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
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