2dx-[,,- 3 X
2 lnx - y + 3 lnty + 3) = 1nC
&$u y=21nx+31n(y+3)-lnc
y = lnx2(y + 3 13C
&I ey.cx2(y+3)3
MA 216 13
ii-Id4 3 /=a&W-n-i y cot x + y = 2 ; y ( ; I=0
?&PI i5-1 y * 2 ~8:: cot x * 0 duniad7uisn~~Xi$l
dY - 2 +tanxdx=@
In 1 G 1 =c
14 MA 216
0= 2+c;cos~
dx
MA 216 15
3.
4.
E!=-” dP F
(xy + XMX = lx y + x + y? + 1 My
xcce2y dx + tany dy = 8
y/ = ysec x
10.
11.
12.
13.
14.
15.
dr ;it = -2rt. ; t = 0 , r = r0
dv v&-g =0; x = x
0’ v = v
sin y q c cos x
-x2 -2 e +Y = c
PV = c
X2 + tan2y = c2
eX( x 1 1- 1) z: - + ---i + c Y 2Y-
arc sin y + arc sin x = arc sin c
X2 - 2y2 =2 xe
Y- __-_-.-- 2eX - x - 1
PIA 216 17
2. ;,-$,,=0
iiw7arrh&.m~Ejunn mwii~iuianiihu%h6yl
dtxy) = xdy + ydx
O X ay
C’(y) = N -
au htirn7mduni7 - = N
ii-J&II42 wGhm7 3x( xy - 2 Mx + (x3 + 2yMy = 0
Yin? M( x, .y 1 = 3x2;y - 6x ttar: N(x,y) = x3 + 2y
G!!! = 3x2 aN
- r M r 3x2y - 6xOU
i3X
I X
( 3x2ydx + x3dy t - 6xdx + 2ydy = 0
Al
Iwmi7%G~n7im~I:1Ry x3y - 3x2 + y2 = tJ
til4 3 wu&Wm e-‘dx - (2~ + x e-Y)dy = 0
sin-l M(x,y) = e-’ , N(x,y) = -(2y + xe-y )
E _ -y i3N ay - -e ’ ax = -e
-Y
. . . ..(16)
+ C’(y)
C’(y) = - 2 y
e-‘dx - 2ydy - xe-‘dy = 0
Ha*n (e-‘dx - xe-‘dy 1 - 2ydy = 0
Gil d(e-Yx) - d(y2) = 0
i%Aiirnm~I:ln” .-yx - y2 IC
n’-JAlJii 4 WMhJWl?- ( xYex*y 2 x+y
+3xe - XMX + (x3ex+y + yMy = 0 . . . ..(17)
%?I? M( x ‘y) - x3ezc+y + 3x2ex+y - x,. -
aM 3 x+y ay=Xe
.& 3x2ex+y
BN ax
= .3,x+y .t 3x2ex+’
&kJ L hju&.J”lalriu~m ;2&& u( x , y 1 2.4
au ax
- = N = x3ex+’au ay
J Y
= x3ex+Y 2
+ 3x2ex+y + C’(x)
AmIll g=n x3ex+Y + 3x2ex+y + C’( x ) = x3,‘+’ + 3x2ex+’ - x
C’(x) = -x
C(x) = - g.
3 x+y 2 2 u(x,y) = x e + k - & = c
ii%uIi+ -wWmi7 ( 17 1 t iim%ai t &I
( x3ex+y + 3x2eX+YMx + x3,‘+’dy - xdx + ydy = 0
?&I
2 )+d+=O
(Y2 - 2xy + 6xMx - (x2 - 2xy + 2)dy = 0
(1 + y2Mx + (x2y + yMy = 0
v( 2uv2 - 3Mu + (3u2v2 -3u*4vMv=0
(Cos 2y - 3x2y2 Mx + (cos 2y - 2x sin 2y - 2x3yMy = 0
(s in 0 - 2rcos2 e Mr + rcos MZrsin 8 + 1 Me = 0
c2x + yc0.4 xy )ldx + xcos( xy My = 0
3y( x2 - lkix + (x 3 + 8y - 3xkly = 0 ) y(0) = 1
10. (1 -xy) -2 dx+Cy 2 + x2( 1 - xd21dy = 0 , y(2) = 1
2 6 MA 216
2
v( A2 - :3u + 2v ) = c
$ sin 2y + xcos 2.y - x3y2 = c
rsin 6 -- r2cos2Q = c
2 X + sintxy) = c
xy4 - y3 + 5xy - 3x = 5
MA 216 27
xdx + ydy
ii-3d-iAl wuhm7 (x2 + y2 + x kh + ydy = 0
GTPl Liiurc~~ni7l~‘2~u~~~n~rneu elii
(x2 + y2Mx + txdx + y2dy, = 0
(x2 + y2 Mx + ; d( x2 l y2, = 0
. . ...(l)
. . ...(2)
. . ...(3)
. . ...(4)
. . ...(5)
. . ...(6)
. . ...(7)
. . ...(8)
. . . ..(?I
,p/, + 1 !! X2 + Y2, - Q 2.2 -tx + y2)
%ii L ma 7”%iWR L nak?atii;;o
:x + ; 1ni:x2 2+ y ~I= c
iGmi7Gi 2 ~wuhmrr ydx -. ( x2y + x kly = 0
ydx - xd,! - y'dy = 0
Y Y2-+---=c x 2
XY - lnlyi = c
i&Id 4 74&duni~7 y( x3 - yMx - x(x3 +yMy=O-
Gin? ~64plun~7%&lti k iira
x3( ydx - xdy) - yfydx + xdy) = 0
KA 216 29
7m,lyjl d( f ) = ydx - xdy uau’ d( xy 1 = ydx + xdy
Y2
3 0 MA 2 1 6
aM - aN 1 dv ay ax_- - z - . . . . .(12)
1 du.- - = u dv
HA 216 3 1
X
Y
[ g-$gYz
2( xN - yM 1
ii-&IL 5 7&i~un17 (x2y+y+l)+x(1 +x2)y’ =0
ish hi%? M=x2y+y+1, 2 N : x(1 + x )
BM 2 aNg--$=x + 1 , 5 & = 1 + 3 x 2
kil7all A( g-g )= 1
I -2x u = exp
= ( 1 + x2 )-l
u~I&p4un77Id 7~Itkiaml7uriua~~~
(x2y+y+ 1X1 + x )2 -1 +x(1+x2)(1+x)2 -1 dy-&0
32
&I 2 2 -1{(x +l)y+lNl+x) dy+x-&=0
MA 216
1 Y + ---.-.
iiwsnr~ ; (!g - g ) = -.-$.-I- ( 1 -4xy+11 2xy -’ y
33
a ’~~ M=y-k,
=0
u3lrlyuMn7hl: ~~Itiz4un17udum~~
Go xdx - t $dx - x dy) =0 Y2
?&I d(x2/2) - dtx/y) = 0
1
%iPl hii M q 2y + 1 1
(x + yj2 9 N = 3y + x +
tx + yj2
3
34 MA 216
n% C2xy + 2y2 + J- Mx + c4xy * x 2 + 3!f 2 + A-.. M y =0
X+Y x+Y
(mydlx + x2dy) + (2y’dx + 4xydy ) + --& d( x+y) + 3y2dy = 0
d(x2y) .+ d(2xy2) + d(lnlx + yl ) + dy3 - 0
tilt+? L fl7R
=c
1. (y + x3y2Mx + x dy =0
2. (4x2 + yMx -xdy=0
3 . (x2+y2+ yklx - x dy = 0
4. y dy = (x dy + y dx)1I + y
5. xy2( xy/ + y) = 1
6. y2dx -. (xy + x3My =0
7. y2dx + (xy + tan xykly =0
8. y(x2 + Y2 + 1Mx + x(x2 + y2 -1kly=0
9. y( x4 - y5dx + x(x4 + y2kiy = 0
10. ycy2 + 1Mx + x(y2 - 1My = 0
i l .
wG53uni7~a 11 - 1 5 aa~hll~=no~~~~nsoli;~?~~llu*~
(x - y2) + 2xyy/ =0
(x + x 4
36 MA 216
4 31 3 3x Y + Y = c x
x( y2 + 11, = cy x-2 2
, Y + x In x = cx -2
Y , Y2 + x = c3
(x + y) ,, x3y + 2 2i!X y 3 + XY q c
1 ---72 ) (x2
MA 216
5GtnaGo
u~Itigwtaanlum7 ( 1)
e x p ( PtxMx)( 2 + P(x)y) I: e x p ( J J
P(XMX)a(X) . . . . .(2)
P( x kix )y
P( x Mx )y’
P( x kix )y’
exp ( J ~(x~x)y = J’ exp tJ ~(x~x)~(xtix + c
38 MA 216
y = cexpC- J
x( psr + I y ) r x3 dx x
Go
&A?I b naw
MA 216 39
Y= - $ + c exp(x2)
++c
M?a (2x )’ = 22y
dxGa -& (3 - 1) 2*------x~~- Y Y
hi3ll?ma¶&G 6 flml;a
Y
. . ...(4)
-Y l c
4 2 MA 226
Y/ = x - 2y
u dx + (1 - 3uIx du = 3udu
Y/ = csc x - y cot x
Y/ 1= - - -
,.-y - x
1.
2.
3.
4.
5.
6.
7.
8.
Y = 2(x + 2) -1 + dx + 2) - 4
-2x 4Y = 2x - 1 + et: *
x u = (u3 4i c),rn
XL1 L ce3u - u - f
y sin x = x + c
x r e-y (:v + c) 2
x - y - 1 + c expGy2)
MA 216 43
f( ax + by + c 1 ~~U~~AYIO~ ax + by + c
(ax + by + c o~h~tir~~rh~~&~rl~ x uas Y)
&mhos f( ax + by + c 1 th
(2x + 3y - 2
“7,un’~sml7n~~~aur~~uPr~auJ?lii
v = ax + by * c
dv& -& = bftv) + a
4 4 MA 216
k&4 1__-- sit&hma 4 = (4x ,t y + 1)’
pm”0 dy dv iri
‘z-4
4*v
umdFh v
. . ...(Z)
45
I& ln(v * 1) = v - x - c
v+l v-x -c =ce1 , c 1
=e
t&l x+y+2 = c2expt y), c2 = cle
. . ...(S)
UmARl v = x - y
q x - y + In I x - y - 2- - +c x - y + 2 I
x - y - 2 y - c = l n - -
I x-y+2 I
&I
%&I dy _ x+y-1 dx - -x + y + 1
. . ...(4)
UmMh v = x + y
l&l x - y - c = lnlx + yl
x + y = clexp(x - y)
* = g( alx + bly
dx a2x + b2y ) #
x * + v - f(v) ‘dx -
%%I dv 1 dx
. . . . .(8)