2yex

)dx+ (y + ex) dy = 0.

M(x, y) = y2

2+2yex N(x, y) = y+ex, pi : M(x, y)

y= y+2ex

N(x, y)x

= ex. ... , M(x, y)y

6=N(x, y)

x. : My Nx

N=y + ex

y + ex= 1 = f(x), ...

pi/ (x) = edx = ex, pi ...

24

(y2ex

2+ 2ye2x

)dx+ (yex + e2x) dy = 0

pi x0 = 0 pi : x0

(y2et

2+ 2ye2t

)dt+

(y + 1) dy = c

y2

2(ex 1) + y (e2x 1)+ y2

2+ y = c y

2

2ex + ye2x = c

pi 1(x)

= ex 6= 0, ... .

17 ...(2xy4ey + 2xy3 + y) dx+ (x2y4ey x2y2 3x) dy = 0.

M(x, y) = 2xy4ey+2xy3+y N(x, y) = x2y4eyx2y23x, pi :M(x, y)

y= 8xy3ey + 2xy4ey + 6xy2 + 1 N(x, y)

x= 2xy4ey 2xy2 3.

... , M(x, y)y

6= N(x, y)x

.

:My NxM =8xy3ey + 8xy2 + 4

y (2xy3ey + 2xy2 + 1) = 4

y= f(y),

... pi/ (y) = e

4ydy = 1

y4, pi ...(

2xey + 2x1

y+

1

y3

)dx+

(x2ey x

2

y2 3xy4

)dy = 0

x0 = 0, : x0

(2tey + 2t

1

y+

1

y3

)dt = c x2ey + x2 1

y+ x

1

y3= c.

y = 0 .

2.5 ... pi

2.5 pi .. pi

25

pi y, y. ... pi :

a1(x)y(x) + a0(x)y(x) = g(x) (2.25)

a1(x), a0(x), g(x) pi a1(x) 6= 0.

(2.25)

a1(x)y(x) + a0(x)y(x) = 0 (2.26)

(i) a0(x) = a1(x), (2.25) pi , pi :

a1(x)y(x) + a1(x)y(x) = g(x) (a1(x)y(x)) = g(x)

pipi pi x:

a1(x)y(x) =

g(x)dx+ c y(x) = 1

a1(x)

[ g(x)dx+ c

].

(ii) , (2.25) pi pi :

y(x) + a(x)y(x) = b(x) (2.27)

pi a(x) = a0(x)a1(x)

b(x) = g(x)a1(x)

.

2.3 ... (2.27) pi/

(x) = ea(x)dx (2.28)

:

y(x) = ea(x)dx

[c+

b(x)e

a(x)dxdx

](2.29)

pi (2.27) :

y(x) + a(x)y(x) = b(x) dy + (a(x)y b(x)) dx = 0 (2.30)

P (x, y) = a(x)y b(x) Q(x, y) = 1, pi

26

P (x, y)

y= a(x) Q(x, y)

x= 0, .

P (x, y)

y Q(x, y)

x

Q(x, y)=a(x)

1= a(x)

x, pi (2.30) pi/ (x) = ea(x)dx.

/ (2.27) pi/ , :

ea(x)dxy(x)+e

a(x)dxa(x)y(x) = e

a(x)dxb(x) (e a(x)dxy(x)) = b(x)e a(x)dx,

pi pi x pipi

ea(x)dxy(x) =

b(x)e

a(x)dxdx+ c

y(x) = ea(x)dx

[ b(x)e

a(x)dxdx+ c

],

pi (2.29).

2.5 pi (2.28), pi 1(x)

= ea(x)dx 6= 0, -

... pi .

2.6 (2.27) . -, (2.29) :

y(x) = cea(x)dx + e

a(x)dx

b(x)e

a(x)dxdx y(x) = yo(x) + y(x),

pi yo(x) = cea(x)dx. ,

(2.27), pi pipi pi (2.29) b(x) = 0. y(x) = e

a(x)dx

b(x)e

a(x)dxdx (2.27),

pi (2.27), :

y(x) + a(x)y(x) =(e

a(x)dx

b(x)e

a(x)dxdx

)+ a(x)e

a(x)dx

b(x)e

a(x)dxdx =

27

a(x)ea(x)dx

b(x)e

a(x)dxdx+ b(x)e

a(x)dxe

a(x)dx+

a(x)ea(x)dx

b(x)e

a(x)dxdx = b(x)

pi (2.27).

2.7 ... pi pi y(x), pi x(y), pi :

x(y) + a(y)x(y) = b(y)

pi/

(y) = ea(y)dy :

x(y) = ea(y)dy

[c+

b(y)e

a(y)dydy

].

2.8 ... (2.25) a(x) b(x) , x0 , - ... x < x0 x > x0 pi x0. pi- pi .

18 ... y + yx

= 2x, x 6= 0

... pi y(x) a(x) = 1x

b(x) = 2x. pi/ (x) = e

1xdx = x pi/

... x, pi:

y +y

x= 2x xy + y = 2x2 (xy) = 2x2

x :

xy =

2x2dx+ c xy = 2

3x3 + c y = 2

3x2 +

c

x,

pi ..

28

19 ... (1+x2)dy+2xydx = cotxdx, y(pi

2

)= 0.

... :

(1 + x2)dy + 2xydx = cotxdx (1 + x2)y + 2xy = cotx (1)

pi a1(x) = 1 + x2, a0 = 2x b(x) = cotx.

a0(x) = a1(x), pi (1) :((1+x2)y

)= cotx (1+x2)y =

cotxdx+c (1+x2)y = ln |sinx|+c (2)

(2) pipi c = 0, pi ..., pipi pi (2) c = 0,:

y =ln |sinx|1 + x2

.

20 ... (x+ 2y3)dydx

= y (1)

... (1) pi y(x). pi :

(x+ 2y3)dy

dx= y (x+ 2y3) = y 1

y x+ 2y3 = ydx

dy

ydx

dy x = 2y3 dx

dy 1yx = 2y2 (2)

... (2) pi x(y), pi/

(y) = e

1ydy =

1

y

pi/ , , :

29

1y

dx

dy 1y2x = 2y

(1

yx(y)

)= 2y 1

yx(y) = y2 + c x(y) = y3 + cy,

pi ... .

21 ... y + y = b(x) =

2, x [0, 1]0, x > 1

,

y(0) = 0.

pi b(x) , ... () x [0, 1] () x > 1.

() x [0, 1] : y + y = 2

... pi/ (x) = edx = ex, pi

y + y = 2 exy + exy = 2ex (exy) = 2ex

exy = 2

exdx+ c

exy = 2ex + c y = cex + 2 (1)

() x > 1 : y + y = 0, pi pi () pi-pi pipi :

y = c1ex (2)

pi x0 = 1, pi (1) (2) c c1. :

limx1y(x) = limx1+y(x) limx1(cex + 2) = limx1+(c1ex) ce1 + 2 = c1e1 c1 = c+ 2e

pi (2)

30

y = (c+ 2e)ex (3)

.. :

y(x) =

cex + 2, x [0, 1]

(c+ 2e)ex, x > 1

y(0) = 0, pi pi (1) : 0 = c + 2 c = 2 - ... :

y(x) =

2 2ex, x [0, 1]

(2e 2)ex, x > 1

22 ... y(x) + ay(x) = bex,pi a 6= a > 0, > 0, b R, x +.

... pi y(x), pi/(x) = e

adx = eax :

y(x) + ay(x) = bex (eaxy) = be(a)x eaxy = be(a)x

a + c

y(x) = ceax +b

a ex

x +, pi-pi pi pi a, b, , eax

x+0 ex

x+0,

pi pipi pi.

2.6 ... pi pi

1) ... Bernoulli

...

y(x) + a(x)y(x) = b(x)yn(x), n 6= 0, 1 (2.31)

31

pi a(x) b(x) . n = 0 (2.31) n = 1 . (2.31), pi :() pi/ (2.31) yn(x), pi :

yn(x)y(x) + a(x)yn+1(x) = b(x) (2.32)

() yn+1(x) = z(x), pi pi x,(n+ 1)yn(x)y(x) = z(x) (2.32), pi :

1

n+ 1z(x) + a(x)z(x) = b(x) z(x) = (n+ 1)a(x)z(x) = (n+ 1)b(x)

() pipi ..., pi pi z(x) y(x) :yn+1(x) = z(x) y(x) = (z(x)) 1n+1 .

2.9 ... Bernoulli pi y(x),pi ... Bernoulli pi x(y), :

x(y) + a(y)x(y) = b(y)xn(y),

pi pi pi (2.31).

23 ... y xy = x3y3.

... Bernoulli pi y(x), pi pi/ y3:

y xy = x3y3 y3y xy2 = x3

y2 = z(x) (2)y3y = z pi-pi .... :

y3y xy2 = x3 12z xz = x3 z + 2xz = 2x3

... pi z(x). pi/ Euler

32

(x) = e2xdx = ex

2, pi :

z + 2xz = 2x3 (ex

2z)

= 2x3ex2 ex2z = c x2ex2 +

2xex2

dx ex2z = c x2ex2 + ex2 z = cex2 + 1 x2

... :

y2 = cex2

+ 1 x2 y2 = 1cex2 + 1 x2 .

24 ... (x2y3 + xy) y = 1, y(

1

2

)= 0.

... Bernoulli pi y(x), pi :

(x2y3 + xy) y = 1 x = x2y3 + xy x xy = x2y3

pi Bernoulli pi x(y) :

x xy = x2y3 x2x yx1 = y3.

x1 = z(y), pi pi y, :x2x = z(y) pipi ..:z yz = y3 z + yz = y3.

.. pi z(y) pi/ (y) = e y2

2 , pi :(ey2

2 z)

= y3ey2

2 e y2

2 z =

ey2

2 y3dy + c e y2

2 z = c y2e y2

2 + 2ey2

2

z(y) = ce y2

2 +2y2 x1(y) = ce y2

2 +2y2 x(y) = 1ce

y2

2 + 2 y2pi y

(1

2

)= 0, pi :

1

2=

1

c+ 2 c = 0

pipi ... :

33

x(y) =1

2 y2 .

2) ... Riccatti

... Riccatti :

y(x) = 0(x) + 1(x)y(x) + 2(x)y2(x) (2.33)

pi i(x) i = 0, 1, 2 . ... pi pi pi pipi :

() y1(x) (2.33),

y(x) = y1(x) + u(x) (2.34)

... (2.33) ... Bernoulli pi u(x)., (2.34) (2.33) pipi :

y1(x)+u(x) = 0(x)+1(x)(y1(x)+u(x))+2(x)(y21(x)+u

2(x)+2y1(x)u(x))

pi y1(x) (2.33), -

y1(x) = 0(x) + 1(x)y1(x) + 2(x)y21(x)

pipi ... :

u(x) = 1(x)u(x) + 2(x)(u2(x) + 2y1(x)u(x))

u(x) [1(x) + 2y1(x)]u(x) = 2(x)u2(x) (2.35) pi ... Bernoulli pi u(x). u(x) pi (2.35), (2.34) pi (2.33).

() y1(x) (2.33),

y(x) = y1(x) +1

u(x)(2.36)

34

... (2.33) ... pi u(x) pi pi pi (2.36) (2.33). , (2.36) (2.33) pi-pi :

y1(x)u(x)u2(x)

= 0(x)+1(x)

(y1(x) +

1

u(x)

)+2(x)

(y21(x) +

1

u2(x)+ 2y1(x)

1

u(x)

) pi y1(x) (2.33), -

y1(x) = 0(x) + 1(x)y1(x) + 2(x)y21(x)

pipi ... :

u(x)

u2(x)= 1(x)

1

u(x)+ 2(x)

(1

u2(x)+ 2y1(x)

1

u(x)

)

u(x) = 1(x)u(x) + 2(x) + 22(x)y1(x)u(x)

u(x) + [1(x) + 22(x)y1(x)]u(x) = 2(x)

pi ... pi u(x).

() y1(x) y2(x) (2.33) pi pi:

y(x) y1(x)y(x) y2(x) = ce

2(x)[y1(x)y2(x)]dx, c = (2.37)

, y1(x) y2(x) (2.33), pi pi :

y(x) y1(x) = 1(x)(y(x) y1(x)) + 2(x)(y2(x) y21(x))y(x) y2(x) = 1(x)(y(x) y2(x)) + 2(x)(y2(x) y22(x))

pi pi pipi :

y(x) y1(x)y(x) y1(x) = 1(x) + 2(x)(y(x) + y1(x))

35

y(x) y2(x)y(x) y2(x) = 1(x) + 2(x)(y(x) + y2(x)).

pi:

y(x) y1(x)y(x) y1(x)

y(x) y2(x)y(x) y2(x) = 2(x)[y1(x) y2(x)]

d

dx

[ln |y(x) y1(x)|

] ddx

[ln |y(x) y2(x)|

]= 2(x)[y1(x) y2(x)].

pi ,:

d

dx

[ln

y(x) y1(x)y(x) y2(x) ] = 2(x)[y1(x) y2(x)].

pi x, pi pi:

ln

y(x) y1(x)y(x) y2(x) = 2(x) [y1(x) y2(x)] dx+ c

y(x) y1(x)y(x) y2(x) = ce

2(x)[y1(x)y2(x)]dx

pi pi .

()

y(x) = 12(x)

u(x)u(x)

(2.38)

pi ... (2.33) , piu(x). y(x) pi pi (2.38) (2.33) :

2(x)22(x)

u(x)u(x)

12(x)

u(x)u(x)

+1

2(x)

(u(x)u(x)

)2=

= 0(x) 1(x) 12(x)

u(x)u(x)

+ 2(x)1

22(x)

(u(x)u(x)

)2

2(x)

22(x)

u(x)u(x)

12(x)

u(x)u(x)

= 0(x) 1(x) 12(x)

u(x)u(x)

36

12(x)

u(x) +[2(x)22(x)

+ 1(x)1

2(x)

]u(x) 0(x)u(x) = 0

pi pi u(x).

2.10 pi - i(x) i = 0, 1, 2. , i(x) pi x, : y1(x) = ax + b. i(x) , y1(x) = aebx. i(x) -, y1(x) = axb. ... pi a, b. pi pi.

25 ... yy2+2exy = e2x+ex.

.. :

y = y2 + 2exy e2x ex

: y1(x) = aebx. pix: y1(x) = abebx ... pi :

abebx (aebx)2 + 2exaebx = e2x + ex abebx a2e2bx + 2abe(b+1)x = e2x + ex , - a b pi pipi . pi b = 1,

aex a2e2x + 2ae2x = e2x + ex {a = 1

a2 + 2a 1 = 0 a = 1

pi : y1(x) = ex. pi

y(x) = ex +1

u(x) y(x) = ex u

(x)u2(x)

... pi:

37

ex u(x)

u2(x) e2x 1

u2(x) 2ex 1

u(x)+ 2e2x + 2ex

1

u(x)= e2x + ex

u(x)

u2(x) 1u2(x)

= 0 u(x) + 1 = 0 du = dx u(x) = x+ c

... :

y(x) = ex +1

x+ c.

26 ... y + 2y2 = 1x2.

y = axb y = abxb1 - ... pi a b. :

abxb1 + 2a2x2b = x2

b = 1, a + 2a2 1 = 0 a = 1a = 1

2

. pi

: y1(x) =1

x y2(x) = 1

2x

... pi pi (2.37). :

y(x) 1x

y(x) +1

2x

= ce 2[ 1

x+ 1

2x]dx

y(x) 1x

y(x) +1

2x

= ce3lnx

y(x) 1x

y(x) +1

2x

=c

x3

y(x) 1x

3

2x

=c

x3 c y(x) =1

x+

3c

2x(x3 c)

27 ... y = (1 y)(

1

x+ 1 y

)

y = 1 .. pi Riccati,

38

pi : y = 1x

+ 1(

1

x+ 2

)y + y2, pi :

y(x) = 1 +1

u(x) y(x) = u

(x)u2(x)

pi pi , pi pi:

u(x)

u2(x)= 1

u(x)

(1

x 1u(x)

) u(x) = u(x)

(1

x 1u(x)

)

u(x) u(x)x

= 1 (x)=1x====(

1

xu(x)

)= 1

x 1

xu(x) = ln|x|+ c

u(x) = cx xln|x| y(x) = 1 + 1cx xln|x| .

39