4 Listing of problems and solutions obtained by Mathematica and Maple

 4.1    ′
y (x) = af (x)
 4.2   y′(x) = y (x )+ x + sin(x)
 4.3   y′(x) = x2 + 2y(x)+ 3cosh(x)
 4.4   y′(x) = a + bx + cy(x)
 4.5   y′(x) = a cos(bx + c)+ ky (x )
 4.6   y′(x) = a sin(bx + c)+ ky(x)
 4.7    ′           kx
y (x) = a + be + cy(x)
 4.8    ′       ( 2      )
y (x) = x x −  y(x )
 4.9            (            )
y′(x) = x ay(x) + e−x2
 4.10    ′       2(  3       )
y (x) = x  ax  + by (x )
 4.11    ′        n
y (x) = ax y(x)
 4.12   y′(x) = y (x )cos(x )+ sin(x) cos(x)
 4.13   y′(x) = y (x )cos(x )+ esin(x)
 4.14   y′(x) = y (x )cot(x)
 4.15   y′(x) = 1 − y(x)cot(x)
 4.16   y′(x) = x csc(x )− y(x)cot(x)
 4.17    ′
y (x) = y (x )(cot(x)+ 2 csc(2x))
 4.18    ′
y (x) = sec(x)− y (x )cot(x )
 4.19    ′                   x
y (x) = y (x )cot(x) + e sin(x)
 4.20   y′(x)+ 2y(x) cot(x) + csc(x) = 0
 4.21   y′(x) = 4x csc(x)sec2(x) − 2y(x)cot(2x)
 4.22            (                           )
y′(x) = 2 cos(2x )cot2(x) − y(x)csc(2x)
 4.23   y′(x) = 4x csc(x)(y(x)+ sin3(x))
 4.24   y′(x) = 4x csc(x)(y(x)− tan2(x)+ 1)
 4.25    ′
y (x) = y (x )sec(x)
 4.26    ′
y (x)+ tan(x) = (1− y(x))sec(x)
 4.27   y′(x) = y (x )tan(x)
 4.28   y′(x) = y (x )tan(x)+ cos(x)
 4.29   y′(x) = cos(x) − y(x)tan(x)
 4.30   y′(x) = sec(x)− y (x )tan(x)
 4.31   y′(x) = y (x )tan(x)+ sin (2x )
 4.32    ′
y (x) = sin(2x)− y (x )tan (x )
 4.33    ′
y (x) = 2y (x )tan(x)+ sin(x )
 4.34    ′
y (x) = 2(y(x) tan (2x )+ sec(2x )+ 1)
 4.35   y′(x) = 3y (x )tan(x)+ csc(x)
 4.36   y′(x) = y (x )(a + sin(log(x))+ cos(log(x)))
 4.37   y′(x) = 6e2x − y(x)tanh(x)
 4.38   y′(x) = y (x )f ′(x)+ f(x)f′(x)
 4.39   y′(x) = f (x)+ g(x)y(x)
 4.40    ′       2      2
y (x) = x − y(x)
 4.41       2    ′      ′         2
f(x)  + y(x) = f (x)+ y(x)
 4.42   y′(x)− x + 1 = y(x)(y(x )+ x)
 4.43   y′(x) = (y(x) + x)2
 4.44   y′(x) = (x − y(x))2
 4.45    ′               2
y (x) = (x − y(x)) + 3(y(x)− x + 1)
 4.46    ′       (  2   )           2
y (x) = − x  + 1 y(x) + y(x) + 2x
 4.47    ′       ( 3    )  (  2       )
y (x) = x x  + 2 −  2x  − y(x) y(x)
 4.48            (      )  (          )
y′(x) = x 2−  x3 +  2x2 − y(x) y(x)+ 1
 4.49   y′(x) = cos(x) − y(x)(sin(x)− y(x))
 4.50   y′(x) = y (x )(y(x) + sin(2x))+ cos(2x)
 4.51   y′(x) = xf (x)y(x)+ f(x) + y(x)2
 4.52   y′(x) = (− 4y(x)+ x + 3)2
 4.53    ′                    2
y (x) = (9y(x) + 4x+ 1)
 4.54    ′       (        2     )
y (x) = 3 a+ by(x)  + bx
 4.55   y′(x) = a + by(x)2
 4.56   y′(x) = ax + by(x)2
 4.57   y′(x) = a + bx + cy(x)2
 4.58   y′(x) = axn −1 + bx2n + cy(x)2
 4.59   y′(x) = ax2 + by(x)2
 4.60    ′                         2
y (x) = a0 + a1y(x) + a2y(x)
 4.61    ′                  2
y (x) = ay (x )+ by(x) + f (x )
 4.62    ′
y (x) = a (x − y(x))y(x)+ 1
 4.63   y′(x) = ay (x )2 + f(x)+ g(x )y (x )
 4.64   y′(x) = xy (x)(y(x) + 3)
 4.65                 (      )
y′(x) = − x3 + 2x2 + 1 y(x)− xy(x)2 − x+ 1
 4.66   y′(x) = x (x2y(x)− y (x )2 + 2)
 4.67   y′(x) = − (1− x )y (x )2 + (1− 2x)y(x) + x
 4.68    ′            2
y (x) = axy (x)
 4.69    ′       n(        2)
y (x) = x  a + by(x )
 4.70   y′(x) = axm + bxny(x)2
 4.71   y′(x) = y (x )(a + by(x)cos(kx))
 4.72                (              )
y′(x) = sin(x) 2 sec2(x)− y (x )
 4.73   y′(x)+ 4 csc(x) = y(x)2sin (x )+ y(x)(3− cot(x))
 4.74   y′(x) = y (x )sec(x) + (sin(x)− 1 )2
 4.75    ′     (        2)
y (x)+  1 − y(x)  tan(x) = 0
 4.76    ′                              2
y (x) = f (x)+ g(x)y(x)+ h (x )y (x )
 4.77    ′          (                2)
y (x) = f (x) a+ by(x) + cy(x)
 4.78   y(x)2(ax + y(x))+ y′(x )
 4.79   y′(x) = y (x )2 (aex + y(x))
 4.80   3a(y(x) + 2x)y(x)2 + y′(x ) = 0
 4.81   y′(x) = y (x )(a+ by(x)2)
 4.82   y′(x) = a0 + a1y(x) + a2y(x)2 + a3y(x)3
 4.83    ′           3
y (x) = xy (x)
 4.84    ′         (         2)
y (x)+ y(x) 1 − xy(x)   = 0
 4.85   y′(x) = y (x )2(a + bxy(x))
 4.86        (               )
y(x)3 a + 4b2x+  3bx2 + y′(x)+ 3xy(x)2 = 0
 4.87                (         )
y′(x) = y (x )2 x3y(x)+ 1
 4.88   2xy (x )(axy(x)2 + 1) + y′(x ) = 0
 4.89   y′(x) = y (x )2 − ax(1 − xn−1) y(x)3
 4.90    ′          2        3(      n−1)
y (x) = ay (x ) + xy(x) b + cx
 4.91    ′         (    2              )
y (x)+ y(x) y (x ) sec(x) + tan (x ) = 0
 4.92   y′(x)+ y(x)3tan(x) sec(x) = 0
 4.93   y′(x) = f0(x)+ f1(x)y(x)+ f2(x)y(x)2 + f3(x)y(x)3
 4.94   y′(x) = y (x )(y(x)3sec(x )+ tan(x))
 4.95    ′        -n-       n
y (x) = ax 1−n + by (x )
 4.96    ′                        k
y (x) = f (x)y(x )+ g(x)y(x)
 4.97   y′(x) = f (x)+ g(x)y(x)+ h (x )y (x )n
 4.98   y′(x) = f (x)y(x )m + g (x )y(x)n
 4.99          ∘  ------
y′(x) =   |y(x)|
 4.100   y′(x) = a + ∘A0-+-B0y-(x)+ by(x)
 4.101   y′(x) = ax + b∘y-(x)
 4.102    3    ′      ∘ --4-------
x  + y (x ) = x x  + 4y(x)
 4.103            (     ∘ ----)
y′(x)+ 2  1− x   y(x) y(x) = 0
 4.104    ′     ∘  --------2-
y (x) =   a+ by(x)
 4.105    ′         ∘ ---------
y (x) = y (x ) a + by(x)
 4.106                   ∘ -------------------
g(x)(f(x)−  y(x ))  (y(x) − a)(y(x) − b)+ y′(x) = 0
 4.107          √ ----
y′(x) =  XY
 4.108             (      )    (        )
y′(x) = R1  x,√X-  R2  y(x),√Y--
 4.109   y′(x) = cos2(x)cos(y(x))
 4.110   y′(x) = sec2(x)Cosy(y(x))cot(y(x))
 4.111   y′(x) = a + bcos(Ax + By (x))
 4.112   y′(x) = − (1 − f′(x))cos(y(x))+ f ′(x)− f (x)sin(y(x))+ 1
 4.113   g(x)sin(ay(x))+ h(x) cos(ay(x))+  f(x)+ y′(x) = 0
 4.114    ′
y (x) = a + bcos(y(x))
 4.115     (             2   2     )    ′
x  sin(2y(x))− x  cos (y(x))  + y(x) = 0
 4.116    ′                    2
y (x)+ tan(x)sec(x)cos (y(x)) = 0
 4.117   y′(x) = cot(x)cot(y(x))
 4.118   y′(x)+ cot(x)cot(y(x)) = 0
 4.119   y′(x) = sin(x)(csc(y(x))− cot(y(x)))
 4.120   y′(x) = tan(x) cot(y(x))
 4.121   y′(x)+ tan(x)cot(y(x)) = 0
 4.122    ′
y (x)+ sin(2x)csc(2y(x )) = 0
 4.123    ′
y (x) = tan(x)(tan(y(x))+ sec(x)sec(y(x )))
 4.124   y′(x) = cos(x)sec2(y(x))
 4.125   y′(x) = sec2(x)sec3(y(x))
 4.126   y′(x) = a + bsin(y(x))
 4.127   y′(x) = a + bsin(Ax + By (x))
 4.128   y′(x) = tan(y(x))(cos(x )sin(y(x))+ 1)
 4.129    ′
y (x)+ csc(2x)sin(2y(x )) = 0
 4.130                          ′
f (x )+ g(x)tan(y(x))+ y (x) = 0
 4.131    ′     ∘  --------------
y (x) =   a+ bcos(y(x))
 4.132   y′(x) = ey(x) + x
 4.133   y′(x) = ey(x)+x
 4.134             (          )
y′(x) = ex a+ be− y(x)
 4.135   y′(x)+ y(x) log(x)log(y(x)) = 0
 4.136   y′(x) = xm −1y(x)1−nf (axm + by(x)n)
 4.137    ′
y (x) = af (y(x))
 4.138    ′
y (x) = f (a+ bx + cy(x))
 4.139   y′(x) = f (x)g(y (x ))
 4.140   y′(x) = Csx (x)y(x)sec(x )+ sec2(x )
 4.141   2y′(x)+ 2 csc2(x) = y(x)csc(x)sec(x )− y(x)2sec2(x)
 4.142   2y′(x) = 2sin2(y(x))tan(y(x))− x sin(2y(x))
 4.143          ′     ∘ --2-2-----2---------
ax + 2y (x) =  a x  − 4bx  − 4cy(x)
 4.144     ′      ∘ -2--------
3y (x) =   x − 3y(x) + x
 4.145      ′     √-------
xy (x) =  a2 − x2
 4.146   xy ′(x)+ y (x )+ x = 0
 4.147   x2 + xy′(x)− y(x) = 0
 4.148   xy ′(x) = x3 − y(x)
 4.149   xy ′(x) = x3 + y(x) + 1
 4.150   xy ′(x) = xm + y(x)
 4.151      ′
xy (x) = xsin(x)− y(x)
 4.152      ′      2
xy (x) = x sin(x )+ y(x)
 4.153   xy ′(x) = xnlog(x)− y(x)
 4.154   xy ′(x) = sin(x) − 2y(x)
 4.155   xy ′(x) = ay(x)
 4.156   xy ′(x) = ay(x)+ x + 1
 4.157   xy ′(x) = ax+  by (x )
 4.158      ′       2
xy (x) = ax + by(x)
 4.159      ′          n
xy (x) = a+ bx  + cy(x)
 4.160      ′
xy (x)+ (3 − x)y(x)+ 2 = 0
 4.161   (ax + 2)y(x)+ xy ′(x)+ x = 0
 4.162   y(x)(a + bx)+ xy′(x) = 0
 4.163                (       )
xy ′(x) = x3 + 1 − 2x2 y(x)
 4.164   xy ′(x) = ax−  (1 − bx2)y(x)
 4.165   (2− ax2) y(x)+ xy′(x)+ x = 0
 4.166    2     ′         2
x  + xy (x)+ y(x) =  0
 4.167      ′      2
xy (x) = x + y(x)(y(x)+  1)
 4.168   xy ′(x)+ y (x )2 − y(x) = x2∕3
 4.169   xy ′(x) = a+ by(x)2
 4.170   xy ′(x) = ax2 + by(x)2 + y(x)
 4.171   xy ′(x) = ax2n + y (x )(by(x)+ n )
 4.172   xy ′(x) = axn + by (x )+ cy(x)2
 4.173      ′       n               2
xy (x) = ax  + by (x )+ cy(x) + k
 4.174         ′          2
a + xy (x)+ xy(x)  = 0
 4.175      ′
xy (x)+ y (x )(1 − xy(x)) = 0
 4.176   xy ′(x) = y(x)(1 − xy(x))
 4.177   xy ′(x) = y(x)(xy (x )+ 1)
 4.178   xy ′(x) = ax3y(x)(1− xy(x))
 4.179   xy ′(x) = x3 + (2x2 + 1)y(x) + xy(x)2
 4.180   xy ′(x) = y(x)(2xy (x)+ 1)
 4.181                            ′
y(x)(axy(x) + 2)+ bx + xy (x ) = 0
 4.182                                    ′
a0 + a1x + y(x)(a2 + a3xy(x))+  xy(x) = 0
 4.183   ax2y (x )2 + xy′(x )+ 2y(x) = b
 4.184   12 (n − m )y(x)+ xm +  xny(x)2 + xy′(x) = 0
 4.185   y(x) (a + bxny(x))+ xy ′(x) = 0
 4.186   xy ′(x) = axm − by(x)− cxny (x )2
 4.187   xy ′(x) = axn(x − y(x))2 − y(x)+ 2x
 4.188                            ′
y(x)(1−  ay(x)log(x )) + xy (x ) = 0
 4.189      ′         ( 2       2)
xy (x) = f(x) x  − y(x)  + y(x)
 4.190   xy ′(x) = y(x)(y(x)2 + 1)
 4.191   xy ′(x)+ y (x )(1− xy (x )2) = 0
 4.192   xy ′(x)+ y (x ) = a (x2 + 1) y(x )3
 4.193      ′             (  2   )    3
xy (x) = ay(x)+ b x  + 1 y (x )
 4.194      ′              2k    k
xy (x)+ 2y (x ) = ax y(x)
 4.195             (       ∘ ----)
xy ′(x) = 4 y(x) −   y(x)
 4.196      ′            ∘ ----2----
xy (x)+ 2y (x ) =  y(x)  + 1
 4.197      ′     ∘ -2------2-
xy (x) =   x + y (x ) + y(x)
 4.198      ′     ∘ ----------
xy (x) =   x2 − y (x )2 + y(x)
 4.199             ∘ ----------
xy ′(x) = x  x2 + y(x)2 + y(x)
 4.200                             ∘ ----------
xy ′(x) = y(x)− x(x − y(x))  x2 + y(x)2
 4.201             ∘ ------------
xy ′(x) = a  b2x2 + y(x)2 + y(x)
 4.202   cos(y(x))(sin(y(x))− 3x2cos(y(x)))+ xy′(x) = 0
 4.203      ′                (y(x))
xy (x)− y (x )+ xcos  -x-- + x = 0
 4.204      ′               2 (y(x))
xy (x) = y(x)− x cos   x
 4.205            (      )
xy ′(x) =  1− 2x2  cot2(y(x))
 4.206   xy ′(x) = y(x)− cot2(y(x))
 4.207   xy ′(x)+ y (x )+ 2xsec(xy(x)) = 0
 4.208      ′               ( y(x))
xy (x)− y (x )+ xsec   x   = 0
 4.209                        (   )
xy ′(x) = y(x)+ x sec2  y(x)-
                       x
 4.210      ′
xy (x) = sin(x − y(x))
 4.211                       (   )
xy ′(x) = y(x)+ x sin  y(xx)-
 4.212   xy ′(x)+ tan(y(x)) = 0
 4.213      ′
xy (x)+ tan(y(x) + x)+ x = 0
 4.214                       ( y(x))
xy ′(x) = y(x)− x tan   -x--
 4.215   xy ′(x) = (y(x)2 + 1)(x2 + tan−1(y(x)))
 4.216              y(x)
xy ′(x) = xe x  + y(x)
 4.217   xy ′(x) = xey(xx) + y(x)+ x
 4.218   xy ′(x) = y(x)log(y (x ))
 4.219      ′
xy (x) = y(x)(− log(y(x))+ log(x)+ 1)
 4.220      ′
xy (x)+ y (x )(− log(y(x))− log (x )+ 1) = 0
 4.221                         (    )
xy ′(x) = y(x)− 2x tanh  y(xx)-
 4.222   ny (x )+ xy′(x) = f (x )g (xny(x))
 4.223      ′            m     n
xy (x) = y(x)f (x y(x) )
 4.224           ′              3
(x + 1)y(x) = (3x+ 4 )x  + y(x)
 4.225           ′                   4
(x + 1)y(x) = 2y(x)+ (x + 1)
 4.226   (x + 1)y′(x) = ny(x) + ex(x + 1)n+1
 4.227   (x + 1)y′(x) = ay(x)+ bxy (x )2
 4.228   (x + 1)y′(x) + (x + 1)4y(x)3 + y(x) = 0
 4.229   (x + 1)y′(x) = y(x)(1 − xy(x)3)
 4.230           ′           ∘ --------
(x + 1)y(x) = (x+ 1)  y(x) + 1+ y (x )+ 1
 4.231           ′
(a + x)y(x) = bx
 4.232   (a + x)y′(x) = bx+  y(x )
 4.233   (a + x)y′(x) + bx2 + y(x) = 0
 4.234   (a + x)y′(x) = 2(a+ x )5 + 3y(x)
 4.235   (a + x)y′(x) = b+ cy(x)
 4.236   (a + x)y′(x) = bx+  cy (x )
 4.237   (a + x)y′(x) = y(x)(1− ay(x))
 4.238           ′         3
(a − x)y(x) = y(x) (b+ cx)+ y (x )
 4.239       ′       3
2xy (x) = 2x − y (x )
 4.240       ′            2
2xy (x)+ 1 = y(x)  + 4ixy (x)
 4.241                 (        )
2xy ′(x) = y(x) y(x)2 + 1
 4.242                 (        )
2xy ′(x)+ y (x ) y(x)2 + 1 = 0
 4.243                 (               )
2xy ′(x) = y(x) − 6y(x)2 + x + 1
 4.244   ∘a2--−-4b−-4cy(x) + a+ 2xy ′(x) + 4y(x) = 0
 4.245            ′
(1 − 2x)y(x) = 2(− 3y (x )+ 16x + 8)
 4.246            ′       −y(x)
(2x + 1)y(x) = 4e     − 2
 4.247                         √ -----
2(1 − x)y′(x) = y(x)+ 4  1 − xx
 4.248   2(x + 1)y′(x) + (x+ 1)4y(x)3 + 2y (x ) = 0
 4.249   3xy ′(x) = 3x2∕3 + (1− 3y (x ))y(x)
 4.250                 (          )
3xy ′(x) = y(x) xy(x)3 + 2
 4.251   3xy ′(x) = y(x)(3xy(x)3log(x)+ 1)
 4.252    2 ′
x y (x) = a− y(x)
 4.253    2 ′                2
x y (x) = a+ bx + cx + xy (x)
 4.254    2 ′                2
x y (x) = a+ bx + cx − xy (x)
 4.255   x2y′(x)+ (1 − 2x)y(x) = x2
 4.256   x2y′(x) = a+ bxy(x)
 4.257   x2y′(x) = y(x )(a + bx)
 4.258   x2y′(x)+ (x + 2)xy(x) = (1− e−2x) x− 2
 4.259   x2y′(x)+ 2(1 − x)xy(x) = ex(2ex − 1)
 4.260    2 ′       2              2
x y (x)+ x  + xy(x) + y(x) = 0
 4.261    2 ′                     2
x y (x) = (− y(x)+ 2x + 1)
 4.262   x2y′(x) = a+ by(x)2
 4.263   x2y′(x) = y(x )(ay (x )+ x)
 4.264   x2y′(x) = y(x )(ax + by(x))
 4.265   ax2 + bxy(x)+  cy (x )2 + x2y′(x ) = 0
 4.266   x2y′(x) = a+ bxn + x2y(x)2
 4.267    2 ′
x y (x)+ xy (x )(xy (x )+ 4)+ 2 = 0
 4.268                    2 ′      2(     2)
ax (1 − xy(x))+ x  y(x)+  x  − y(x )  + 2 = 0
 4.269    2 ′           2    2
x y (x) = a+ bx y(x)
 4.270   x2y′(x) = a+ bxn + cx2y(x)2
 4.271   x2y′(x) = a+ bxy(x) + cx2y(x)2
 4.272   x2y′(x) = a+ bxy(x) + cx4y(x)2
 4.273   x2y′(x)+ y(x) (x2 + y(x)2 − x) = 0
 4.274   x2y′(x) = 2y(x)(x−  y(x )2)
 4.275    2 ′        2    2       3
x y (x) = ax y(x) − ay(x)
 4.276        2     2    3   2 ′
ay(x)  + bx y(x) + x y (x) = 0
 4.277                 (          )
x2y′(x) = y(x ) ax + by(x )3
 4.278                    ∘ ----
x2y′(x)+ xy (x )+   y(x) = 0
 4.279   x2y′(x) = 3xtan(y(x))+ sec(y(x))
 4.280   (1− x2) y′(x) = − x2 + y(x)+ 1
 4.281   (1− x2) y′(x)+  1 = xy (x )
 4.282   (    2)  ′
 1− x   y(x) = 5− xy (x )
 4.283       ( 2    ) ′
a +  x +  1 y(x) + xy(x) = 0
 4.284   a + (x2 + 1)y′(x) − xy(x) = 0
 4.285   a + (1− x2 )y′(x) − xy(x) = 0
 4.286   (1− x2) y′(x)+  xy(x)− x = 0
 4.287   (    2)  ′      2
 1− x   y(x)−  x + xy(x) = 0
 4.288   (    2)  ′      2
 1− x   y(x)+  x + xy(x) = 0
 4.289   ( 2   )  ′      ( 2    )
 x + 1  y(x) = x x  + 1 − xy (x)
 4.290   (     )         (          )
 x2 + 1 y′(x) = x 3x2 − y(x)
 4.291   (     )
 1− x2  y′(x)+  2xy(x) = 0
 4.292   (     )
 x2 + 1 y′(x) = 2x(x−  y(x ))
 4.293   ( 2   )  ′       (  2   )2
 x + 1  y(x) = 2x x  + 1  + 2xy (x)
 4.294   (    2)  ′
 1− x   y(x)+  cos(x) = 2xy(x)
 4.295   ( 2   )  ′
 x + 1  y(x) = tan(x)− 2xy(x)
 4.296   (     )
 1− x2  y′(x) = a+ 4xy (x)
 4.297   (     )
 x2 + 1 y′(x) = y(x)(a+ 2bx)
 4.298   (     )
 x2 + 1 y′(x) = y(x)2 + 1
 4.299   (1− x2) y′(x) = 1− y(x)2
 4.300   (1− x2) y′(x) = 1− (2x − y(x))y(x)
 4.301   (    2)  ′      (    2            )
 1− x   y(x) = n y(x)  − 2xy(x)+ 1
 4.302   ( 2   )  ′
 x + 1  y(x)+  x(1− y(x))y(x) = 0
 4.303   (     )
 1− x2  y′(x) = xy(x)(ay(x)+ 1)
 4.304   (     )                  (        )
 x2 + 1 y′(x) = y(x)2 − 2x y(x)2 + 1 y(x) + 1
 4.305   (      )                              (      )
 x2 + 1 y′(x)+ x sin(y(x))cos(y(x)) = x x2 + 1 cos2(y (x ))
 4.306   (x2 + 1) y′(x) = x2 − y (x )cot− 1(x )+ 1
 4.307   (4− x2) y′(x)+  4y(x ) = (x + 2)y(x)2
 4.308   ( 2   2)  ′
 a + x   y(x) = b+ xy(x)
 4.309   (      )        (√ -------   )
 a2 + x2 y′(x) =    a2 + x2 + x (b+ y(x))
 4.310   (a2 + x2) y′(x)+ (x − y(x))y(x) = 0
 4.311   ( 2   2)  ′      2       2
 a + x   y(x) = a − 2y(x)  + 3xy(x)
 4.312   ( 2   2)  ′           2
 a + x   y(x)+  bxy (x) + xy(x) = 0
 4.313            ′
(1 − x)xy(x) = a + (x+ 1)y(x)
 4.314   (1 − x)xy′(x) = 2(xy(x) + 1)
 4.315   (1 − x)xy′(x) = 2(xy(x) − 1)
 4.316   x(x + 1)y′(x) = (1− 2x )y(x )
 4.317   (1 − x)xy′(x) + (2x+ 1)y(x) = a
 4.318   (1 − x)xy′(x) = a + 2(2− x)y(x)
 4.319            ′
(1 − x)xy(x) − 3xy(x)+  y(x )+ 2 = 0
 4.320            ′     ( 2       )              ( 2    )
x(x + 1)y(x) =  x  + x− 1  y(x)+ (x + 1) x  − 1
 4.321   x2 + (x− 3)(x − 2)y′(x) + 3xy(x)− 8y (x ) = 0
 4.322   x(a + x)y′(x ) = y(x)(b+ cy(x))
 4.323   (a + x)2y′(x) = 2(a+ x )(b + y(x))
 4.324   k(− a+ y (x )+ x)2 + (x − a)2y′(x) + y(x)2 = 0
 4.325   (x − a)(x− b)y′(x )+ ky(x) = 0
 4.326                 ′
(x − a)(x − b)y (x ) = y(x)(− a − b+ 2x)+ (x − a)(x − b)
 4.327                 ′          2
(x − a)(x− b)y (x ) = cy(x)
 4.328                                       ′
k(y(x) − a)(y (x )− b)+ (x − a)(x − b)y(x) = 0
 4.329   k(− a + y(x)+ x)(− b+ y (x )+ x)+ (x − a)(x− b)y′(x)+ y(x)2 = 0
 4.330   2x2y′(x) = y(x)
 4.331   2x2y′(x)+ 2x2y (x )cot(x )+ xcot(x) − 1 = 0
 4.332   2x2y′(x)+ x2 (− y (x )2) + 2xy(x)+ 1 = 0
 4.333   2x2y′(x) = (x2 − y(x)2)(1 − xcot(x))+ 2xy (x )
 4.334   2 (1− x2) y′(x) = √1-−-x2-+ (x+ 1)y(x)
 4.335   (1 − 2x)xy′(x )+ (1− 4x )y(x) + 1 = 0
 4.336             ′         2
(1 − 2x)xy (x ) = y(x) − (4x + 1)y(x)+ 4x
 4.337             ′
2(1 − x)xy (x )+ (1− 2x )y(x) + x = 0
 4.338   2(1 − x)xy′(x )+ (1− x)y(x)2 + x = 0
 4.339     (         )
2  x2 + x + 1 y′(x) = 8x2 − (2x + 1)y(x)+ 1
 4.340     (     )
4  x2 + 1 y′(x) − x2 − 4xy (x ) = 0
 4.341   ax2y ′(x) = axy(x)+ b2y(x)2 + x2
 4.342   (a+ bx2) y′(x) = A + By (x )2
 4.343   (     2)  ′
 a+ bx   y(x) = cxy(x)log(y(x))
 4.344             ′
x(ax + 1)y (x )+ a − y(x) = 0
 4.345           2 ′        3               2
(a + bx) y(x)+  y(x )(a + bx)+ cy(x)  = 0
 4.346   x3y′(x) = a+ bx2y(x)
 4.347   x3y′(x) = x2y(x)− x2 + 3
 4.348   x3y′(x) = x4 + y(x)2
 4.349   x3y′(x) = y(x )(x2 + y (x ))
 4.350   x3y′(x) = x2(y (x )− 1)+ y(x)2
 4.351    3 ′                2
x y (x) = (x + 1)y(x)
 4.352    3 ′       2    (    2    )
x y (x)+ x y (x ) 1− x y(x)  + 20 = 0
 4.353      (      )
x6  − y(x)2 + x3y′(x) + (3− 2x)x2y(x) + 3 = 0
 4.354                 (          )
x3y′(x) = y(x ) 2x2 + y (x )2
 4.355                     (                       )
x3y′(x) = cos(y(x)) cos(y(x))− 2x2 sin(y(x))
 4.356   x (x2 + 1) y′(x) = ax2 + y(x)
 4.357   x (1− x2) y′(x) = ax2 + y(x)
 4.358     ( 2   )  ′       3
x  x + 1  y(x) = ax  + y(x)
 4.359     ( 2   )  ′          2
x  x + 1  y(x) = a − x y(x)
 4.360     ( 2   )  ′     (     2)
x  x + 1  y(x) =  1−  x  y(x)
 4.361     (     )        (         )
x  1− x2  y′(x) =  x2 − x+ 1  y(x)
 4.362     (     )              (      )
x  1− x2  y′(x) = ax3 +  1− 2x2  y(x)
 4.363     (     )        (       )      (      )
x  1− x2  y′(x) =  1−  2x2 y(x)+  1 − x2 x3
 4.364   x (x2 + 1) y′(x) = 2(1 − 2x2y(x))
 4.365   x (x2 + 1) y′(x) = x − (5x2 + 3) y(x)
 4.366   (    2)   ′     (    2)     2   2
 1− x   xy(x) +  1− x   y(x) + x  = 0
 4.367           2 ′                       2
(1 − x)x y(x) = (2− x )xy (x )− y(x)
 4.368                  (         )
2x3y′(x) = y(x) x2 − y (x )2
 4.369                  (           )
2x3y′(x) = y(x) ay(x)2 + 3x2
 4.370   6x3y′(x) = 4x2y(x)+ (1 − 3x)y(x)4
 4.371   xy′(x)(a + bx+ cx2) − y(x)(a + bx+ cx2) + x2 = y(x)2
 4.372   x4y′(x) = y(x )(x3 + y (x ))
 4.373    2    4 ′      4    2
a  + x y (x )+ x y(x) =  0
 4.374    4 ′       3
x y (x)+ x y (x )+ csc(xy (x )) = 0
 4.375   (    4)  ′       (        2)
 1− x   y(x) = 2x 1 − y(x)
 4.376     (     )             (      )
x  1− x3  y′(x) = 2x −  1− 4x3  y(x)
 4.377     (     )
x  1− x3  y′(x) = x2 + y(x)(1 − 2xy(x))
 4.378      (     )            (           )
x2  1− x2  y′(x) = y(x) x − 3x3y(x)
 4.379   x (1− 2x3) y′(x) = 2(1 − x3)y(x)
 4.380   ( ′        2) (          2)2
 y(x)+  y(x )   a+ bx + cx   + A =  0
 4.381   x5y′(x) = 1− 3x4y(x)
 4.382   x (1− x4) y′(x) = (1−  x4)y(x)+ 2x (x2 − y(x )2)
 4.383   x7y′(x)+ 5x3y (x )2 + 2(x2 + 1)y(x)3 = 0
 4.384   xny ′(x) = a+ bxn− 1y(x)
 4.385    n  ′      2n−1      2
x  y(x) = x    − y (x )
 4.386        ( n−1      )    n ′      2
y(x) x    + y(x)  + x y (x )+ x  = 0
 4.387           n−1    2n−2   n  ′        2
(1 − n)x    + x    + x  y(x)+  y(x ) = 0
 4.388   xny ′(x) = a2x2n−2 + b2y(x)2
 4.389                 (                     )
xny ′(x) = xn−1 ax2n − by(x)2 + ny (x)
 4.390                              (                     )
x2ny′(x) = − nxn− 1 + xny (x ) x2ny(x)2 − 3xny (x )+ 1 + 1
 4.391   xky ′(x) = axm + by(x)n
 4.392   √x2--+-1y′(x) = 2x − y(x)
 4.393   √ -----2 ′         2
  1 − x y(x) = y(x) + 1
 4.394   (    √------)              ∘ ---------
 x −  x2 + 1  y′(x) = y(x)+   y(x)2 + 1
 4.395   √a2-+--x2y′(x) + y(x)+ x = √a2-+-x2-
 4.396   √ -2----2 ′     ∘ -2------2-
  b + x y (x) =   a + y(x)
 4.397   √ -2----2 ′     ∘ -2------2-
  b − x y (x) =   a − y(x)
 4.398    √ -------           ∘ ----------
x  a2 + x2y′(x) = y(x)  b2 + y (x )2
 4.399    √ -------           ∘ ----------
x  x2 − a2y′(x) = y(x)  y(x)2 − b2
 4.400   √ --       √ --
  Xy ′(x )+   Y = 0
 4.401   √ --       √ --
  Xy ′(x ) =  Y
 4.402   x3∕2y′(x ) = a + bx3∕2y (x )2
 4.403   √ -3---- ′     ∘ ----3----
  x  + 1y(x) =   y(x) + 1
 4.404   ∘ ---------------- ′     ∘ ------------------------
  (1 − x)x(1− ax )y (x) =   (1− y(x))y(x)(1− ay(x))
 4.405   √ ------       ∘ ---------
  1 − x4y′(x) =   1 − y(x )4
 4.406   √ -----------       ∘ ----------------
  x4 + x2 + 1y′(x) =  y(x)4 + y(x)2 + 1
 4.407   √ --
  Xy ′(x ) = 0
 4.408     --         --
√ Xy ′(x )+ √ Y = 0
 4.409   √Xy- ′(x ) = √Y-
 4.410   ( 3   )2∕3 ′     (    3   )2∕3
 x + 1    y (x)+  y(x) + 1    =  0
 4.411   (              )          (                   )
 a0 + a1x + 4x3 2∕3y′(x) +  a0+ a1y (x )+ 4y(x)3 2∕3 = 0
 4.412   X2 ∕3y′(x) = Y 2∕3
 4.413   (       2(x))  ′            (x-)(             2(x )   )
 a + cos  2   y(x) = y(x)tan  2  a − y(x)+ cos  2  + 1
 4.414   (       2   ) ′          (    2      )
 1− 4 cos (x) y (x) = y(x ) 4cos (x)+ 1  tan (x )
 4.415               ′
(1 − sin(x))y(x) + y(x)cos(x) = 0
 4.416   y′(x)(cos(x) − sin(x))+ y (x )(sin(x )+ cos(x)) = 0
 4.417        (             )      (             )
y′(x ) a0+ a1 sin2(x) + a2x  a1sin2(x)+ a3  + a1y(x)sin(2x) = 0
 4.418   (x − ex)y′(x)+ (1 − ex)y(x)+ exx = 0
 4.419   x log(x)y′(x) = ax(log(x) + 1)− y(x)
 4.420   y(x)y′(x)+ x = 0
 4.421    x2         ′
e  x + y(x)y(x) = 0
 4.422    3        ′
x  + y(x)y(x) + y(x) = 0
 4.423   ax + by(x)+ y (x )y′(x) = 0
 4.424   y(x)y′(x)+ e−xx (y (x )+ 1) = 0
 4.425   f (x )+ y(x)y′(x) = g(x)y(x)
 4.426   y(x)y′(x)+ y(x)2 + 4x(x+ 1) = 0
 4.427   y(x)y′(x) = ax + by(x)2
 4.428        ′          2
y(x)y (x) = ay (x ) + bcos(c+ x)
 4.429        ′                         2
y(x)y (x) = a0 + a1y(x)+ a2y(x)
 4.430        ′                2
y(x)y (x) = ax + bxy(x)
 4.431   y(x)y′(x) = csc2(x)− y(x)2 cot(x)
 4.432   y(x)y′(x) = ∘a2-+-y(x)2
 4.433   y(x)y′(x) = ∘y-(x-)2-−-a2
 4.434         ( 2      2)        ′
g(x)f  x + y(x)   + y(x)y(x)+  x = 0
 4.435             ′
(y(x)+  1)y (x) = y(x)+ x
 4.436             ′       2
(y(x)+  1)y (x) = x (1− y(x))
 4.437   (y(x)+  x)y′(x)+  y(x ) = 0
 4.438   (x − y(x))y′(x) = y(x)
 4.439   (y(x)+  x)y′(x)−  y(x )+ x = 0
 4.440   (y(x)+  x)y′(x) = x− y (x )
 4.441   1 − y′(x) = y(x)+  x
 4.442              ′
(x − y(x))y(x) = y(x)(2xy (x)+ 1)
 4.443              ′
(y(x)+  x)y(x)+  tan (y(x)) = 0
 4.444                    (  -x-    )
(x − y(x))y′(x) =  e−y(x) + 1 y(x)
 4.445   (y(x)+  x+ 1)y′(x)+ 3y(x) + 4x+ 1 = 0
 4.446                 ′
(y(x)+  x+ 2)y (x) = − y(x)− x + 1
 4.447                   ′
(− y(x)− x + 3)y(x) = − 3y(x)+ x + 1
 4.448   (y(x)−  x+ 3)y′(x) = 3y(x) − 4x + 11
 4.449   (y(x)+  2x)y′(x)−  2y(x)+ x = 0
 4.450   (− y(x)+ 2x + 2)y′(x) + 3(− y(x) + 2x+ 1) = 0
 4.451   (− y(x)+ 2x + 3)y′(x) + 2 = 0
 4.452   (− y(x)+ 2x + 4)y′(x) − 2y(x)+ x + 5 = 0
 4.453                    ′
(− y(x)− 2x + 5)y(x) − 2y(x)− x + 4 = 0
 4.454                  ′
(y(x)−  3x+ 1)y (x) = 2(x − y(x))
 4.455                  ′
(y(x)−  3x+ 2)y (x)− 3y(x) − 2x+ 5 = 0
 4.456   (4x − y(x))y′(x)−  5y(x)+ 2x = 0
 4.457   (− y(x)− 4x + 6)y′(x) = 2x − y(x)
 4.458   (− y(x)+ 5x + 1)y′(x) − 5y(x)+ x + 5 = 0
 4.459   y′(x)(a+ bx + y(x))+ a − bx− y(x) = 0
 4.460   (x2 − y(x))y ′(x)+ x = 0
 4.461   ( 2      )  ′
 x − y(x) y (x) = 4xy(x)
 4.462    ′
y (x )(y(x) − cot(x )csc(x))+  y(x )csc(x )(y(x)cos(x)+  1) = 0
 4.463   x2 + 2y(x)y′(x) + y(x)2 + 2x = 0
 4.464   2y(x)y′(x) = x3 + xy(x)2
 4.465   (x − 2y(x))y′(x) = y(x)
 4.466   (2y(x)+  x)y′(x)−  y(x )+ 2x = 0
 4.467   (x − 2y(x))y′(x)+  y(x )+ 2x = 0
 4.468                    ′
(− 2y(x)+ x + 1)y(x) = − y(x)+ 2x + 1
 4.469                  ′
(2y(x)+  x+ 1)y (x)− 2y(x) − x+ 1 = 0
 4.470                  ′
(2y(x)+  x+ 1)y (x)− 4y(x) + x+ 7 = 0
 4.471   x2 + 2(y(x)+ x)y′(x)+ 2y(x) = 0
 4.472   (− 2y(x)+ 2x + 3)y′(x) = − 2y(x)+ 6x + 1
 4.473   (− 2y(x)− 4x + 1)y′(x) + y(x)+ 2x = 0
 4.474   (6x − 2y(x))y′(x) = − y (x )+ 3x + 2
 4.475   (2y(x)+  9x+ 19)y′(x)− 6y(x) − 2x+  18 = 0
 4.476   ( 3       )  ′
 x + 2y(x) y (x) = 3x(2− xy (x ))
 4.477    ′
y (x)(tan(x) sec(x) − 2y(x))+ sec(x )(2y(x) sin(x)+ 1) = 0
 4.478   (e−xx − 2y(x))y′(x) = 2e−2xx− (− 2y(x)+ e− xx+ e−x )y(x)
 4.479   3y(x)y′(x)+ 5 cot(x) cos2(y(x))cot(y(x)) = 0
 4.480   3(2 − y(x))y ′(x)+ xy (x) = 0
 4.481   (x − 3y(x))y′(x)−  y(x )+ 3x + 4 = 0
 4.482   (− 3y(x)− x + 4)y′(x) − 3y(x)− x + 3 = 0
 4.483                   ′
(3y(x)+  2x+ 2)y (x) = − 3y(x)− 2x+ 1
 4.484                     ′
(− 3y(x)− 2x + 5)y(x) − 3y(x)− 2x + 1 = 0
 4.485                     ′
(− 3y(x)+ 9x + 1)y(x) − y(x)+ 3x + 2 = 0
 4.486   (4y(x)+  x)y′(x)−  y(x )+ 4x = 0
 4.487   (4y(x)+  2x+ 3)y′(x) = 2y (x )+ x + 1
 4.488   (− 4y(x)+ 2x + 5)y′(x) = − 2y(x)+ x + 3
 4.489   (− 4y(x)+ 3x + 5)y′(x) = − 3y(x)+ 7x + 2
 4.490   4(− y(x)− x + 1)y′(x) − x+  2 = 0
 4.491                       ′
(− 4y(x)− 11x + 11)y (x ) = − 25y(x)− 8x + 62
 4.492                   ′
(5y(x)+  3x+ 6)y (x) = 7y (x )+ x + 2
 4.493   (5y(x)+  7x)y′(x) + 8y(x)+ 10x = 0
 4.494   (              )
 4x3 + 5y(x) + x y′(x)+ 7x3 + 3x2y(x)+ 4y(x ) = 0
 4.495   (6y(x)−  x+ 5)y′(x) = 4y (x )− x + 3
 4.496   3(2y(x) + x)y′(x) = − 2y (x )− x + 1
 4.497   (7y(x)−  3x+ 3)y′(x)+ 3y(x) − 7x+ 7 = 0
 4.498                  ′
(9y(x)+  x+ 1)y (x)+ 5y(x) + x+ 1 = 0
 4.499                      ′
(− 12y(x)+ 5x + 8)y (x ) = − 5y(x)+ 2x + 3
 4.500                        ′
(− 16y(x)+ 7x + 140)y (x )+ y(x)+ 8x + 25 = 0
 4.501   (21y(x) + 9x+ 3)y′(x) = − 5y(x) + 7x+ 45
 4.502   y′(x)(ax+ by(x))+  x = 0
 4.503   y′(x)(ax+ by(x))+  y(x ) = 0
 4.504   y′(x)(ax+ by(x))+  ay(x)+ bx = 0
 4.505   y′(x)(ax+ by(x)) = ay(x)+ bx
 4.506         ′
a1 + y (x )(a2 + bx+ c2y(x)) + by(x )+ b1x = 0
 4.507    ′
y (x)(a2+ b2y(x) + c2y(x)) = a1 + b1x + c1y(x)
 4.508   xy (x )y′(x)+ y(x)2 + 1 = 0
 4.509   xy (x )y′(x) = y(x )2 + x
 4.510   x2 + xy(x)y′(x )+ y(x)2 = 0
 4.511   x4 + xy(x)y′(x )− y(x)2 = 0
 4.512   xy (x )y′(x) = ax3cos(x)+ y(x)2
 4.513         ′       2              2
xy (x )y (x) = x − xy (x )+ y(x)
 4.514     2         ′                  2
2x  + xy(x)y (x )− 2xy(x) − y(x) = 0
 4.515         ′              2
xy (x )y (x) = a+ by(x)
 4.516   xy (x )y′(x) = axn + by (x )2
 4.517                (     ) (        )
xy (x )y′(x) =  x2 + 1 1 − y(x)2
 4.518           (    )
x2 cot− 1 y(x) +  xy(x)y′(x) − y(x)2 = 0
           x
 4.519   x2e− 2y(xx)+ xy(x)y′(x)− y(x)2 = 0
 4.520   (xy(x) + 1)y′(x)+  y(x )2 = 0
 4.521               ′
x(y(x) + 1)y(x)−  (1 − x)y(x) = 0
 4.522               ′
x(1 − y(x))y(x)+  (x + 1)y(x) = 0
 4.523               ′
x(1 − y(x))y(x)+  (1 − x)y(x) = 0
 4.524   ax + x(y(x)+  2)y ′(x) = 0
 4.525   (x(− y(x))+ 3x + 2)y′(x) + y(x) = 0
 4.526   x(y(x) + 4)y′(x) = y(x)2 + 2y(x) + 2x
 4.527   x(a + y(x))y′(x) + bx+ cy(x) = 0
 4.528   x(a + y(x))y′(x) = y(x)(A + Bx )
 4.529   x(y(x) + x)y′(x) + y(x)2 = 0
 4.530   x(x − y(x))y′(x) + y(x)2 = 0
 4.531   x(y(x) + x)y′(x) = x2 + y(x)2
 4.532     2               ′                 2
2x  + x(x − y(x))y (x)+ 3xy (x)− y(x) =  0
 4.533    ∘  -2------2-             ′
x   x − y(x) +  x(y(x )+ x)y (x )− y(x)(y(x)+ x) = 0
 4.534                    ′
(a + x(y(x)+ x))y (x ) = by(x)(y(x)+ x)
 4.535   x(y(x) + 2x)y′(x) = x2 + xy(x)− y(x)2
 4.536   4x2 + x(4x − y(x))y ′(x)− 6xy (x)− y(x)2 = 0
 4.537     (        )        (        )
x  x3 + y(x) y′(x) =  x3 − y (x ) y(x)
 4.538   x (2x3 + y (x )) y′(x) = (2x3 − y(x )) y(x)
 4.539   x (2x3 + y (x )) y′(x) = 6y(x)2
 4.540               ′
(1 − x)y(x)y(x)+  x(1− y(x)) = 0
 4.541                 ′
(a + x)(b+ x)y (x ) = xy(x )
 4.542   − 2x3 + 2xy (x)y′(x) − y(x)2 + 1 = 0
 4.543   a + 2xy(x)y′(x )+ y(x)2 = 0
 4.544   2xy (x )y ′(x) = ax+ y (x )2
 4.545   x2 + 2xy(x)y′(x )+ y(x)2 = 0
 4.546   2xy (x )y ′(x) = x2 + y(x)2
 4.547           ′              2       2
2xy (x )y (x) = 4(2x + 1)x  + y(x)
 4.548    2 (  3   )          ′          2
x   ax + 1  + 2xy(x)y (x) = 6y(x)
 4.549     2                   ′         2
3x  + (2xy(x)− x + 3)y (x )+ y(x) − y (x ) = 0
 4.550   x(x − 2y(x))y′(x) + y(x)2 = 0
 4.551   x(2y(x) + x)y′(x) + (2x− y(x))y(x) = 0
 4.552   x(x − 2y(x))y′(x) + (2x− y(x))y(x) = 0
 4.553   x(− 2y(x)+ x + 1)y′(x )+ y(x)(y(x)− 2x + 1) = 0
 4.554   x(− 2y(x)− x + 1)y′(x )+ y(x)(y(x)+ 2x + 1) = 0
 4.555      (  2      )  ′         (   2       )
2x  2x + y (x ) y(x) + y(x) 12x  + y(x) = 0
 4.556       2               ′         2
− 3x + 2 (x + 1)y(x)y(x) + y(x) + 2x = 0
 4.557   x(3y(x) + 2x)y′(x) = y(x)2
 4.558   x(3y(x) + 2x)y′(x) + 3(y(x)+ x)2 = 0
 4.559   (              )
 x2 + 6xy (x )+ 3 y′(x)+ 3y(x)2 + 2xy(x)+ 2x = 0
 4.560   x3 + 3x(2y(x)+ x )y ′(x)+ 3y (x )(y(x) + 2x) = 0
 4.561   axy (x )y ′(x) = x2 + y(x)2
 4.562           ′      2      2
axy (x )y (x)+ x  − y(x) =  0
 4.563      ′
xy (x)(a+ by(x)) = cy(x)
 4.564                ′
x(x − ay(x))y(x) = y(x)(y(x)− ax )
 4.565   y′(x)(x(Ax + By (x))+ a0 + a1x + a2y(x)) = y(x)(Ax + By (x))+ b0 + b1x + b2y(x)
 4.566   y′(x)(x(a1 + b2x + c2y(x )) + a1+ b1x + c1y(x)) = y(x)(a2+ b2x + c2y(x))+ a3 + b3x + c2y(x)
 4.567   xy ′(x)(ay(x) + xn)+ y(x)2(b+ cy(x)) = 0
 4.568   (    2    )  ′          2
 1− x y(x) y (x)− xy (x) + 1 = 0
 4.569   (    2    )  ′          2
 1− x y(x) y (x)+ xy (x) − 1 = 0
 4.570   x(1 − xy(x))y′(x) + y(x)(xy (x)+ 1) = 0
 4.571   x(xy (x )+ 2)y′(x) = 2x3 − xy(x)2 − 2y(x) + 3
 4.572   x(2 − xy(x))y′(x) − x(xy(x)+ 1)y(x)2 + 2y(x) = 0
 4.573   x(3 − xy(x))y′(x) = y(x)(xy(x)− 1)
 4.574   x2(1 − y(x))y ′(x)+ (1 − x)y(x) = 0
 4.575    2           ′               2
x (1 − y(x))y (x)+ (x + 1)y(x) = 0
 4.576   ( 2   )      ′      (       2)
 x + 1  y(x )y (x)+ x  1−  y(x )  = 0
 4.577   (1− x2) y(x )y ′(x)+ 2x2 + xy(x)2 = 0
 4.578   2x2y(x)y′(x) = x2(2x + 1) − y(x)2
 4.579   x(1 − 2xy(x))y′(x) + y(x)(2xy(x)+ 1) = 0
 4.580                 ′
x(2xy (x )+ 1)y(x) + y(x)(3xy(x)+ 2) = 0
 4.581       (   2    2            )                ′
y(x) − x y(x) +  2xy(x)+ 1  + x(2xy(x)+ 1)y (x) = 0
 4.582    2            ′       3         2      3
x (x − 2y(x))y(x) = 2x  − 4xy(x) + y(x)
 4.583   2x(x + 1)y(x)y′(x) = y(x)2 + 1
 4.584   3x2y(x)y′(x)+ 2xy (x )2 + 1 = 0
 4.585                            (                    )
x2(4x − 3y(x))y′(x) = y(x) 6x2 − 3xy(x)+  2y(x )2
 4.586   (1− x3y(x))y ′(x) = x2y(x)2
 4.587   a + 2x3y(x)y′(x )+ 3x2y(x)2 = 0
 4.588     (      2   )  ′       2    2
x  3− 2x y (x ) y(x) = 3x y(x) − 3y (x )+ 4x
 4.589     (  2       )  ′         (  2        )
x  2x y(x)+ 3  y(x) + y(x) 3x y(x)+  4 = 0
 4.590   3x4 + 8x3y(x)y′(x )− 6x2y(x)2 − y (x )4 = 0
 4.591         (      )
xy (x ) a+ bx2  y′(x) = A + By (x)2
 4.592   3x4y(x)y′(x) = 1− 2x3y(x)2
 4.593   x7y(x)y′(x) = 5x3y (x)+ 2(x2 + 1)
 4.594   √ -2----     ′      ∘ ----2----
  x  + 1y(x)y(x) + x  y(x) + 1 = 0
 4.595   √ -2----          ′         3
  x  + 1(y (x )+ 1)y(x) = y(x)
 4.596   y′(x)(g0(x )+ g1(x)y(x)) = f0(x)+ f1(x)y(x)+ f2(x)y(x)2 + f3(x)y(x)3
 4.597   y(x)2y′(x)+ x(2 − y(x)) = 0
 4.598                 (        )
y(x)2y′(x) = x y(x)2 + 1
 4.599   (y(x )2 + x)y ′(x)+ y (x ) = a + bx
 4.600   (x− y(x)2)y ′(x) = x2 − y(x)
 4.601   ( 2      2) ′
 x + y(x)  y (x)+ xy (x ) = 0
 4.602   ( 2      2) ′
 x + y(x)  y (x) = xy(x)
 4.603   (         )
 x2 − y(x)2 y′(x) = 2xy(x)
 4.604   (         )
 x2 − y(x)2 y′(x)+ x (2y(x) + x) = 0
 4.605   (         )
 x2 + y(x)2 y′(x)+ 2x (y(x) + 2x) = 0
 4.606   (              )
 − x2 + y(x)2 + 1 y′(x) = x2 − y (x )2 + 1
 4.607   (a2 + x2 + y(x)2)y′(x )+ 2xy(x) = 0
 4.608   ( 2   2       2) ′      2   2
 a + x  + y(x)  y (x )+ b + x  + 2xy(x) = 0
 4.609   ( 2      2    ) ′
 x + y(x) +  x y (x) = y(x)
 4.610   (  2      2) ′
 3x − y(x)  y (x) = 2xy(x)
 4.611   (         )
 x4 + y(x)2 y′(x) = 4x3y(x)
 4.612   y(x)(y(x)+ 1)y′(x) = x(x+ 1)
 4.613   (                )
 y(x)2 + 2y(x)+ x  y′(x)+  y(x )2(y(x) + x)2 + y(x)(y(x)+ 1) = 0
 4.614   (x2 + y(x)2 + 2y(x))y′(x) + 2x = 0
 4.615   (x3 − y(x)2 + 2y(x))y′(x) + 3x2y(x) = 0
 4.616   (   2                  ) ′
 y(x ) + xy(x)+  y(x )+ 1 y (x)+ y(x) + 1 = 0
 4.617             2 ′      2
(y(x)+  x)y (x) = a
 4.618   (x − y(x))2y ′(x) = a2
 4.619   (                  )
 x2 + 2xy (x )− y(x)2 y′(x )+ x2 − 2xy(x)+ y(x)2 = 0
 4.620   (x − y(x))2y ′(x) = (− y(x)+  x+ 1)2
 4.621   (y(x)+  x)2y ′(x) = (y (x )+ x + 2)2
 4.622   (y(x)+  x)2y ′(x) = x2 − 2xy (x )+ 5y(x)2
 4.623    ′                   2             2
y (x)(a+ b + y(x)+ x)  = 2(a+ y(x))
 4.624   (  2               2) ′      2                 2
 2x + 4xy (x )− y(x)  y (x) = x − 4xy(x) − 2y(x)
 4.625              2 ′
(y(x)+  3x)y (x) = 4y(x)(2y (x )+ 3x)
 4.626   (− y(x)− 3x + 1)2y′(x) = (− 4y(x) − 6x + 3)(1 − 2y(x))
 4.627   y′(x)(cot(x)− 2y(x)2) = y(x)3csc(x )sec(x)
 4.628   3y(x)2y′(x) = ay (x )3 + x+ 1
 4.629   ( 2       2) ′
 x − 3y(x)  y (x)+ 2xy (x)+ 1 = 0
 4.630   (  2       2)  ′
 2x + 3y(x)  y (x)+ x (y (x )+ 3x) = 0
 4.631     ( 2      2) ′          3                    x
3  x − y(x)  y (x)− 2y(x)  + 6x(x+  1)y(x) + 3e =  0
 4.632   (                     )
 3x2 + 2xy(x) + 4y(x)2 y′(x )+ 2x2 + 6xy(x)+ y(x)2 = 0
 4.633   (2y(x)−  3x+ 1)2y′(x) = (− 3y(x)+ 2x + 4)2
 4.634   (                    )
 − 3x2y(x) + 6y(x)2 + 1 y′(x)+ x2 − 3xy (x )2 = 0
 4.635   a + (x− 6y (x ))2y′(x)− 6y(x)2 + 2xy(x) = 0
 4.636   (ay(x)2 + x2)y ′(x) = xy(x)
 4.637   (     2   2        ) ′        2              2
 ay(x) + x  + xy(x) y (x) = ax + xy(x) + y(x)
 4.638   (   2       2         ) ′                2      2
 ax  − ay(x)  + 2xy(x) y (x)− 2axy (x )+ x −  y(x ) = 0
 4.639        (                                  )
y′(x)  x2(a + 2b)+ 2x (2b + c)y(x)+ 3cy(x)2 + 2x (a + 2b)y(x)+ 3ax2 + (2b+ c)y(x)2 = 0
 4.640        (                     )
y′(x ) ax2 + 2bxy (x)+ cy(x)2 + 2axy (x)+ by(x)2 + kx2 = 0
 4.641   x (1− y(x)2)y ′(x) = (x2 + 1) y(x)
 4.642   x (3x−  y(x )2) y′(x) + y(x)(5x − 2y(x)2) = 0
 4.643     ( 2      2)  ′         ( 4    2      2)
x  x + y(x)   y(x) = y(x) x  + x + y(x)
 4.644     (   2      2   )  ′         ( 2       2   )
x  − x + y(x) + 1  y(x) + y(x) x  − y(x ) + 1  = 0
 4.645     (     2      2) ′         (     2       2)
x  a− x  − y(x)  y (x)+ y(x) a + x  + y(x)  = 0
 4.646     (          )            (            )
x  2x2 + y (x )2 y′(x) = y(x) 2x2 + 3y(x)2
 4.647   (  (             )      )            (             )
 x  a− x2 − y(x)2 + y(x)  y′(x)−  y(x ) a− x2 − y(x)2 + x = 0
 4.648   x(a + y(x))2y′(x) = by(x)2
 4.649   x (x2 − xy(x) + y(x)2)y′(x )+ y(x)(x2 + xy(x)+ y (x )2) = 0
 4.650   x (x2 − xy (x )− y(x)2)y′(x) = y(x) (x2 + xy(x) − y(x)2)
 4.651     (          2      2) ′          (         2       2)
x  axy(x)+ x  + y(x)  y (x) = y(x) bxy (x)+ x  + y(x)
 4.652     ( 2       2)  ′         (  2       2)
x  x − 2y (x )  y(x) = y(x) 2x  − y(x)
 4.653     (          )            (            )
x  x2 + 2y (x )2 y′(x) = y(x) 2x2 + 3y(x)2
 4.654      (          )
2x  5x2 + y (x )2 y′(x) = x2y(x)− y(x)3
 4.655     (                   )
x  axy(x)+  x2 + 2y(x)2 y′(x) = y(x )2(ax + 2y(x))
 4.656   3xy (x )2y′(x) = 2x− y(x)3
 4.657   (3xy(x)2 − 4x + 1)y′(x) = y(x )(2− y(x)2)
 4.658     (        2)  ′         (         2)
x  x− 3y (x )  y(x) + y(x) 2x − y(x)  = 0
 4.659    3     (    2    ) ′                   3
x  + 3x y (x ) + x y (x)− 3xy (x )− 2y(x) =  0
 4.660     (    3       3        2)  ′          3
x  − 3x y(x )+ x  + 4y(x)  y(x) = 6y(x)
 4.661   6xy (x )2y′(x)+ 2y(x)3 + x = 0
 4.662     (         )
x  6y(x)2 + x y′(x) − 3y(x)3 + xy(x) = 0
 4.663     (          )             (           )
x  x2 − 6y (x )2 y′(x) = 4y(x) x2 + 3y(x)2
 4.664   x (3x−  7y(x )2) y′(x) + y(x)(5x − 3y(x)2) = 0
 4.665   x3 + x2y(x)2y′(x) − x+  1 = 0
 4.666   (    2    2) ′           3
 1− x y(x)  y (x) = xy(x)
 4.667   (    2    2) ′          2
 1− x y(x)  y (x) = y(x)(xy (x )+ 1)
 4.668     (         )
x  xy(x)2 + 1 y′(x) + y(x) = 0
 4.669     (         )            (           )
x  xy(x)2 + 1 y′(x) = y(x) 2 − 3xy(x)2
 4.670                       (     ) (         )
x2(a + y(x))2y ′(x) =  x2 + 1  a2 + y(x)2
 4.671   (x2 + 1) (y(x )2 + 1)y′(x)+ 2xy (x )(1− y(x)2) = 0
 4.672   (x2 + 1) (y(x )2 + 1)y′(x)+ 2xy (x )(1 − y(x))2 = 0
 4.673   (  3     2    2   )  ′      (       3            )
− x  + 6x y(x) + 1  y(x) = x − 4y(x) + 3xy (x)+ 6
 4.674   x (4x2y(x)− 12xy (x)2 + 5x + 3)y′(x)+ y(x)(6x2y (x )− 8xy(x)2 + 10x+ 3) = 0
 4.675   x3 (y(x)2 + 1)y′(x)+ 3x2y (x ) = 0
 4.676        ( 2   2    )              2 ′
y(x) x  y(x ) + 1 + x (1 − xy(x)) y (x ) = 0
 4.677   (    4    2) ′       3    3
 1− x y(x)  y (x) = x y(x)
 4.678   (        3)  ′      2
 3x− y(x)  y (x) = x − 3y(x)
 4.679   (         )
 x3 − y(x)3 y′(x)+ x2y (x ) = 0
 4.680                   (          )
x2(ax + 3y(x))+  x3 + y(x)3 y′(x ) = 0
 4.681   (                  )
 − x2y(x) − y(x)3 + x y′(x) = x3 + xy (x)2 − y(x)
 4.682   (a2x + y(x)(x2 − y(x)2))y′(x)+ x (x2 − y(x)2) = a2y(x)
 4.683   y(x) (a + x2 + y(x)2)y′(x ) = x (a− x2 − y(x)2)
 4.684        (  2      2) ′       ( 2       2)
y(x) 3x  + y(x)  y (x)+ x  x + 3y (x )  = 0
 4.685        (     2       2) ′      (     2       2)
y(x) a − 3x  − y(x)  y (x )+ x a − x  + y(x)  = 0
 4.686   2y(x)3y′(x) = x3 − xy(x)2
 4.687        (         )         (       )
y(x) 2y (x )2 + 1 y′(x) = x 2x2 + 1
 4.688            (            )
x3 + y(x) 3x2 + 2y(x)2 y′(x ) = 0
 4.689   y(x) (5x2 + 2y(x)2)y′(x)+ x (x2 + 5y (x )2) = 0
 4.690     3     2       (  3    2         2       3) ′         3      2
2x  + 3x y(x)+  − x  + x + 3xy (x) + 2y(x)  y (x)− y(x) +  y(x ) = 0
 4.691     3     2      (  3     2            2         3) ′           2       3
4x + 9x  y(x)+  3x  + 6x y(x)− 3xy (x ) + 20y(x)  y (x)+ 6xy(x)  − y(x) = 0
 4.692   (          )
 ay(x)3 + x3 y ′(x) = x2y(x)
 4.693        (                             )
y′(x) ax2 − bx3 + 3cxy (x )2 + 2cy(x)3 + ay(x)2 + 2bx3 + 3bx2y(x) − cy(x )3 = 0
 4.694   xy (x )3y′(x) = (1 − x2)(y(x)2 + 1)
 4.695   x (x− y (x )3) y′(x) = y(x)(y(x)3 + 3x )
 4.696     (  3      3)  ′         (  3    2          3)
x  2x + y (x )  y(x) = y(x) 2x  − x y(x)+ y(x)
 4.697     (  3      3)  ′         ( 3        3)
x  2x − y (x )  y(x) = y(x) x  − 2y(x)
 4.698     (                   )             (          )
x  x3 + 3x2y (x )+ y(x)3 y′(x) = y(x)2 3x2 + y(x)2
 4.699     (          )            (           )
x  x3 − 2y (x )3 y′(x) = y(x) 2x3 − y(x)3
 4.700     (          )            (          )
x  x4 − 2y (x )3 y′(x)+  y(x ) 2x4 + y (x )3 = 0
 4.701   x (2y(x)3 + y(x) + x)y′(x) = (x − y(x))y(x)
 4.702   (− 7xy (x )3 − y(x)+ 5x) y′(x) − y(x)4 + 5y(x) = 0
 4.703        (     3    )    (          3) ′
y(x) 1 − 2x y(x)  + x 1 − 2xy(x)  y (x) = 0
 4.704     (        3       2    ) ′
x  − 2xy (x) − xy(x) + 2 y (x )+ 2y(x)+ 1 = 0
 4.705   (    2    3        2   )  ′      (     4    )
 − 10x y(x) + 3y(x) + 2 y (x) = x 5y(x)  + 1
 4.706         (           )       (          )
xy ′(x) a + bxy(x)3 + y(x) a + cx3y(x)  = 0
 4.707        (           )    (            )
y(x) 1 − 2x3y(x)2 +  x 1 − 2x2y(x)3 y′(x ) = 0
 4.708               (          )                      (          )
x(1− xy (x)) 1− x2y(x)2 y ′(x)+ y (x )(xy(x) + 1) x2y(x)2 + 1 = 0
 4.709   (x2 − y(x)4)y′(x) = xy(x)
 4.710   (x3 − y(x)4)y′(x) = 3x2y(x)
 4.711   (       (         )2)
 a2x2 + x2 + y(x)2   y′(x) = a2xy(x)
 4.712   2 (x− y(x)4)y ′(x) = y(x)
 4.713   (− 2y(x)4 − xy(x)3 + 4x) y′(x) = y(x)(y(x)3 + 2 )
 4.714        ′   (           3     3)    (           3       3)
y(x)y(x)  (ax + by(x)) + ax   + x  (ax + by(x)) + by(x)   = 0
 4.715   (   2   3        4           ) ′          (    4   )
 2x y (x ) + xy(x) + 2y (x )+ x y (x)+ y(x)  y(x) + 1  = 0
 4.716      (         )            (           )
2x  x3 + y (x )4 y′(x) = y(x) x3 + 2y(x)4
 4.717     (          )
x  1− x2y (x )4  y′(x)+  y(x ) = 0
 4.718   (         )
 x2 − y(x)5 y′(x) = 2xy(x)
 4.719   x (x3 + y(x)5) y′(x) = y(x)(x3 − y(x)5)
 4.720   y(x)3(3x5y (x )5 − 1)+ x3 (5x3y(x)7 + 1) y′(x) = 0
 4.721   y′(x )(f1(x,y(x))+ xf2(x,y(x))) = y(x)f2(x,y(x))+ f3(x,y(x))
 4.722    ′                  n             n
y (x)(a(y (x )+ x)+ 1)  + a(y(x)+ x)  = 0
 4.723      ′            n
xy (x)(a + xy(x) )+ by(x) = 0
 4.724           m  ′             m+1           n
f (x )y(x)  y(x) + g(x)y(x)    + h(x)y(x) = 0
 4.725   ∘ ----------       √-------
  b2 + y(x)2y′(x) =  a2 + x2
 4.726   ∘ ----------       √-------
  b2 − y(x)2y′(x) =  a2 − x2
 4.727   √ --       √ --
  Y y′(x) =   X
 4.728   (∘ --------   )  ′
   y(x)+ x + 1  y(x) + 1 = 0
 4.729   ∘ ------                 ∘ ------
  xy (x )y ′(x)− y (x )+ x =   xy(x)
 4.730   (            )
 x − 2∘xy-(x)-y ′(x) = y(x)
 4.731   (     )   (       ∘ --------)
 x2 + 1 3∕2 y(x) +  y(x)2 + 1 y′(x ) = y(x)2 + 1
 4.732   (     )3∕2(       ∘ --------)
 x2 + 1    y(x) +   y(x)2 + 1 y′(x ) = y(x)2 + 1
 4.733   (    ∘ -2------2)  ′
 x −   x + y(x)   y(x) = y(x)
 4.734     (               )
x  1 − ∘x2-−-y(x)2- y′(x) = y(x)
 4.735     (∘ ----------   )        ∘ ----------
x    x2 + y(x)2 + x y′(x )+   x2 + y(x)2y (x ) = 0
 4.736         (∘ ----------   )                (          )3∕2
xy(x)    x2 − y(x )2 + x  y′(x) = xy(x)2 − x2 − y(x)2
 4.737   (        2∘ ----2----2) ′          ( ∘ ----2----2   )
  x− y (x )  y(x)  − x  y (x) = y(x) x  y (x ) − x + 1
 4.738   (    --------------                 )          --------------
 x ∘ x2 + y(x)2 + 1− y(x) (x2 + y(x)2) y ′(x) = ∘ x2 + y (x )2 + 1y(x)+ x (x2 + y (x )2)
 4.739    ′   ( 2                  )
y (x ) x sec(x)cos(y(x )) + 1 − y(x) tan (x )+ sec(x )sin(y(x)) = 0
 4.740    ′
y (x)cos(y(x))(cos(y(x))− sin (A )sin(x))+ cos(x )(cos(x)− sin(A) sin(y(x))) = 0
 4.741   y′(x)(a cos(ay(x) + bx)− bsin(ax+  by (x )))−  asin (ax + by(x))+ b cos(ay(x) + bx) = 0
 4.742   y′(x)(a sin(xy(x))− sin(y(x))+ cos(y(x )+ x))+ y (x )sin(xy(x))+  cos(y(x)+  x)+ cos(x) = 0
 4.743   y′(x )(cos(x)sec(y(x))+ x)+ tan(y(x))−  y(x )sin(x) sec(y(x)) = 0
 4.744    ′   ( 2                      )                    2
y (x) x  + 2y(x)sin (x )sec(y(x)) + 2x tan(y(x))+ y(x) cos(x)sec(y(x)) = 0
 4.745        (                        )
y′(x ) x2sec2(y(x))− 6xy(x)+  2 − 3y(x)2 + 2xtan(y(x)) = 0
 4.746   y′(x)((y(x) + x)tan(y(x))+ 1) + 1 = 0
 4.747         (            (   ) )       (        (   )    )
xy′(x) x − y(x)tan  y(x)-  + y(x)  y(x)tan  y(x)- + x  = 0
                     x                      x
 4.748   (         )
 xey(x) + ex y′(x) + exy(x)+ ey(x) = 0
 4.749   y′(x)(− log(y(x))− 2x + 1)+ 2y (x ) = 0
 4.750   y′(x )(x cosh(y(x))+ sinh (x )) + sinh(y(x))+  y(x )cosh(x) = 0
 4.751   (sinh (x )+ 1)y′(x) sinh(y(x))+ cosh(x)(cosh(y(x))− 1) = 0
 4.752   y′(x)2 = axn
 4.753    ′   2
y (x) = y(x)
 4.754    ′   2
y (x) = x − y(x)
 4.755    ′   2    2
y (x) = x  + y(x)
 4.756   x2 + y′(x )2 = 4y(x)
 4.757   3x2 + y′(x )2 = 8y(x)
 4.758   ax2 + by(x)+ y′(x)2 = 0
 4.759   y′(x)2 = y(x)2 + 1
 4.760   y′(x)2 = 1 − y(x)2
 4.761    ′   2   2       2
y (x) = a  − y(x)
 4.762    ′   2   2    2
y (x) = a y(x)
 4.763    ′   2           2
y (x) = a + by(x)
 4.764   y′(x)2 = x2y (x )2
 4.765   y′(x)2 = (y(x) − 1)y(x )2
 4.766   y′(x)2 = (y(x) − a)(y(x) − b)(y(x) − c)
 4.767   y′(x)2 = a2y(x)n
 4.768   y′(x)2 = a2y(x)2 (1− log2(y(x )))
 4.769                             ′  2
f (x )(y(x) − a)(y(x) − b)+ y (x ) = 0
 4.770                 2            ′  2
f (x )(y(x) − a)(y(x) − b)+ y (x ) = 0
 4.771   f (x )(y(x) − a)(y(x) − b)(y(x) − c)+ y′(x )2 = 0
 4.772   f (x )(y(x) − a)2(y(x) − b)(y(x) − c)+ y′(x )2 = 0
 4.773   f(x)(y(x)− a1)(y(x)−  a2)(y(x) − a3)(y(x) − a4)+ y′(x)2 = 0
 4.774   y′(x)2 = f (x )2(y(x) − a)(y(x) − b)(y(x) − c)2
 4.775   y′(x)2 = f (x )2(y(x) − a)(y(x) − b)(y(x) − u(x))2
 4.776    ′   2    ′
y (x) + 2y (x)+ x = 0
 4.777                 ′   2    ′
a(x − y(x))+ y (x) − 2y (x) = 0
 4.778    ′   2    ′         2
y (x) − 2y (x)− y(x)  = 0
 4.779   y′(x)2 − 5y′(x)+ 6 = 0
 4.780   y′(x)2 − 7y′(x)+ 12 = 0
 4.781   ay′(x)+ b + y′(x)2 = 0
 4.782   ay′(x)+ bx + y′(x )2 = 0
 4.783   ay′(x)+ by(x) + y′(x)2 = 0
 4.784    ′   2    ′
y (x) + xy (x)+ 1 = 0
 4.785    ′   2    ′
y (x) + xy (x)− y(x) = 0
 4.786   y′(x)2 − xy′(x)+ y(x) = 0
 4.787   y′(x)2 − xy′(x)− y(x) = 0
 4.788   y′(x)2 + xy′(x)− y(x) + x = 0
 4.789   y′(x)2 + (1− x)y′(x)+  y(x ) = 0
 4.790   y′(x)2 − (x + 1)y′(x)+ y(x ) = 0
 4.791    ′   2          ′
y (x) − (2−  x)y(x)−  y(x )+ 1 = 0
 4.792           ′      ′  2
(a + x)y(x) + y(x)  − y(x) = 0
 4.793    ′   2     ′
y (x) − 2xy (x)+ 1 = 0
 4.794   − 3x2 + 2xy ′(x) + y′(x)2 = 0
 4.795   y′(x)2 + 2xy′(x)− y(x) = 0
 4.796   y′(x)2 + 2xy′(x)− y(x) = 0
 4.797   y′(x)2 − 2xy′(x)+ 2y (x ) = 0
 4.798   y′(x)2 − (2x + 1)y′(x) − (1 − x)x = 0
 4.799    ′   2           ′
y (x) + 2(1 − x)y(x) − 2(x − y(x)) = 0
 4.800    ′   2     ′
y (x) + 3xy (x)− y(x) = 0
 4.801   y′(x)2 − 4(x + 1)y′(x) + 4y(x) = 0
 4.802   axy ′(x)+  y′(x)2 = bcx2
 4.803   − axy′(x)+ ay(x) + y′(x)2 = 0
 4.804   axy ′(x)+  bx2 + cy(x)+ y ′(x)2 = 0
 4.805   (a + bx)y′(x) + c+ y′(x)2 = by(x)
 4.806       2 ′        ′      ′   2
− 2x y (x )+ 2xy (x)+ y (x) = 0
 4.807      2 ′              ′  2
ax y (x)+ bxy (x )+ y (x ) = 0
 4.808   ax3y ′(x)− 2ax2y (x)+ y′(x)2 = 0
 4.809   − 2ax3y′(x )+ 4ax2y(x) + y′(x)2 = 0
 4.810   4x5y′(x)− 12x4y (x )+ y′(x )2 = 0
 4.811    ′   2           ′
y (x) − 2 cosh(x )y (x)+ 1 = 0
 4.812    ′   2       ′
y (x) + y(x)y (x) = x(y(x) + x)
 4.813    ′   2       ′      x
y (x) − y(x)y (x)+ e  = 0
 4.814   y′(x)2 + (y(x)+ x )y ′(x)+ xy (x) = 0
 4.815   y′(x)2 − 2y(x)y′(x)− 2x = 0
 4.816   y′(x)2 + (2y(x)+ 1 )y′(x)+ (y(x) − 1)y (x ) = 0
 4.817   y′(x)2 − 2(x − y(x))y ′(x)− 4xy (x) = 0
 4.818   y′(x)2 − (4y(x)+ 1 )y′(x)+ y(x)(4y(x)+ 1) = 0
 4.819    ′   2              ′
y (x) − 2(1 − 3y(x))y (x)− (4 − 9y(x))y(x ) = 0
 4.820                                    ′      ′  2
y(x)(3a + b+ 9y(x))+ (a + 6y(x))y(x) + y(x)  = 0
 4.821   ay(x)y′(x)− ax + y′(x)2 = 0
 4.822   − ay(x)y′(x) − ax + y′(x)2 = 0
 4.823   y′(x)(ax+ by(x))+  abxy(x)+ y′(x)2 = 0
 4.824   y(x)2log(ay(x))− xy (x )y ′(x)+ y′(x)2 = 0
 4.825   y′(x)2 − (2xy(x) + 1)y′(x)+ 2xy(x) = 0
 4.826    ′   2  (    2    ) ′         2
y (x) −  y(x) + 4  y(x) + y(x) + 4 = 0
 4.827                   ′      ′  2        3
− (x− y(x))y(x)y (x )+ y (x ) − xy(x) =  0
 4.828        2 ′      ′   2      3
xy (x )y (x)+ y (x) + y(x)  = 0
 4.829   − 2x3y(x)2y′(x) − 4x2y(x)3 + y′(x)2 = 0
 4.830              (          )
x4y(x)4 − x x2 + y(x)2 y(x)y′(x )+ y′(x )2 = 0
 4.831   2xy (x )3y′(x)+ y′(x)2 + y(x)4 = 0
 4.832   y′(x)2 + 2y(x)cot(x)y′(x )− y(x)2 = 0
 4.833   y′(x)2 − 3xy(x)2∕3y′(x) + 9y(x)5∕3 = 0
 4.834    ′   2   4x−2y(x)  ′
y (x) = e        (y (x)− 1)
 4.835     ′   2    ′
2y (x) + xy (x)− 2y (x ) = 0
 4.836   2y′(x)2 − (1 − x)y′(x) − y(x) = 0
 4.837   − 2x2y′(x )+ 2y′(x )2 + 3xy(x) = 0
 4.838   2y′(x)2 + 2(6y(x)− 1)y ′(x)+ 3y (x )(6y(x) − 1) = 0
 4.839   3y′(x)2 − 2xy ′(x)+ y (x ) = 0
 4.840   x2 + 4xy′(x)+ 3y′(x)2 − y(x) = 0
 4.841     ′   2
4y (x) = 9x
 4.842     ′   2     − 2y(x) ′      −2y(x)
4y (x) + 2xe      y(x) − e     =  0
 4.843     ′   2    2x−2y(x) ′      2x−2y(x)
4y (x) + 2e       y (x)− e        = 0
 4.844   5y′(x)2 + 3xy ′(x)− y (x ) = 0
 4.845   5y′(x)2 + 6xy ′(x)− 2y (x ) = 0
 4.846   3xy (x )4y′(x)+ 9y′(x)2 + y(x)5 = 0
 4.847   xy ′(x)2 = a
 4.848   xy ′(x)2 = a− x2
 4.849      ′  2
xy (x) =  y(x )
 4.850      ′  2
xy (x) − 2y(x) + x = 0
 4.851   xy ′(x)2 + y′(x) = y(x )
 4.852   xy ′(x)2 + 2y′(x)− y(x) = 0
 4.853   xy ′(x)2 − 2y′(x)− y(x) = 0
 4.854   xy ′(x)2 + 4y′(x)− 2y (x ) = 0
 4.855   xy ′(x)2 + xy ′(x)− y(x ) = 0
 4.856     ( 2   )  ′       ′  2
−  x + 1  y(x) + xy (x ) + x = 0
 4.857         ′   2       ′
a + xy (x) + y(x)y (x ) = 0
 4.858   a + xy′(x)2 − y(x)y′(x ) = 0
 4.859   ax + xy′(x)2 − y(x)y′(x) = 0
 4.860   − x2 + xy ′(x)2 + y(x )y ′(x) = 0
 4.861    3     ′   2       ′
x  + xy (x) + y(x)y (x ) = 0
 4.862             ′  2        ′
ay(x) + xy (x ) − y(x)y(x) = 0
 4.863        ′       ′   2      4
y(x)y (x)+ xy (x) − y(x)  = 0
 4.864   (a − y(x))y ′(x)+ b + xy′(x)2 = 0
 4.865   xy ′(x)2 + (x − y(x))y′(x) − y(x)+ 1 = 0
 4.866   (a − y(x)+ x)y′(x)+ xy′(x)2 − y(x) = 0
 4.867   xy ′(x)2 − (3x − y(x))y′(x) + y(x) = 0
 4.868   a − by(x)+ bx + xy′(x)2 − y(x) = 0
 4.869         ′   2        ′
a + xy (x) − 2y(x)y (x ) = 0
 4.870      ′  2        ′
xy (x) + 2y(x)y (x)− x = 0
 4.871   ax + xy′(x)2 − 2y(x)y′(x) = 0
 4.872   xy ′(x)2 − 2y(x)y′(x)+ 2y(x) + x = 0
 4.873   9x2 + xy′(x)2 − 3y(x)y′(x ) = 0
 4.874   xy ′(x)2 − (3y(x) + 2x)y′(x) + 6y(x) = 0
 4.875   − ay(x)y′(x) + b+ xy ′(x)2 = 0
 4.876         ′            ′   2
ay(x)y (x)+ bx + xy (x) = 0
 4.877      ′  2              ′
xy (x) − (xy(x) + 1)y(x) + y(x) = 0
 4.878               ′       ′  2       2
(1 − x)y(x)y(x)+  xy(x)  − y(x) = 0
 4.879   (         )
 1− x2y(x) y ′(x)+ xy ′(x)2 − xy(x) = 0
 4.880   (x + 1)y′(x)2 = y(x)
 4.881   (x + 1)y′(x)2 − (y(x) + x)y′(x) + y(x) = 0
 4.882   (a − x)y′(x)2 − b+ y(x)y′(x ) = 0
 4.883   (a+ x )y′(x)2 + y′(x)(a1+ b1x + c1y(x))+ a2 + b2x + c2y(x) = 0
 4.884       ′  2              ′
2xy (x) + (2x − y(x))y(x) − y(x)+ 1 = 0
 4.885       ′  2        ′
3xy (x) − 6y (x )y (x)+ 2y(x) + x = 0
 4.886   (3x + 1)y′(x)2 − 3(y (x )+ 2)y′(x) + 9 = 0
 4.887   (3x + 5)y′(x)2 − (3y (x )+ 3)y′(x) + y(x) = 0
 4.888   4xy ′(x)2 = (a − 3x)2
 4.889   4xy ′(x)2 + 2xy ′(x) − y(x) = 0
 4.890   4xy ′(x)2 − 3y (x )y′(x)+ 3 = 0
 4.891       ′  2        ′
4xy (x) + 4y (x )y (x) = 1
 4.892         ′         ′  2      4
4y(x)y (x)+ 4xy (x) − y(x)  = 0
 4.893            ′  2
4(2 − x)y(x)  + 1 = 0
 4.894   8y(x)y′(x)+ 16xy ′(x)2 + y(x)6 = 0
 4.895   x2y′(x)2 = a2
 4.896   x2y′(x)2 = y (x )2
 4.897   x2y′(x)2 + x2 − y(x)2 = 0
 4.898   x2y′(x)2 = (x − y(x))2
 4.899    2 ′   2      4       2
x y (x) − y(x)  + y(x) = 0
 4.900    2 ′   2     ′
x y (x) − xy (x)+ (1 − y(x))y(x ) = 0
 4.901   a2 + 2axy′(x)− 2ay(x) + x2y′(x)2 + x2 = 0
 4.902   x2y′(x)2 − 2xy (x )y′(x)+ y(x)(y(x)+ 1)− x = 0
 4.903                   (      )
− x4 + x2y ′(x)2 + 1 − x2 y(x)2 − 2xy(x)y′(x ) = 0
 4.904   x2y′(x)2 − (2xy(x) + 1)y′(x) + y(x)2 + 1 = 0
 4.905   − (a+ 2xy (x ))y′(x)+ x2y′(x)2 + y(x)2 = 0
 4.906    2 ′   2               ′         2
x y (x) − x(x − 2y(x))y(x) + y(x) = 0
 4.907           2 ′  2                ′         2
− 4a+ x  y(x) + 2x (y(x )+ 2x)y (x )+ y(x) =  0
 4.908   x (x3 − 2y (x )) y′(x) − (2x3 − y(x))y (x )+ x2y′(x )2 = 0
 4.909   x2y′(x)2 + 3xy (x )y′(x)+ 2y (x )2 = 0
 4.910   x3 + x2y′(x )2 − 3xy(x)y′(x )+ 2y(x)2 = 0
 4.911    2 ′   2         ′          2
x y (x) + 4xy (x )y (x)− 5y (x ) = 0
 4.912    2 ′   2               ′
x y (x) − 4x(y(x) + 2)y(x) + 4y(x)(y(x) + 2) = 0
 4.913    2 ′   2         ′          2
x y (x) − 5xy (x )y (x)+ 6y (x ) = 0
 4.914              (                  )              (        )
x2y′(x )2 + x x2 + xy(x) − 2y(x) y′(x) + (1− x)  x2 − y(x) y (x ) = 0
 4.915   x2y′(x)2 + (y(x)+ 2x)y(x )y ′(x)+ y (x )2 = 0
 4.916   x2y′(x)2 + (2x − y(x))y(x )y ′(x)+ y (x )2 = 0
 4.917   y′(x)(a + bx2y(x)3) + aby(x)3 + x2y ′(x)2 = 0
 4.918   (    2)  ′  2          2
 1− x   y(x) =  1− y(x)
 4.919   (    2)  ′  2     2         ′
 1− x   y(x) + 4x  + 2xy(x)y (x) = 0
 4.920   (      )
 a2 + x2 y′(x)2 = b2
 4.921   (      )
 a2 − x2 y′(x)2 + b2 = 0
 4.922   (      )
 a2 − x2 y′(x)2 = b2
 4.923   (a2 − x2) y′(x)2 = x2
 4.924   (a2 − x2) y′(x)2 + x2 + 2xy(x)y′(x) = 0
 4.925   ( 2   2)  ′  2          ′        2
 a − x   y(x) − 2xy (x)y(x)−  y(x ) = 0
 4.926   ( 2   2)  ′  2             ′         2
 a + x   y(x) + b − 2xy(x)y (x )+ y(x) =  0
 4.927   (  2    ) ′   2  ( 2               2    ) ′          2
 2x  + 1 y (x) +  x  + 2xy(x)+ y(x)  + 2 y (x)+ 2y(x) +  1 = 0
 4.928   4x2y′(x)2 − 4xy (x )y ′(x) = 8x3 − y(x)2
 4.929   ax2y ′(x)2 + (1 − a)ax2 − 2axy (x)y′(x) + y(x)2 = 0
 4.930   (     )
 1− a2 x2y ′(x)2 − a2x2 − 2xy(x)y′(x)+ y(x)2 = 0
 4.931   x3y′(x)2 = a
 4.932   x3y′(x)2 + xy ′(x)− y (x ) = 0
 4.933        3 ′  2    2     ′
a + x y (x ) + x y(x)y (x ) = 0
 4.934    3 ′   2  (  2        ) ′          2
x y (x) −  2x y(x) + 1 y (x)+ xy(x)  = 0
 4.935     (     )          (      )            (         )
x  1− x2  y′(x)2 − 2 1− x2  y(x)y′(x) + x 1 − y(x)2 = 0
 4.936                          (                     )
4x(a − x)(b− x)y′(x)2 = − 2x(a + b)+ ab+ 2x2  2
 4.937   x4y′(x)2 − xy ′(x)− y (x ) = 0
 4.938   x4y′(x)2 + 2x3y(x)y′(x)− 4 = 0
 4.939    4 ′   2       2 ′         3
x y (x) + xy (x )y (x)− y(x)  = 0
 4.940    2 ( 2   2)  ′  2
x   a − x   y(x) +  1 = 0
 4.941   3x4y′(x)2 − xy (x )− y(x) = 0
 4.942   4x5y′(x)2 + 12x4y (x )y′(x)+ 9 = 0
 4.943   x6y′(x)2 − 2xy ′(x)− 4y (x ) = 0
 4.944   x8y′(x)2 + 3xy ′(x)+ 9y (x ) = 0
 4.945   y(x)y′(x)2 = a
 4.946        ′   2   2
y(x)y (x) = a x
 4.947        ′   2   2x
y(x)y (x) = e
 4.948        ′                 ′   2
2axy (x) − ay(x)+ y(x)y (x) = 0
 4.949   − 4a2xy′(x )+ a2y(x)+  y(x )y ′(x)2 = 0
 4.950   axy ′(x)+  by (x )+ y(x)y′(x)2 = 0
 4.951   − (a− 2bx)y′(x)− by(x)+  y(x )y ′(x)2 = 0
 4.952   x3y′(x)− x2y (x )+ y(x)y′(x)2 = 0
 4.953   y(x)y′(x)2 + (x − y(x))y′(x)− x = 0
 4.954        ′   2            ′
y(x)y (x) − (y(x)+ x )y (x)+ y(x) = 0
 4.955        ′   2             ′
y(x)y (x) − (xy(x)+  1)y (x)+ x = 0
 4.956   y(x)y′(x)2 + (x − y(x)2)y′(x )− xy(x) = 0
 4.957   y(x)y′(x)2 + y(x) = a
 4.958   (y(x)+  x)y′(x)2 + 2xy ′(x) − y(x) = 0
 4.959               ′  2           ′
(2x − y(x))y(x) − 2 (1 − x)y (x )− y(x)+ 2 = 0
 4.960         ′   2           ′
2y(x)y (x) + (5−  4x)y(x)+  2y(x) = 0
 4.961     3 ′        2            ′  2
4x y (x)− 4x  y(x )+ 9y(x)y (x ) = 0
 4.962   (1 − ay(x))y ′(x)2 = ay(x)
 4.963   y′(x)2(a0 + b0x + c0y(x))+ y′(x)(a1 + b1x + c1y(x))+ a2 + b2x + c2y(x) = 0
 4.964   (x2 − ay(x)) y′(x)2 − 2xy (x)y′(x) = 0
 4.965   xy (x )y′(x)2 + (y(x)+ x)y′(x)+  1 = 0
 4.966   ( 2      2) ′            ′  2
 x + y(x)  y (x)+ xy (x )y (x) + xy (x ) = 0
 4.967   ( 2      2) ′            ′  2
 x − y(x)  y (x)+ xy (x )y (x) − xy (x ) = 0
 4.968     (         )
−  x2 − y (x )2 y′(x)+  xy(x)y′(x)2 − xy(x) = 0
 4.969   (             )
 a+ x2 − y(x)2 y′(x )+ xy(x)y′(x)2 − xy(x) = 0
 4.970          (              )
− y′(x)  a− bx2 + y(x)2 − bxy(x) + xy(x)y′(x)2 = 0
 4.971   (3x2 − 2y(x)2)y ′(x)+ xy (x)y′(x)2 − 6xy (x) = 0
 4.972   − 2xy(x)y′(x) + x(x−  2y(x))y′(x)2 + y(x)2 − 2xy(x) = 0
 4.973          ′                  ′  2       2
6xy (x )y (x)+ x (x − 2y(x))y (x ) + y(x) − 2xy(x) = 0
 4.974       2 ′   2   2
y(x) y (x) = a
 4.975      2      2 ′   2      2
− a + y(x) y (x) + y(x) =  0
 4.976   y(x)2y′(x)2 − 3xy′(x)+ y(x) = 0
 4.977   − 6x3y′(x )+ 4x2y(x)+  y(x )2y′(x)2 = 0
 4.978   4a2 − 4ay(x)y′(x )− 4ax + y(x)2y′(x)2 + y(x )2 = 0
 4.979   y(x)2y′(x)2 − (x+ 1)y(x)y′(x)+ x = 0
 4.980   x2 + 2xy(x)y′(x )+ y(x)2y′(x)2 = 0
 4.981           2 ′  2          ′         2
a + y(x) y(x)  + 2xy(x)y(x) − y(x) = 0
 4.982      2          ′         2 ′  2       2
− x − 2xy (x)y(x) + y(x)y (x) + 2y(x)  = 0
 4.983   a − x2 − 2xy(x)y′(x) + y(x)2y′(x)2 + 2y (x )2 = 0
 4.984   (a− 1)b + ax2 + 2axy (x)y′(x) + (1− a)y(x)2 + y (x )2y′(x)2 = 0
 4.985   (        )
 1− y(x)2 y′(x)2 = 1
 4.986   (a2 − y(x)2)y′(x)2 = y(x)2
 4.987   (a2 − 2axy(x) + y(x)2)y′(x )2 + 2ay(x)y′(x )+ y(x)2 = 0
 4.988   ( 2 2      2) ′   2  ( 2    ) 2          ′
 a x − y(x)  y (x) +  a  − 1 x  − 2xy(x)y (x ) = 0
 4.989   (        2      2) ′   2           ′               2   2
 (1 − a)x + y(x)  y (x) + 2axy (x )y (x)+ (1 − a)y(x) + x  = 0
 4.990   ((      )          )                       (       )
  1− 4a2 x2 + y(x)2 y′(x)2 − 8a2xy(x)y′(x)+ 1 − 4a2 y(x)2 + x2 = 0
 4.991   ((     )          )                       (      )
  1− a2 x2 + y(x)2 y′(x)2 + 2a2xy(x)y′(x)+ 1 − a2 y(x)2 + x2 = 0
 4.992   (y(x)+  x)2y ′(x)2 = y(x )2
 4.993     ( 2               2) ′              2 ′   2
−  x − xy (x)− 2y(x)  y (x)+ (y(x)+ x )y (x) − (x − y(x))y(x ) = 0
 4.994     2 ′     ( 2            2) ′   2   2            2
2a y (x)+  a  − (x − y(x))  y (x) + a  − (x − y(x)) =  0
 4.995   x2 + 2xy(x)y′(x )+ 2y(x)2y′(x )2 + y(x)2 − 1 = 0
 4.996   − x2 − 2xy (x)y′(x) + 3y(x)2y ′(x)2 + 4y(x)2 = 0
 4.997   3x3 + 2(3x+  1)xy (x)y′(x)+  4y(x )2y′(x)2 = 0
 4.998   (x2 − 4y(x)2)y′(x)2 − 4x2 + 6xy(x)y′(x)+ y(x)2 = 0
 4.999   9y(x)2y′(x)2 − 3xy′(x)+ y(x) = 0
 4.1000             2 ′   2
(2 − 3y(x))y (x) = 4 (1 − y(x))
 4.1001    2(   2)     2      ′     (    2)     2 ′  2      2
a  − x   − 3a xy(x)y(x) +  1− a   y(x)y (x) + y(x)  = 0
 4.1002   (a − b)y(x)2y′(x)2 − ab+ ay(x)2 − bx2 − 2bxy(x)y′(x) = 0
 4.1003   a2y′(x)2(b2 − (cx − ay(x))2) + c2(b2 − (cx − ay(x))2)+ 2ab2cy′(x ) = 0
 4.1004    2        3    ′           2 ′  2
a x + y(x) (− y(x))+ xy (x)y (x) =  0
 4.1005   (    3       3) ′      2            2 ′  2
 a− x  − y(x)  y (x )+ x y(x)+  xy(x) y(x)  = 0
 4.1006      (  2       2)       3 ′          2 ′   2
− x x  − 2y(x)  − 2y(x) y (x)+ xy(x) y (x) = 0
 4.1007   − a+ y (x )3 (− y′(x))+ 2xy(x)2y′(x )2 = 0
 4.1008                   (          )
4x2y (x )2y′(x)2 = x2 + y(x)2 2
 4.1009   − 2x(x3 + 2y(x)2)y(x)y′(x)+ (2x3 + y(x)2)y(x)2 + 4x2y(x)2y′(x)2 = 0
 4.1010        3 ′   2     ′
4y(x) y (x) − 4xy (x)+ y(x) = 0
 4.1011         ( 2      2)  ′  2        ( 2      2)   ( 4   2    2       4) ′
xy (x ) x + y(x)  y (x) − xy (x ) x + y(x)   −  x + x y(x)  + y(x)  y (x ) = 0
 4.1012    (          (          ))        (           (         ) )                   (         )
x a2x + y(x) x2 − y(x)2  y′(x )2 +  2a2xy(x )+  x2 − y (x )2 2 y′(x) + a2y(x)2 − x x2 − y(x)2 y(x) = 0
 4.1013                                    (                       )
x(a2x + y(x)(x2 − y(x)2))y′(x )2 −  2a2xy(x )− (x2 − y (x )2)2 y′(x) + a2y(x)2 − x (x2 − y(x)2)y(x) = 0
 4.1014   (  2      2( 2       2)) ′  2          ′         2
 x  − y(x)  x  + y(x)   y (x ) − 2xy(x)y (x )+ y(x) =  0
 4.1015       (           )           (          (         ))
− 2x x2 + 2y(x)2 y(x)y′(x)+  x4 + y(x)2 x2 − y (x )2 y ′(x)2 + y(x)4 = 0
 4.1016   y(x)5(− y′(x )) + 3xy(x)4y′(x)2 + 1 = 0
 4.1017   − a− 3y (x )5y′(x)+ 9xy (x )4y′(x)2 = 0
 4.1018   9 (1− x2) y(x)4y ′(x)2 + 4x2 + 6xy(x)5y′(x) = 0
 4.1019   y′(x )2 (a2r(x,y (x )) − x2)+ a2r(x,y(x))+  2xy(x)y′(x) − y(x)2 = 0
 4.1020    ′  2 (             2)                      ′         2
y (x )  ar(x,y(x))− x  +  ar(x, y(x )) + 2xy(x)y (x )− y(x) =  0
 4.1021    ′   3
y (x) = a + bx
 4.1022    ′   3     n
y (x) = ax
 4.1023   y′(x)3 − y(x) + x = 0
 4.1024                (                 )
y′(x)3 = f (x ) a+ by(x) + cy(x)2
 4.1025   y′(x)3 = (y(x) − a)2(y(x) − b)2
 4.1026   f (x )(y(x) − a)2(y(x) − b)2 + y′(x )3 = 0
 4.1027   f (x )(y(x) − a)2(y(x) − b)2(y(x) − c)2 + y′(x)3 = 0
 4.1028             ′  3    ′
a − bx+  y(x) + y (x) = 0
 4.1029    ′   3   ′
y (x) + y (x)− y(x) = 0
 4.1030   y′(x)3 + y′(x) = ey(x)
 4.1031   y′(x)3 − 7y′(x)+ 6 = 0
 4.1032   ay(x) + y′(x)3 − xy′(x ) = 0
 4.1033   y′(x)3 + 2xy ′(x)− y (x ) = 0
 4.1034   y′(x)3 − 2xy ′(x)− y (x ) = 0
 4.1035        ′      3    ′  3
− axy (x)+ x  + y(x)  = 0
 4.1036       ′             ′   3
axy (x)−  ay(x)+ y (x) = 0
 4.1037             ′              ′  3
− (a+ bx)y (x)+ by(x)+  y(x) =  0
 4.1038   y′(x)3 − y(x)y′(x)− x = 0
 4.1039   y′(x)3 − (y(x)+ 3 )y′(x)+ x = 0
 4.1040   y′(x)3 − 2y(x)y′(x)+ y(x)2 = 0
 4.1041   − axy(x)y′(x )+ 2ay(x)2 + y′(x)3 = 0
 4.1042   − (x3 + xy (x )+ y(x)2)y′(x )+ y′(x )3 + xy(x)(y(x)+ x) = 0
 4.1043          4 ′      ′  3       5
− xy(x) y(x) + y(x)  − y(x) = 0
 4.1044    ′   3   3x−2y(x)  ′
y (x) + e       (y (x)− 1) = 0
 4.1045   y′(x)3 + (e2x + e3x) e−2y(x)y′(x) − e3x− 2y(x) = 0
 4.1046   y′(x)3 + y′(x)2 − y(x) = 0
 4.1047   y′(x)3 − y′(x)2 + y(x)2 = 0
 4.1048    ′   3   ′   2    ′
y (x) − y (x) + xy (x)− y(x) = 0
 4.1049           ′   2          ′   3
abx − ay (x) + by(x)+ y (x) =  0
 4.1050      ′   2     ′                    ′  3
a0y (x) + a1y (x)+ a2 + a3y(x)+ y (x) =  0
 4.1051   y′(x)3 + xy′(x)2 − y(x) = 0
 4.1052   − x3 − (1 − 3x)xy′(x )+ y′(x )3 + (1− 3x )y′(x)2 − 1 = 0
 4.1053   y′(x)3 − y(x)y′(x)2 + y(x)2 = 0
 4.1054   y′(x)3 + y′(x)2(cos(x)cot(x)− y (x )) − y′(x)(y(x)cos(x)cot(x)+ 1) + y(x) = 0
 4.1055    ′   3  (         2) ′  2         2 ′
y (x) +  2x − y(x)  y (x ) − 2xy(x) y (x ) = 0
 4.1056   ( 2         2      2)  ′        2 ( 2      2)    ′  3  (    2     ) ′   2
 x  + 2xy(x) − y(x)   y(x)− y (x )  x − y(x)   + y(x) −  y (x ) + 2x y (x) = 0
 4.1057                (                 )            (                 )
− x3y (x )3 + x x2 + xy(x) + y(x)2 y(x)y′(x) −  x2 + xy (x )+ y(x)2 y′(x )2 + y′(x )3 = 0
 4.1058   − x3y (x )6 + x(x2 + y(x)4 + xy (x )2) y(x)2y ′(x)− (x2 + y(x)4 + xy (x)2) y′(x)2 + y′(x)3 = 0
 4.1059     ′   3     ′
2y (x) + xy (x)− 2y (x ) = 0
 4.1060     ′   3   ′   2
2y (x) + y (x) − y(x) = 0
 4.1061    4    ′        3         ′  3
x  (− y (x))+ 2x y(x)+ 3y(x)  = 0
 4.1062   4y′(x)3 + 4y′(x) = x
 4.1063   8y′(x)3 + 12y′(x)2 = 27 (y(x) + x)
 4.1064   a + xy′(x)3 − y(x)y′(x )2 = 0
 4.1065   xy′(x)3 + (2x − y(x))y ′(x)2 + (2 − 2y(x))y′(x)− y(x )+ 1 = 0
 4.1066     ( 2          ) ′   2  ( 2              )  ′       ′  3
−  x + y(x )+ x y (x) +  x  + xy(x)+  y(x ) y(x) + xy (x ) − xy(x) = 0
 4.1067      2    ′   3        ′  2
4x  + xy (x) − 2y(x)y (x ) = 0
 4.1068   2xy ′(x)3 − 3y (x )y′(x)2 − x = 0
 4.1069   4xy ′(x)3 − 6y (x )y′(x)2 + 3y(x) − x = 0
 4.1070   8xy ′(x)3 − 12y (x )y ′(x)2 + 9y(x) = 0
 4.1071   x2y′(x)3 − 2xy (x )y ′(x)2 + y(x)2y′(x)+ 1 = 0
 4.1072   bx (a2 − x2) y′(x)2 + (a2 − x2)y ′(x)3 − bx − y′(x ) = 0
 4.1073     ( 5       2)  ′       5        2     ′   2    ′   3      3
x  x + 3y (x )  y(x)−  2x y(x)− 3x y(x)y (x) + xy (x) − y(x)  = 0
 4.1074     3 ′   3    2     ′   2                  ′          3
2x y (x) + 6x y(x)y (x) − (1− 6xy (x))y(x)y (x)+ 2y(x)  = 0
 4.1075                              (           )
8x3y′(x)3 + 12x2y(x)y′(x)2 − 1 − 6xy(x)2 y′(x )+ y(x)3 = 0
 4.1076   x4y′(x)3 − x3y(x)y′(x)2 − x2y(x)2y′(x)+ xy(x)3 = 1
 4.1077   x6y′(x)3 − xy ′(x)− y (x ) = 0
 4.1078   y(x)y′(x)3 − 3xy′(x)+ 3y(x) = 0
 4.1079   2y(x)y′(x)3 − 3xy′(x)+ 2y (x ) = 0
 4.1080         ′   3        ′
2y(x)y (x) + 3y(x)y (x)+ x = 0
 4.1081         ′   3       ′   2     ′
2y(x)y (x) − y(x)y (x) + 2xy (x)− x = 0
 4.1082   (2y(x)+ x )y′(x)3 + 3(y(x) + x)y′(x)2 + (y(x) + 2x)y′(x) = 0
 4.1083   y(x)2y′(x)3 − xy′(x)+ y(x) = 0
 4.1084   y(x)2y′(x)3 + 2xy′(x)− y(x) = 0
 4.1085   4y(x)2y′(x)3 − 2xy′(x)+ y(x) = 0
 4.1086   16y(x)2y′(x)3 + 2xy′(x)− y(x) = 0
 4.1087     ( 2   )  ′      2          3(   ′  2)        2 ′  3
x  x + 1 y (x)− x  y(x )+ y(x)  − y(x)   + xy(x) y(x)  = 0
 4.1088    7    2 ′  3   (     6    3)  ′  2     5   4 ′      4    5
x y(x) y(x)  +  1− 3x y(x)  y (x) + 3x y (x )y (x)− x y (x ) = 0
 4.1089   y(x)3y′(x)3 − xy′(x)+ y(x) = 0
 4.1090   x3 + 3x2y(x)y′(x) + y(x )3y ′(x)3 − (1 − 3x)y(x)2y′(x)2 − y (x )2 = 0
 4.1091   y(x)4y′(x)3 − 6xy′(x)+ 2y(x) = 0
 4.1092    ′   4            3        2
y (x) = (y(x) − a) (y(x) − b)
 4.1093                 3        2    ′  4
f (x )(y(x) − a)(y(x) − b) + y (x ) = 0
 4.1094                 3        3    ′  4
f (x )(y(x) − a)(y(x) − b) + y (x ) = 0
 4.1095   f (x )(y(x) − a)3(y(x) − b)3(y(x) − c)2 + y′(x)4 = 0
 4.1096   y′(x)4 + xy′(x)− 3y (x ) = 0
 4.1097   y′(x)4 − 3(1 − x)y′(x)2 + 3(1 − 2y(x))y′(x) + 3x = 0
 4.1098   − 4x2y(x)y′(x)2 + y′(x)4 + 16xy(x)2y′(x) − 16y(x)3 = 0
 4.1099    ′   4        ′   3       2 ′  2   (        3)  ′         (       3)
y (x) + 4y(x)y (x) + 6y(x) y(x)  −  1− 4y(x)  y (x)− y (x ) 3− y(x)   = 0
 4.1100     ′   4       ′
2y (x) − y(x)y (x)− 2 = 0
 4.1101   12x3 + xy′(x)4 − 2y(x)y′(x)3 = 0
 4.1102   ay′(x)3 + by′(x)2 + y′(x)5 = cy(x)
 4.1103   ay′(x)4 + by′(x)3 + cxy′(x)2 + y′(x)5 = cy(x)
 4.1104   3y′(x)5 − y(x)y′(x)+ 1 = 0
 4.1105   y′(x)6 = (y(x) − a)4(y(x) − b)3
 4.1106                 4        3    ′  6
f (x )(y(x) − a)(y(x) − b) + y (x ) = 0
 4.1107                 5        3    ′  6
f (x )(y(x) − a)(y(x) − b) + y (x ) = 0
 4.1108                 5        4    ′  6
f (x )(y(x) − a)(y(x) − b) + y (x ) = 0
 4.1109   f (x )(y(x) − a)5(y(x) − b)4(y(x) − c)3 + y′(x)6 = 0
 4.1110      (                         )
x2  y′(x)6 + 3y(x)4 + 3y(x)2 + 1 = a2
 4.1111   y′(x)n = axr + by′(x)s
 4.1112   y′(x)n = f(x)n(y(x )− a)n+1(y(x)− b)n−1
 4.1113   y′(x)n = f(x)(y (x )− a)n+1
 4.1114    ′   n                n−1         n−1
y (x) =  f(x)(y (x )− a)   (y(x)− b)
 4.1115               ′  n
f (x )g(x) + y(x)  = 0
 4.1116   f (x,y (x )) + y′(x)n = 0
 4.1117   ay′(x)+ y′(x)n = by(x )
 4.1118   y′(x)n + xy ′(x)− y(x ) = 0
 4.1119   ay′(x)m + y′(x )n = by(x)
 4.1120   Y1 (y(x))y ′(x)n−1 + y′(x)n = 0
 4.1121               ′  n−1    ′  n
X1 (x,y(x))y(x)    + y(x)  = 0
 4.1122    n− 1 ′  n      ′
x    y(x)  − nxy (x)+ y(x) = 0
 4.1123               ′  n              ′  n− 1
X0 (x,y(x))y(x)  + X1(x,y(x))y(x)    = 0
 4.1124    ∘ ------
2  ay ′(x)+  xy′(x) − y(x) = 0
 4.1125             ∘ -----
(x − y(x))  y′(x) = a(y′(x)+ 1)
 4.1126   3xy ′(x)+  2(y(x) + 1)3∕2 − 3y(x) = 0
 4.1127             ---------
ay′(x)+ ∘ y ′(x)2 + 1 = x
 4.1128   ay′(x)+ ∘y--′(x)2 +-1 = y(x)
 4.1129   ∘ -′---2---     ′
  y (x) + 1 = xy (x)
 4.1130           ′          ∘ -′--2----
− ay(x)y(x) − ax +   y(x)  + 1 = 0
 4.1131              ∘ ---------
− xy′(x )2 +   y′(x )2 + 1+  y(x ) = 0
 4.1132   ∘ ------------
  a2 + b2y′(x)2 + xy′(x) − y(x) = 0
 4.1133   √ -------                ∘ -------------------
  ac−  b2 (xy ′(x) − y(x))+   a + 2by′(x) + cy′(x)2 = 0
 4.1134    ∘ ---------
a  y ′(x)2 + 1 + xy′(x)− y(x) = 0
 4.1135   ax ∘y-′(x)2 +-1+ xy′(x)− y(x) = 0
 4.1136           ′              ∘ -′--2----
− ay(x)y(x) − ax + y(x)  y(x)  + 1 = 0
 4.1137       ∘ -′---2---          ′
y(x)  y (x) + 1 = f (y (x )y (x)+ x )
 4.1138   ∘ (ax2-+-y(x)2)(y′(x)2-+-1)− ax − y(x)y′(x) = 0
 4.1139   a 3∘y-′(x)3 +-1+ xy′(x)− y(x) = 0
 4.1140    ′   (     ∘ -′---2---)       ∘ -′---2---
y (x) a + x  y (x) + 1  = y(x)  y (x) + 1
 4.1141   xy ′(x)+ cos (y ′(x)) = y (x )
 4.1142   a cos(y′(x))+  by′(x)+ x = 0
 4.1143   y′(x)+ sin(y′(x )) = x
 4.1144   y′(x)sin (y′(x)) + cos(y′(x)) = y(x)
 4.1145    ′   2     ′
y (x) sin (y(x)) = y(x)
 4.1146    ′   2      ′
y (x) (sin (y(x))+ x ) = y(x)
 4.1147   ( ′  2    )  2          ′
 y(x) + 1  sin  (y (x )− xy (x)) = 1
 4.1148     ∘ ---------       (               )
−   1−  y′(x)2 + y ′(x) cos−1(y′(x))− x + y(x) = 0
 4.1149   (         )(                )
 y′(x)2 + 1  ax + tan −1(y′(x ))  + y′(x) = 0
 4.1150   − y′(x)2 + ey′(x)− y(x) + 1 = 0
 4.1151    ′          ′
y (x)+ log(y (x)) = x
 4.1152         ′          ′
a + xy (x)+ log(y (x )) = 0
 4.1153   a + xy′(x)+ log(y′(x )) = y(x)
 4.1154   a + by(x)+ xy ′(x)+ log (y′(x)) = 0
 4.1155   4xy ′(x)+  log (y′(x))− 2y(x) = 0
 4.1156   a (xy′(x) − y(x))+ log(y′(x )) = 0
 4.1157   a (log (y ′(x))− y′(x))+ y(x)− x = 0
 4.1158    ′              ′
y (x)+ y(x) log (y (x))− xy(x) − y(x)log(y(x)) = 0
 4.1159            ′      ′        ′
− (x+ 1)y (x)+ y (x)log(y (x )) + y(x) = 0
 4.1160         ′             ′
log (xy (x) − y(x)) = y (x)
 4.1161           (∘ ----------      )           ∘ ---------
y′(x )log    a + y′(x)2 + y′(x)  − xy′(x )−   y′(x)2 + 1 + y(x) = 0
 4.1162    ′        ′               ′
y (x)tan(y (x))+ log(cos(y (x ))) = y(x)
 4.1163       ′
f (y(x)) = 0
 4.1164         ′
f (x,y(x)) = 0
 4.1165     (      )
f  xy′(x )2  = y(x)−  2xy′(x)
 4.1166   f (y(x),y′(x)) = 0
 4.1167   f (y′(x))+  xy′(x) = y(x)
 4.1168   f (y′(x))+  xg(y′(x )) = y(x)
 4.1169   f (y′(x),y(x)−  xy′(x)) = 0
 4.1170        ′
f (xy (x ),y(x)) = 0
 4.1171       (         )
xnf  y′(x), y(xx) = 0
 4.1172   f (y(x)y′(x) + x) = y(x)2 (y′(x)2 + 1)
 4.1173         (y′(x)  )
y(x)f  -y(x),x  = 0
 4.1174   f (x,y(x),y′(x)) = 0
 4.1175   y′′(x) = 0
 4.1176    ′′
y (x) = x + sin(x)
 4.1177    ′′
y (x) = c1cos(ax)+  c2 sin(bx)
 4.1178    ′′      x
y (x) = e x
 4.1179   y′′(x) = c1eax + c2e−bx
 4.1180   y′′(x) + y(x) = 0
 4.1181   y′′(x) − y(x) = 0
 4.1182   y′′(x) + y(x) = ax
 4.1183   y′′(x) + y(x) = a cos(bx )
 4.1184    ′′
y (x) + y(x) = 8 cos(x )cos(2x )
 4.1185    ′′
y (x) + y(x) = sec(x)
 4.1186   y′′(x) + y(x) = a sin(bx)
 4.1187   y′′(x) + y(x) = sin(ax)sin(bx)
 4.1188   y′′(x) + y(x) = 4x sin (x )
 4.1189    ′′
y (x) + y(x) = x (cos(x)− xsin(x ))
 4.1190    ′′              2
y (x) + y(x) = tan (x)
 4.1191    ′′            − x
y (x) + y(x) = e
 4.1192                    (     )
y′′(x) + y(x) = ex x2 − 1
 4.1193   y′′(x) + y(x) = ex sin(2x)
 4.1194   y′′(x) + y(x) = e2x cos(x)
 4.1195   y′′(x) − 2y(x) = 0
 4.1196   y′′(x) − 2y(x) = 4ex2x2
 4.1197    ′′
y (x) + 4y(x) = 0
 4.1198    ′′                 2
y (x) + 4y(x) = x sin (x)
 4.1199   y′′(x) + 4y(x) = 2 tan(x)
 4.1200   y′′(x) + 4y(x) = 2 tan(x)
 4.1201   y′′(x) − a2y(x) = x + 1
 4.1202   y′′(x) = ax + by(x)
 4.1203   a2y(x) + y′′(x) = x2 + x + 1
 4.1204    2        ′′
a y(x) + y (x) = cos(bx)
 4.1205    2        ′′
a y(x) + y (x) = cot(ax)
 4.1206    2        ′′
a y(x) + y (x) = sin(bx)
 4.1207   y′′(x) + xy(x) = 0
 4.1208   y(x)(a + bx)+ y′′(x) = 0
 4.1209   (     )
 a+ x2  y(x)+ y′′(x) = 0
 4.1210   (a− x2) y(x)+ y′′(x) = 0
 4.1211   y′′(x) = (a+  x2)y(x)
 4.1212        (    2 2)    ′′
y(x)  a+ b x  +  y (x ) = 0
 4.1213        (          2)   ′′
y(x)  a+ bx + cx  + y (x) = 0
 4.1214        (             )
y(x)  a0+ a1x2 + x4 + y ′′(x ) = 0
 4.1215   axky (x )+ y′′(x) = 0
 4.1216       (                                 )
y(x) a0 + a1x + a2x2 + a3x3 + a4x4 + x8 + y′′(x) = 0
 4.1217   y(x)(a + bcos(2x))+ y′′(x) = 0
 4.1218   y(x)(a + bcos(2x)+ k cos(4x)) + y′′(x) = 0
 4.1219        ∑m                     ′′
y(x) (  n=0a(n)cos(2nx)) + y (x) = 0
 4.1220    ′′             2
y (x) = 2y(x)csc (x)
 4.1221           2       ′′
ay(x) csc (x) + y (x ) = 0
 4.1222        (                        )
y(x)  a0+ a1 cos2(x) + a2csc2(x ) + y′′(x) = 0
 4.1223               (                                   )
y′′(x) = y(x) a2 + (p− 1)pcsc2(x)+ (q − 1)qsec2(x)
 4.1224        (           )
y(x)  a+ bsin2(x) + y′′(x) = 0
 4.1225   y′′(x) = y(x)(2 tan2(x) + 1)
 4.1226   y′′(x) − y(x)(a2 − bex) = 0
 4.1227    ′′     ( 2   2x)
y (x) −  a − e   y(x) = 0
 4.1228        (     x    2x)    ′′
y(x)  a+ be +  ce    + y (x) = 0
 4.1229   aebxy(x)+  y′′(x ) = 0
 4.1230        (            )
y(x)  a+ bcosh2(x) +  y′′(x ) = 0
 4.1231        (            )
y(x)  a+ bsinh2(x) + y′′(x) = 0
 4.1232   y(x) (a+ bsin2(x))+ y′′(x) = 0
 4.1233   (a+b)y(x)-  ′′
  x2   + y (x) = 0
 4.1234    ′′      ′
y (x) − y(x) + xy(x) = 0
 4.1235   y′′(x) + 2y′(x) + y(x) = 0
 4.1236   y′′(x) − 2y′(x) + y(x) = (x − 6)x2
 4.1237   y′′(x) − 2y′(x) + y(x) = ex
 4.1238    ′′       ′            x (  2        )
y (x) − 2y(x) + y(x) = e  3x + 2x + 1
 4.1239    ′′       ′            x
y (x) − 2y(x) + y(x) = e sin(x)
 4.1240    ′′       ′             2    2x
y (x) + 2y(x) + y(x) = x + 3e  − cos(x)
 4.1241   y′′(x) − 2y′(x) + y(x) = 8e3xx2
 4.1242   y′′(x) − 2y′(x) + y(x) = 50 cos(x )cosh(x)
 4.1243   y′′(x) + 2y′(x) + 3y(x) = 0
 4.1244   y′′(x) + 2y′(x) + y(x) = e− xcos(x)
 4.1245   y′′(x) + 2y′(x) + 5y(x) = 0
 4.1246    ′′       ′
y (x) + 2y(x) + 5y(x) = 8sinh (x )
 4.1247             ′        2          ′′
− 2tan(a)y (x )+ csc (a)y(x) + y (x) = 0
 4.1248   − 2tan(a)y′(x )+ csc2(a)y(x) + y′′(x) = x2ex tan(a)
 4.1249   y′′(x) + 3y′(x) + 2y(x) = 0
 4.1250   y′′(x) + 3y′(x) + 2y(x) = cos(ax)
 4.1251   y′′(x) + 3y′(x) + 2y(x) = ex + sin (x )
 4.1252   y′′(x) − 3y′(x) + 2y(x) = x2 + 2e−x
 4.1253    ′′       ′               ax
y (x) − 3y(x) + 2y(x) = xe
 4.1254    ′′       ′
y (x) − 3y(x) − 4y(x) = 0
 4.1255    ′′       ′
y (x) − 3y(x) − 4y(x) = 10cos(2x)
 4.1256   y′′(x) − 4y′(x) + 4y(x) = 0
 4.1257   y′′(x) − 4y′(x) + 4y(x) = e2x cos2(x)
 4.1258   y′′(x) + 4y′(x) + 5y(x) = 0
 4.1259   y′′(x) + 4y′(x) + 5y(x) = sin(x)
 4.1260   y′′(x) − 4y′(x) + 13y(x) = 0
 4.1261    ′′       ′
y (x) − 5y(x) + 6y(x) = 0
 4.1262    ′′       ′               x 2
y (x) − 5y(x) + 6y(x) = 4e x
 4.1263   y′′(x) − 5y′(x) + 6y(x) = eax
 4.1264   y′′(x) + 6y′(x) + 9y(x) = 0
 4.1265   y′′(x) + 6y′(x) + 9y(x) = e− 3xcosh(x)
 4.1266   y′′(x) − 7y′(x) + 12y(x) = 0
 4.1267   y′′(x) − 7y′(x) + 12y(x) = x
 4.1268    ′′       ′
y (x) + 8y(x) + 16y(x) = 0
 4.1269    ′′       ′                x   2x
y (x) + 8y(x) + 16y(x) = 4e − e
 4.1270    ′′       ′
y (x) − 9y(x) + 20y(x) = 0
 4.1271   y′′(x) − 9y′(x) + 20y(x) = e3xx2
 4.1272   2ay ′(x)+ b2y(x) + y′′(x) = 0
 4.1273   2ay ′(x)+ b2y(x) + y′′(x) = c sin(kx)
 4.1274   a2y(x) − 2ay′(x )+ y′′(x) = ex
 4.1275   ( 2   2)2           ′      ′′
 a + b   y(x) − 4aby(x) + y (x) = 0
 4.1276     ′              ′′
ay (x)+ by(x) + y (x) = 0
 4.1277     ′              ′′
ay (x)+ by(x) + y (x) = f (x )
 4.1278   ay′(x)+ y(x)(b+  cx )+ y′′(x) = 0
 4.1279                (      )
ay′(x)+ y(x)  b+ cx2 +  y′′(x ) = 0
 4.1280   ay′(x)+ y(x) (b + cex)+ y′′(x) = 0
 4.1281   be2axy(x)+ ay′(x)+ y′′(x) = 0
 4.1282   ay′(x)+ bekxy(x) + y′′(x ) = 0
 4.1283    ′′       ′
y (x) + xy (x )+ y(x) = 0
 4.1284    ′′       ′
y (x) + xy (x )− y(x) = 0
 4.1285   y′′(x) − xy′(x )+ 2y(x) = 0
 4.1286   ny (x )+ y′′(x)− xy ′(x) = 0
 4.1287   − ay(x)+ y ′′(x) − xy′(x ) = 0
 4.1288    ′′       ′
y (x) − xy (x )− (1− x )y(x) = 0
 4.1289    ′′        ′
y (x) − 2xy (x )+ 6y(x) = 0
 4.1290    ′′        ′
y (x) + 2xy (x )− 8y(x) = 0
 4.1291   2ny (x )+ y′′(x)− 2xy ′(x) = 0
 4.1292     (           )
−  − x2 − x + 1 y(x) + y′′(x )− (2x + 1)y ′(x) = 0
 4.1293     (      )
2  2x2 + 1 y(x)+ y′′(x)+  4xy′(x) = 0
 4.1294   − (3− 4x2 )y(x)+ y′′(x) − 4xy′(x ) = 0
 4.1295     (      2)       ′′        ′      x2
−  3− 4x   y(x)+ y (x) − 4xy (x ) = e
 4.1296    2 2           ′      ′′
a x y(x) − 2axy (x)+ y (x) = 0
 4.1297   axy ′(x)+  by (x )+ y′′(x) = 0
 4.1298   (a + bx)y′(x) + cy(x )+ y′′(x) = 0
 4.1299   (a0 + b0x)y′(x )+ y(x)(a1+ b1x )+ y′′(x) = 0
 4.1300   (a0 + b0x)y′(x )+ y(x)(a1 + b1x + c1x2)+ y′′(x) = 0
 4.1301   − 2a(1 − 2ax2)y (x )− 4axy′(x)+ y′′(x) = 0
 4.1302    2    ′       ′′
x  (− y (x))+ y (x)+ xy (x) = 0
 4.1303    2    ′       ′′
x  (− y (x))+ y (x)+ xy (x) = x
 4.1304    2 ′      ′′
x y (x)+ y (x) − 4xy(x) = 0
 4.1305   x4y′(x)− x3y (x )+ y′′(x) = 0
 4.1306   a(k + 1)xk−1y(x)+ axky ′(x) + y′′(x) = 0
 4.1307   akxk −1y(x)+ axky ′(x)+  y′′(x ) = 0
 4.1308   − axk−1y(x) + axky′(x )+ y′′(x) = 0
 4.1309   axky ′(x)+  bxk −1y(x)+ y′′(x) = 0
 4.1310    ′′           ′
y (x) − cot(x )y (x)+ 2y(x) = 0
 4.1311                  ′′           ′
k(k + 1)y(x)+ y (x) + cot(x)y (x) = 0
 4.1312   y′′(x) + cot(x )y′(x)− y(x) csc2(x) = 0
 4.1313        (                  )
y(x) p (p + 1)− k2 csc2(x)  + y′′(x)+ cot(x)y′(x) = 0
 4.1314       (                          )
y(x) a0 + 4a1sin2(x)− a2 csc2(x) + y ′′(x )+ cot(x)y′(x) = 0
 4.1315   y′′(x) + 2cot(x)y′(x) + 3y(x) = 0
 4.1316   y′′(x) + 2cot(x)y′(x) + 3y(x) = ex csc(x)
 4.1317           ′          (    2   2   )    ′′
a cot(x )y (x)+ y(x)  b+ k cos (x) + y (x) = 0
 4.1318        (    2                        2   )           ′      ′′
y(x) a cot (x )+ bcot(x)csc(x)+ ccsc (x) + k cot(x )y (x)+ y  (x ) = 0
 4.1319   y′′(x) − cot(2x )y′(x)+ 2y (x ) = 0
 4.1320   ay(x) tan2 (x )+ y′′(x)− 2 cot(2x )y′(x) = 0
 4.1321   a cot(bx)y′(x)+ cy(x) + y′′(x ) = 0
 4.1322   (b2 − a2)y(x) + 2acot(ax)y′(x )+ y′′(x) = 0
 4.1323   ay(x) tan2 (x) + y′′(x)− csc(x)y′(x) = 0
           2
 4.1324    ′′      ′
y (x) + y(x)(cot(x)+ csc(x)) = a csc(x) + 1
 4.1325    ′′            ′          (  2      )   2
y (x) − csc(2x)y (x)+ y(x) sin (x)+ 2  csc (x) = 0
 4.1326   ay(x) csc2(x) + y′′(x )+ (cos(x )+ 2)csc(x)y′(x) = 0
 4.1327   y′′(x)− (3 cos(x) + 2)csc(x )y ′(x)− 2y (x )(cos(x)+ 1) sec(x) = 0
 4.1328   y′′(x) − y′(x)(cot(x)− sin(x))+ y(x)sin2(x ) = 0
 4.1329   y′′(x) − sin(x)y′(x )+ y(x)(− cos(x )) = a−  x+ x log (x )
 4.1330   − 2csc(2x)(1− a sin2(x))y ′(x)+ by(x) tan2 (x )+ y′′(x) = 0
 4.1331           2       ′′            ′
ay(x) cos(x) + y (x)+ tan(x)y (x ) = 0
 4.1332           2       ′′            ′
ay(x) cot(x) + y (x)+ tan(x)y (x ) = 0
 4.1333                   2       ′′            ′
− a(a+ 1)y(x) csc (x)+ y  (x )− tan(x)y(x) = 0
 4.1334   y(x) (acos2(x )− sec2(x )) + y′′(x)− tan(x)y′(x) = 0
 4.1335   y′′(x) + 2tan(x)y′(x )− y(x) = 0
 4.1336   y′′(x) + 2tan(x)y′(x )− y(x) = (x + 1)sec(x)
 4.1337    ′′             ′
y (x) + 2tan(x)y (x )+ 3y(x) = 0
 4.1338            ′′             ′
by(x) + y (x)− 2tan(x)y (x) = 0
 4.1339     ( 2   )        ′′             ′
−  a + 1  y(x)+ y (x) − 2tan(x)y(x) = 0
 4.1340     (     )
−  a2 + 1 y(x)+ y′′(x) − 2tan(x)y′(x) = sin (x )
 4.1341   a tan(x)y′(x)+  by (x )+ y′′(x) = 0
 4.1342   y′′(x) − (2ex + 1)y′(x)+ e2xy(x) = 0
 4.1343   y(x)(a0 + 4a1sinh2(x)− a2csch2(x))+ y′′(x) + coth(x )y ′(x) = 0
 4.1344   y(x)(a0 + 4a1cosh2(x)− a2sech2(x))+  y′′(x )+ tanh(x)y′(x ) = 0
 4.1345            ′′               ′
by(x) + y (x)+ 2tanh (x )y (x) = 0
 4.1346             ′             ′′
a tanh(x)y(x) + by(x)+ y (x) = 0
 4.1347   f (x )y′(x)+ y′′(x) = 0
 4.1348   akxk −1y(x)+ 2axky ′(x)+  2y′′(x ) = 0
 4.1349   3y′′(x) − 10y′(x) + 3y(x) = 0
 4.1350   4y′′(x) = (a + x2)y(x)
 4.1351   (4a− x2 + 2)y(x) + 4y′′(x) = 0
 4.1352     ′′       ′
4y (x) − 8y(x) + 3y(x) = 0
 4.1353      ′′
xy  (x )+ y(x) = 0
 4.1354                   ′′
(a + x)y(x)+ xy (x) = 0
 4.1355   xy ′′(x )+ y′(x ) = 0
 4.1356   xy ′′(x )+ y′(x ) = xn
 4.1357   xy ′′(x )+ y′(x )− y(x) = 0
 4.1358   xy ′′(x )+ y′(x )− (x+ 1 )y(x) = 0
 4.1359   4x3y (x )+ xy′′(x)−  y′(x) = 0
 4.1360      2 3         ′′      ′
− a x y(x)+ xy  (x )− y (x ) = 0
 4.1361      (  2    )
x3  ex − k2  y(x)+ xy ′′(x )− y′(x ) = 0
 4.1362   xy ′′(x )+ 2y′(x ) = 0
 4.1363   xy ′′(x )+ 2y′(x ) = 0
 4.1364      ′′       ′              x
xy  (x )+ 2y (x )− xy(x) = e
 4.1365              ′′       ′
axy (x )+ xy (x) + 2y(x) = 0
 4.1366   ax2y (x )+ xy′′(x)+  2y′(x) = 0
 4.1367   ay′(x)+ xy ′′(x ) = 0
 4.1368   (a + 1)y′(x) + xy′′(x)+ y(x) = 0
 4.1369   (1 − a)y′(x) + xy′′(x)+ y(x) = 0
 4.1370   (a + 1)y′(x) + xy′′(x)− y(x) = 0
 4.1371       ′       ′′
2ny (x)+  xy (x)− y(x) = 0
 4.1372     ′               ′′
ay (x)+ by(x) + xy (x) = 0
 4.1373     ′                ′′
ay (x)+ bxy (x )+ xy (x) = 0
 4.1374   ay′(x)+ y(x)(b1 + b2x) + xy′′(x) = 0
 4.1375                (              )
ay′(x)+ y(x)  a1+ b1x + c1x2  + xy′′(x) = 0
 4.1376   ay′(x)+ bxky (x )+ xy′′(x) = 0
 4.1377   xy ′′(x )− (x+ 1 )y′(x)+ y(x) = 0
 4.1378   ny (x )+ xy′′(x) + (1 − x)y′(x ) = 0
 4.1379              ′               ′′
(k − x+  1)y (x)+ ny (x)+ xy (x) = 0
 4.1380      ′′            ′
xy  (x )− (x+ 1 )y (x)+ 2(1 − x)y(x) = 0
 4.1381   xy ′′(x )− (2− x )y′(x)− y(x) = 0
 4.1382   xy ′′(x )− (x+ 3 )y′(x)+ y(x) = 0
 4.1383   xy ′′(x )− (x+ 3 )y′(x)+ 3y(x) = 0
 4.1384   (a + x)y′(x) + by(x)+ xy′′(x) = 0
 4.1385   − ay(x)+ (c − x)y′(x) + xy′′(x) = 0
 4.1386      ′′             ′
xy  (x )+ (1− 2x )y (x)− (1 − x)y(x) = 0
 4.1387      ′′             ′
xy  (x )− (2x+  1)y (x)+ (x + 1)y(x) = 0
 4.1388      ′′             ′                    2
xy  (x )− (2x+  1)y (x)+ (x + 1)y(x) = x − x − 1
 4.1389   (a + bx)y′(x) + cy(x )+ xy′′(x) = 0
 4.1390   (a1 + b1x)y′(x )+ y(x)(a2+ b2x )+ xy′′(x) = 0
 4.1391   − 2(a+ bx)y′(x)+ y(x)(2a + bx)+ xy′′(x) = 0
 4.1392   y′(x )(x(a + b)+ m  + n)+ y(x)(abx + an+  bm ) + xy′′(x) = 0
 4.1393   xy ′′(x )− (1− x2 )y′(x ) = 0
 4.1394     (     2)  ′       ′′
−  4− x   y(x) + xy (x)+ 2xy (x) = 0
 4.1395    3       (  2    ) ′       ′′
x y(x) −  2x + 1  y(x) + xy (x) = 0
 4.1396              (       )
− 8x3y(x)−  2x2 + 1 y ′(x)+ xy ′′(x) = 0
 4.1397              (       )
− 8x3y(x)−  2x2 + 1 y ′(x)+ xy ′′(x) = 4e− x2x3
 4.1398   (4x2 + 1) y′(x) + 4x(x2 + 1)y(x) + xy′′(x) = 0
 4.1399   − (1− 2ax3 )y′(x )+ ax2(ax3 + 1) y(x )+ xy′′(x) = 0
 4.1400               ′                  ′′
(xf (x )+ 2)y(x) + f(x)y(x)+ xy  (x ) = 0
 4.1401           ′′       ′
(1 − x)y (x)+ xy (x)− y(x) = 0
 4.1402           ′′       ′                  2
(1 − x)y (x)+ xy (x)− y(x) = (1− x )
 4.1403   (3 − x)y′′(x)− (9−  4x)y′(x)+  3(2 − x)y(x) = 0
 4.1404   (a − x)y′′(x)− 2y′(x) = 0
 4.1405   (a + x)y′′(x)+ (a1 + b1x)y′(x) + y(x)(a2+ b2x ) = 0
 4.1406   2xy ′′(x )+ y′(x ) = 0
 4.1407   ay(x) + 2xy′′(x)+  y′(x) = 0
 4.1408                ′′      ′
− ay(x)+ 2xy  (x)+ y (x) = 0
 4.1409                    ′′      ′
y(x)(a + bx)+ 2xy (x) + y(x) = 0
 4.1410     (       )
−  2x2 + 1 y′(x) + 2xy′′(x)− xy (x) = 0
 4.1411   (1 − 2x)y′′(x)− (x + 2)y′(x)−  y(x ) = 0
 4.1412   (1 − 2x)y′′(x)− (4−  3x)y′(x) + (3 − x)y(x) = 0
 4.1413   4y′′(x) + 2y′(x) + y(x) = 0
 4.1414   4y′′(x) − 2y′(x) − y(x) = 0
 4.1415       ′′              ′
4xy  (x )+ 4coth(x)y (x)+ y(x) = 0
 4.1416                     ′′       ′
y(x)(a + bx)+ 16xy (x) + 8y (x ) = 0
 4.1417            ′′       ′
(a + bx)y (x)+ cy (x ) = 0
 4.1418   (a0 + b0x)y′′(x)+ (a1 + b1x)y′(x )+ y(x)(a2+ b2x ) = 0
 4.1419   (1 − xcot(x))y′′(x)− xy′(x)+ y(x) = 0
 4.1420   x2y′′(x) = a + bx
 4.1421   x2y′′(x) = 2y(x)
 4.1422   x2y′′(x) = 6y(x)
 4.1423    2 ′′
x y (x) = 12y(x)
 4.1424            2 ′′
ay(x) + x y (x) = 0
 4.1425   y(x)(a + bx)+ x2y′′(x) = 0
 4.1426             (     )
x2y′′(x) −  2− x2  y(x) = 0
 4.1427             (     )
x2y′′(x) −  2− x2  y(x) = x4
 4.1428   x2y′′(x) − (a2x2 + 2) y(x) = 0
 4.1429   x2y′′(x) − (6− a2x2) y(x) = 0
 4.1430    2 ′′         ( 2 2          )
x y (x) − y(x) a x  + n(n+  1) = 0
 4.1431    2 ′′         (           2  2)
x y (x) − y(x) (n − 1)n− a x   = 0
 4.1432        (          2)   2 ′′
y(x)  a+ bx + cx  + x y (x) = 0
 4.1433   x2y′′(x) − y(x)((a−  1)a − bxk) = 0
 4.1434   xky (x )(a+ bxk) + x2y′′(x) = 0
 4.1435   − by(x)(a + bx2 )+ ay′(x)+ x2y′′(x) = 0
 4.1436    2 ′′       ′
x y (x) + xy (x )+ y(x) = 0
 4.1437    2 ′′       ′
x y (x) + xy (x )− y(x) = 0
 4.1438    2 ′′       ′
x y (x) − xy (x )+ y(x) = 0
 4.1439   x2y′′(x) + xy′(x )− y(x) = ax2
 4.1440   x2y′′(x) − xy′(x )+ y(x) = x2(x+ 3)
 4.1441   x2y′′(x) − xy′(x )+ y(x) = 3x3
 4.1442   x2y′′(x) + xy′(x )+ y(x) = log(x)
 4.1443   x2y′′(x) − xy′(x )+ 2y(x) = 0
 4.1444    2 ′′       ′
x y (x) − xy (x )+ 2y(x) = xlog(x)
 4.1445    2 ′′       ′
x y (x) − xy (x )− 3y(x) = 0
 4.1446   a2(− y(x))+ x2y′′(x)+  xy′(x) = 0
 4.1447   y(x)(a + bx)+ x2y′′(x) + xy′(x) = 0
 4.1448     (      )
−  p2 − x2 y(x)+ x2y′′(x) + xy′(x ) = 0
 4.1449   − (p2 + x2) y(x)+ x2y′′(x) + xy′(x ) = 0
 4.1450   − (p2 + ix2)y (x )+ x2y′′(x)+ xy ′(x) = 0
 4.1451         ( 2    2 2)   2 ′′       ′
− y(x) p −  a x  + x y (x) + xy (x ) = 0
 4.1452         (          2)    2 ′′        ′
− y(x) a + bx+ cx   + x y (x)+ xy (x) = 0
 4.1453         (   2    2    4)   2 ′′       ′
− y(x) 4ax  + n −  x  + x y (x) + xy (x ) = 0
 4.1454         (               )
− y(x) a2 + b2x2 + c2x4 + x2y ′′(x )+ xy′(x) = 0
 4.1455   (m + 1 )a(m )xmy (x)+ x2y′′(x) + xy′(x) = 0
 4.1456   (a + x)y′(x) + x2y′′(x)− y(x) = 0
 4.1457   x2y′′(x) − 2xy′(x)+ 2y(x) = 0
 4.1458   x2y′′(x) − 2xy′(x)+ 2y(x) = 4x3
 4.1459    2 ′′        ′              3
x y (x) − 2xy (x)+ 2y(x) = x sin(x)
 4.1460    2 ′′        ′
x y (x) − 2xy (x)+ 2y(x) = 2xlog(x)
 4.1461   x2y′′(x) − 2xy′(x)+ 2y(x) = x5log(x)
 4.1462   x2y′′(x) + 2xy′(x)− 6y(x) = 0
 4.1463   x2y′′(x) + 2xy′(x)− 6y(x) = 2− x
 4.1464   (a2x2 + 2) y(x)+ x2y′′(x) − 2xy′(x) = 0
 4.1465   − y(x)(n(n + 1)− a2x2) + x2y′′(x)+ 2xy ′(x) = 0
 4.1466        (     2)    2 ′′         ′
y(x)  a+ bx   + x y (x)+ 2xy (x) = 0
 4.1467            2 ′′              ′
ay(x) + x y (x)− 2(1 − x)y(x) = 0
 4.1468    2 ′′        ′
x y (x) + 3xy (x)+ y(x) = 0
 4.1469   x2y′′(x) + 3xy′(x)+ y(x) = x
 4.1470   x2y′′(x) + 3xy′(x)+ y(x) = a− x + x log(x)
 4.1471   x2y′′(x) − 3xy′(x)+ 4y(x) = 0
 4.1472   x2y′′(x) − 3xy′(x)+ 4y(x) = 5x
 4.1473   x2y′′(x) − 3xy′(x)− 5y(x) = 0
 4.1474    2 ′′        ′              2
x y (x) − 3xy (x)− 5y(x) = x log(x)
 4.1475    2 ′′        ′
x y (x) + 4xy (x)+ 2y(x) = 0
 4.1476   x2y′′(x) + 4xy′(x)+ 2y(x) = ex
 4.1477   x2y′′(x) + 4xy′(x)+ 2y(x) = log(x + 1)
 4.1478   x2y′′(x) − 4xy′(x)+ 6y(x) = 0
 4.1479   x2y′′(x) − 4xy′(x)+ 6y(x) = x2(x2 − 1)
 4.1480   x2y′′(x) + (2− x2) y(x)+ 4xy ′(x) = 0
 4.1481    2 ′′     ( 2   )           ′
x y (x) +  x + 6  y(x)+ 4xy (x) = 0
 4.1482    2 ′′        ′
x y (x) + 5xy (x)+ 13y(x) = 0
 4.1483   x2y′′(x) − 7xy′(x)+ 16y(x) = 0
 4.1484   y(x) (a(a + 1)+ b2x2) − 2axy′(x)+ x2y′′(x) = 0
 4.1485   a1xy ′(x)+  a2y(x)+ x2y′′(x) = 0
 4.1486        ′                      2 ′′
a1xy (x)+  y(x )(a2 + b2x) + x y (x) = 0
 4.1487        ′         (        2)   2 ′′
a1xy (x)+  y(x ) a2+ b2x   + x y (x) = 0
 4.1488        ′         (             2)    2 ′′
a1xy (x)+  y(x ) a2+ b2x + c2x   + x y (x) = 0
 4.1489                 (      )
axy ′(x)+  y(x ) b+ cx3  + x2y′′(x) = 0
 4.1490                   (         )
axy ′(x)+  x2y(x) a1+  b1x2 + x2y ′′(x) = 0
 4.1491                 (       )
axy ′(x)+  y(x ) b+ cx2k  + x2y′′(x) = 0
 4.1492   (a + bx)y′(x) + cy(x )+ x2y′′(x) = 0
 4.1493   − 2axy′(x)+ a(a + 1)y(x)+ x2y′′(x) = 0
 4.1494         ′                    2 ′′      x a+2
− 2axy (x)+ a(a + 1)y(x)+ x y (x) = e x
 4.1495        (          2 2)       ′      2 ′′
y(x)  a(a + 1)+ b x   − 2axy (x)+ x y (x) = 0
 4.1496   y(x)(a + bx)+ 2axy ′(x)+ x2y ′′(x ) = 0
 4.1497   x2y′′(x) − x2y′(x )− 2x2y(x) = 2x2log(x)+ x + 1
 4.1498        (      )
y(x)  a+ bx2  + x2y′′(x)+ x2y′(x) = 0
 4.1499   x2y′′(x) − (1− x2) y′(x) − y(x) = 0
 4.1500   x2y′′(x) + (1− x)xy ′(x)− (1 − x)y(x) = 0
 4.1501    2 ′′              ′
x y (x) − (x+ 2)xy (x)+ (x + 2)y(x) = 0
 4.1502    2 ′′              ′                   3
x y (x) − (x+ 2)xy (x)+ (x + 2)y(x) = x
 4.1503    2 ′′              ′
x y (x) + (2− x)xy (x)− (3x + 2)y(x) = 0
 4.1504   x2y′′(x) + (x+ 3)xy ′(x)− y (x ) = 0
 4.1505   x2y′′(x) − 2(x+ 1)xy ′(x)+ 2 (x + 1)y(x) = 0
 4.1506   ax2y ′(x)+ x2y ′′(x )− 2y(x) = 0
 4.1507   − x(ax + 5)y′(x)+  (3ax + 5)y(x)+  x2y′′(x) = 0
 4.1508   x(a1 + b1x)y′(x )+ y(x)(a2 + b2x + c2x2)+ x2y′′(x) = 0
 4.1509    3 ′       2 ′′     (     2)
x y (x)+ x y  (x )−  2− x   y(x) = 0
 4.1510    2 ′′     (     2)   ′     ( 2    )
x y (x) +  1− x   xy (x )−  x +  1 y(x) = 0
 4.1511                       (            )
4x3y ′(x)+ x2y ′′(x )+  4x4 + 2x2 + 1 y(x) = 0
 4.1512         (         )       (                )
xy′(x) a0 + b0xk + y (x ) a1+ b1xk + c1x2k  + x2y′′(x) = 0
 4.1513                                 (                                       )
xy′(x )(a0+ a1xr + a2xs)+ y (x ) b0+ b1xr + b2x2r + b3xs + b4x2s + b5xr+s + x2y′′(x) = 0
 4.1514   ay(x) + x2y′′(x)+ 2x2 cot(x )y ′(x) = 0
 4.1515                         2 ′′                     ′
− y(x)(a − x cot(x))+  x y (x)+ x(2x cot(x) + 1)y(x) = 0
 4.1516            2 ′′        2       ′
ay(x) + x y (x)− 2x  tan(x)y(x) = 0
 4.1517                         2 ′′                     ′
− y(x)(a + x tan (x )) + x y (x)+ x(1 − 2xtan(x))y (x) = 0
 4.1518       (    (             )                                  )        (                 )
y(x) f(x) a1 + b1xk − 1 + a2 + b2xk + c2x2k + f′(x) + f(x)2 + xy′(x) a1 + b1xk + 2f(x) + x2y′′(x) = 0
 4.1519   (     )
 x2 + 1 y′′(x )− 2y(x) = 0
 4.1520   a + (x2 + 1)y′′(x)− xy ′(x) = 0
 4.1521   (1− x2) y′′(x )+ xy′(x) = x
 4.1522   ( 2   )  ′′       ′
 x + 1  y (x )− xy (x)+ y(x) = 0
 4.1523   (    2)  ′′       ′
 1− x   y (x )+ xy (x)− y(x) = 0
 4.1524   (     )
 1− x2  y′′(x )− xy′(x)+ y(x) = 0
 4.1525   (     )
 1− x2  y′′(x )− xy′(x)− y(x) = 0
 4.1526   (1− x2) y′′(x )+ xy′(x)− y(x) = x(1 − x2)3∕2
 4.1527   (    2)  ′′       ′
 1− x   y (x )+ xy (x)+ 3y(x) = 0
 4.1528   ( 2   )  ′′       ′
 x + 1  y (x )+ xy (x)− 4y(x) = 0
 4.1529   ( 2   )  ′′       ′
 x + 1  y (x )+ xy (x)− 4y(x) = 0
 4.1530   n2y (x )+ (1− x2 )y′′(x)− xy ′(x) = 0
 4.1531   a2y(x) + (x2 + 1) y′′(x)+ xy′(x) = 0
 4.1532   a2y(x) + (1− x2) y′′(x)− xy′(x) = 0
 4.1533        (     2)   (    2)  ′′       ′
y(x)  a+ bx   +  1− x   y (x )− xy (x) = 0
 4.1534        (∑n         2m )  (     2) ′′       ′
y(x)    m=0 a(m)x    +  1 − x  y (x) − xy(x) = 0
 4.1535   (    2)  ′′        ′
 1− x   y (x )− 2xy (x) = 0
 4.1536       (      )
a +  1− x2  y′′(x)− 2xy ′(x) = 0
 4.1537   (     )
 x2 + 1 y′′(x )+ 2xy′(x)− 2y(x) = 0
 4.1538   (     )
 x2 + 1 y′′(x )− 2xy′(x)+ 2y(x) = 0
 4.1539   (    2)  ′′        ′             (     2)2
 1− x   y (x )+ 2xy (x)− 2y(x) =  1−  x
 4.1540                 (     2)  ′′        ′
n (n + 1)y(x)+  1 − x  y (x) − 2xy (x) = 0
 4.1541                 (      )                  2((−n−1)xP (x)+(n+1 )P   (x))
n (n + 1)y(x)+  1 − x2 y′′(x) − 2xy′(x ) = --------n-x2−-1----n+1----
 4.1542                   (     )
− p(p+ 1)y(x) +  x2 + 1 y′′(x )+ 2xy′(x) = 0
 4.1543   p(p + 1)y(x)+ (1 − x2)y′′(x)− 2xy ′(x) = 0
 4.1544   n (n + 2)y(x)+ (1 − x2)y ′′(x) − 3xy′(x) = 0
 4.1545            (     2) ′′        ′
− ay(x)+  1 − x  y (x) − 3xy (x ) = 0
 4.1546   ( 2   )  ′′        ′
 x + 1  y (x )+ 4xy (x)+ 2y(x) = 2(cos(x )− x)
 4.1547   (     )
 x2 + 1 y′′(x )− 4xy′(x)+ 6y(x) = 0
 4.1548   (     )        (      )
 1− x2  y′′(x )−  x2 + 1 y(x)− 4xy ′(x) = 0
 4.1549   (     )
 1− x2  y′′(x )− 6xy′(x)− 4y(x) = 0
 4.1550   ny (x )(a + b+ n + 1)+  (− x(a + b+ 2)− a + b)y′(x) + (1− x2) y′′(x) = 0
 4.1551                            ′     (     2)  ′′
p(2k + p)y(x)− (2k + 1)xy(x) +  1− x   y (x) = 0
 4.1552                                               ′     (     2)  ′′
ny (x )(a + b+ n + 1)+  (− x(a + b+ 2)− a + b)y(x) +  1− x   y (x) = 0
 4.1553                                            (      )
−  (k − p)(k + p + 1)y(x)− 2(k + 1)xy′(x )+ 1− x2  y′′(x) = 0
 4.1554                             (      )
− 2axy′(x)+ (1−  a)ay (x)+  1 − x2 y′′(x) = 0
 4.1555                         (      )
axy ′(x)−  (2 − a)y(x)+  x2 + 1 y′′(x) = 0
 4.1556   axy ′(x)+  by (x )+ (1− x2 )y′′(x) = 0
 4.1557   axy ′(x)+  y(x )(a0+ b0x + c0x2) + (1− x2 )y′′(x) = 0
 4.1558            ′            (     2) ′′
(a + bx)y(x) + cy(x )+  1−  x  y (x) = 0
 4.1559   ( 2   2)  ′′         ( 2   2  2)     ′
 a − x   y (x )+ y(x) b +  cx   − xy (x) = 0
 4.1560   (      )
 a2 − x2 y′′(x )− 8xy′(x)− 12y(x) = 0
 4.1561   (1 − x)xy′′(x)+ 2y′(x)+ y(x) = 0
 4.1562   (1 − x)xy′′(x)− 2y′(x)+ 2y (x ) = 0
 4.1563   (1 − x)xy′′(x)+ 2y′(x)+ 6y (x ) = 0
 4.1564   (1 − x)xy′′(x)− 2y′(x)+ 6y (x ) = 0
 4.1565            ′′       ′
(1 − x)xy (x)+ 3y (x)+ 2y (x ) = 0
 4.1566            ′′       ′
(1 − x)xy (x)− 3y (x)+ 2y (x ) = 0
 4.1567            ′′       ′               (  3   )
(1 − x)xy (x)− 3y (x)+ 2y (x ) = x 3x + 1
 4.1568   − ay′(x )+ (1− x )xy ′′(x )+ 2y(x) = 0
 4.1569   x(x + 1)y′′(x)+ (1 − x)y′(x) + y(x) = 0
 4.1570   (1 − x)xy′′(x)− (x + 1)y′(x) + y(x) = 0
 4.1571   (1 − x)xy′′(x)− (x + 4)y′(x) + 4y(x) = 0
 4.1572   (1 − x)xy′′(x)+ 2xy ′(x)−  2y(x) = 0
 4.1573            ′′              ′
(1 − x)xy (x)+ (1 − 2x)y(x) + 6y(x) = 0
 4.1574                          ′′              ′
p(p + 1)y(x)+ (1− x )xy  (x)+ (1−  2x)y(x) = 0
 4.1575   (1 − x)xy′′(x)+ (1 − x)y′(x) + 2y(x) = 0
 4.1576   (1 − x)xy′′(x)− 3xy ′(x)−  y(x ) = 0
 4.1577   x(x + 1)y′′(x)+ (3x + 2)y′(x) + y(x) = 0
 4.1578   (1 − x)xy′′(x)+ (1 − 4x)y′(x) − 2y(x) = 0
 4.1579            ′′               ′
(1 − x)xy (x)− 2(2x + 1)y(x) − 2y(x) = 0
 4.1580            ′′               ′
(1 − x)xy (x)− 2(1 − 2x)y(x) − 6y(x) = 0
 4.1581                                         ′             ′′
(p− k)(k + p+ 1)y(x) + (k + 1)(1− 2x )y(x)+ (1 − x)xy (x) = 0
 4.1582   (c− (a + 1)x)y′(x) + n(a+  n)y(x )+ (1− x )xy ′′(x) = 0
 4.1583   (a + bx)y′(x) + cy(x )+ x(x + 1)y′′(x ) = 0
 4.1584   y′(x)(c− x(a + b+ 1))− aby(x) + (1− x)xy′′(x) = 0
 4.1585   − (a− (2 − a)x)y′(x) − ay(x)+ x(x + 1)y′′(x) = 0
 4.1586   − (a− (2 − a)x)y′(x) − ay(x)+ x(x + 1)y′′(x) = 0
 4.1587            ′                     ′′
(a + bx)y(x) + cy(x )+ (1− x )xy  (x ) = 0
 4.1588        ∑n          m              ′             ′′
y(x)(  m=0 a(m )x  )+ a(1− 2x )y(x)+  x(x+ 1)y (x) = 0
 4.1589   (           )
− x2 − x+ 2  y′′(x)+ (1 − x)xy′(x) + x(6x + 7)y (x ) = 0
 4.1590   (2 − x)xy′′(x)+ 2(1 − x)y′(x) + 2y(x) = 0
 4.1591     (     )
−  2− x2  y′(x) + (2− x)xy′′(x) + 2(1− x)y(x) = 0
 4.1592   (1 − x)2y′′(x )− 4(1− x )y ′(x)+ 2y (x ) = 0
 4.1593   (1 − x)2y′′(x )− 4(1− x )y ′(x)+ 2y (x ) = cos(x)
 4.1594          2 ′′              ′
(x + 1) y (x )− 4(x + 1)y (x)+ 6y (x ) = 0
 4.1595          2 ′′              ′
(x + 1) y (x )− 4(x + 1)y (x)+ 6y (x ) = x
 4.1596     (  2        ) ′           2 ′′
−  − x − x + 1 y (x )+ (x+  1)y (x) − (x+ 2)y(x) = 0
 4.1597   (1 − x)2y′′(x )− 2(1− x )2y′(x)+ (1 − x)2y(x) = ex
 4.1598   (           )        (         )
 x2 + 3x + 4 y′′(x) +  x2 + x + 1 y′(x)− (2x + 3)y(x) = 0
 4.1599   (x + 2)2y′′(x )− (x + 2)y ′(x)+ 2y (x ) = 0
 4.1600   (2 − x)2y′′(x )+ (2− x )y ′(x)− 3y (x ) = 0
 4.1601   x(a0 + x)y′′(x)+ (a1 + b1x)y′(x )+ a2y(x) = 0
 4.1602             ′             2 ′′
− 4(a+ x )y (x)+ (a0 + x) y (x)+ 6y(x) = 0
 4.1603      2 ′′       ′             2
2x y  (x )− xy (x)+ y(x) = x
 4.1604   2x2y ′′(x )+ xy′(x)− 3y(x) = 0
 4.1605   2x2y ′′(x )− (2x+ 7 )xy ′(x)+  2(x + 5)y(x) = 0
 4.1606   2x2y ′′(x )− (1− 4x )xy ′(x)−  2(1 − 3x)y(x) = 0
 4.1607   2x2y′′(x) − (1− 4x)xy ′(x)− 2(1 − 3x)y(x) = x3(x+ 1)
 4.1608   (2x2 + 1) y′′(x)+ 3xy′(x)− 3y(x) = 0
 4.1609     2        (     2) ′′       ′
2a y(x) + 2 1 − x  y (x)−  xy(x) = 0
 4.1610             ′′      ′
2x (x + 1)y (x)+ y (x)− 4y (x ) = 0
 4.1611             ′′             ′
2(1 − x)xy (x)+ (x + 1)y(x) − y(x) = 0
 4.1612   2(1 − x)xy′′(x)+ (1 − x)y′(x) + y(x) = 0
 4.1613   2(1 − x)xy′′(x)+ (1 − 2x)y′(x) − 2y(x) = 0
 4.1614   2(1 − x)xy′′(x)+ (1 − 2x)y′(x) + 8y(x) = 0
 4.1615   ay(x) + 2(1− x)xy ′′(x )− (1− 2x )y′(x) = 0
 4.1616   y(x)(a + bx)+ 2(1−  x)xy′′(x)+ (1 − 2x)y′(x) = 0
 4.1617                            ′′              ′
2a(a + 1)y(x)+ 2(1 − x)xy (x)− (3x + 1)y(x) = 0
 4.1618                  ′′              ′
(1 − 2x)(1− x)y (x)+  2(1 − 2x)y (x)+ 4y(x) = 0
 4.1619   (1 − 2x)(1− x)y′′(x)+  2(3 − 4x)y′(x)+ 12y(x) = 0
 4.1620   2(x + 1)2y′′(x)− (x + 1)y ′(x)+ y (x ) = 0
 4.1621   2(x + 1)2y′′(x)− (x + 1)y ′(x)+ y (x ) = x
 4.1622   4x2y ′′(x )+ y(x) = 0
 4.1623   4x2y ′′(x )+ y(x) = √x-
 4.1624        (        2   2    )    2 ′′
y(x)  4kx− 4p  − x  + 1 + 4x y (x) = 0
 4.1625   (  2 2   )         2 ′′
 4a x + 1  y(x)+ 4x y (x) = 0
 4.1626   − (a2 − x )y(x)+ 4x2y ′′(x )+ 4xy′(x) = 0
 4.1627   4x2y ′′(x )− (4x2 + 1)y(x)+ 4xy ′(x) = 4exx3∕2
 4.1628   − ((2n+ 1)2 − 4x2)y(x) + 4x2y′′(x)+ 4xy ′(x) = 0
 4.1629     ( 2 2   )         2 ′′        ′
−  a x + 1  y(x)+ 4x y (x) + 4xy (x) = 0
 4.1630      2 ′′        ′
4x y  (x )− 8xy (x)+ 5y(x) = 0
 4.1631   4x2y ′′(x )− 2(x+ 2 )xy ′(x)+  (x + 3)y(x) = 0
 4.1632              (             )
4x2y′′(x) −  − 4x2 + 4x + 1 y(x) + 4(1 − 2x)xy′(x) = 0
 4.1633                        (       )
4x3y ′(x)+ 4x2y ′′(x )−  3−  2x2 y(x) = 0
 4.1634   4x3y ′(x)+ 4x2y ′′(x )+ (x4 + 2x2 + 1) y(x) = 0
 4.1635   (a+ x4 + 2x2)y(x) + 4x3y′(x )+ 4x2y′′(x) = 0
 4.1636     (    2)  ′′        ′
4  1− x   y (x)− 8xy (x)− y(x) = 0
 4.1637     (  2   )         (    2)  ′′        ′
−  4p + 1  y(x)+ 4 1 − x  y  (x )− 8xy (x) = 0
 4.1638     ( 2   )  ′′      2      ′
4  x + 1  y (x) = x + 4xy (x)
 4.1639                             (      )
4axy ′(x) − a(a+ 2)y(x)+  4 1− x2  y′′(x) = 0
 4.1640   4(1 − x)xy′′(x)+ 2(1 − x)y′(x) + y(x) = 0
 4.1641   y(x)(a + bx)+ 4(1−  x)xy′′(x)+ 2(1 − 2x)y′(x) = 0
 4.1642   y(x)(a + bx + cx2)+ 4(1 − x)xy′′(x)+ 2(1 − 2x)y′(x) = 0
 4.1643   y(x)(∑n    a(m )xm )+ 4(1− x )xy ′′(x )+ 2(1− 2x )y ′(x) = 0
       m=0
 4.1644                                            ′              ′′
− (k − p)(k + p+ 1)y(x) + 2(1 − (3− 2k )x )y (x)+ 4(1 − x)xy (x) = 0
 4.1645                       (       )
2(ax + 1)y′(x) + y(x) b+ k2x  + 4(1 − x)xy′′(x) = 0
 4.1646   (2x + 1)2y′′(x)− 2(2x + 1)y′(x)−  12y(x) = 0
 4.1647   (2x + 1)2y′′(x)− 2(2x + 1)y′(x)−  12y(x) = 3x + 1
 4.1648   (1 − 3x)2y′′(x)− 3(1−  3x)y′(x)−  9y(x) = 0
 4.1649   16x2y ′′(x )+ (4x+  3)y(x) = 0
 4.1650       2 ′′         ′
16x y  (x )+ 32xy (x)− (4x + 5)y(x) = 0
 4.1651   (  2   )  ′′         ′     2
 ax + 1  y (x)+ axy (x)+ b y(x) = 0
 4.1652   (  2   )  ′′        ′
 ax + 1  y (x)+ bxy (x)+ cy(x) = 0
 4.1653   (       )
 1− a2x2  y′′(x )− 2a2xy′(x) = 0
 4.1654   (       )
 1− a2x2  y′′(x )− 2a2xy′(x)+ 2a2y(x) = 0
 4.1655   x(a + bx)y′′(x)+ 2ay ′(x)− 2by(x) = 0
 4.1656   y′′(x)(a0 + b0x + c0x2)+ (a1 + b1x)y′(x)+ 2a2y(x) = 0
 4.1657   a1(a + bx)y′(x) + (a+ bx)2y′′(x)+ a2y (x ) = 0
 4.1658    3 ′′
x y (x) = a + bx
 4.1659    3 ′′       ′
x y (x) + xy (x )− y(x) = 0
 4.1660   x3y′′(x) + xy′(x )− 2y(x) = 0
 4.1661   x3y′′(x) + 2xy′(x)− y(x) = 0
 4.1662   a1xy ′(x)+  y(x )(a2 + b2x) + x3y′′(x) = 0
 4.1663   y(x) (a+ bx + cx2)+ x3y′′(x) + x2y′(x ) = 0
 4.1664   x3y′′(x) + 3x2y′(x )+ xy(x) = 0
 4.1665    3 ′′       2 ′
x y (x) + 3x y (x )+ xy(x) = 1
 4.1666      2 ′                   3 ′′
ax y (x)+ y (x )(b + cx)+ x y (x) = 0
 4.1667   (        2) ′                3 ′′
 a1+ b1x   y(x) + a2xy(x)+  x y (x) = 0
 4.1668   x(a1 + b1x)y′(x)+ a2y(x) + x3y′′(x) = 0
 4.1669   x(a1 + b1x)y′(x)+ y(x)(a2+ b2x )+ x3y′′(x) = 0
 4.1670   (     )
 1− x3  y′′(x )+ 6xy(x) = 0
 4.1671   x (1− x2) y′′(x)− y′(x) = 0
 4.1672   x3 + (1− x2 )xy′′(x)−  y′(x) = 0
 4.1673      3      (     2)  ′′      ′
ax y (x )+  1−  x  xy (x)−  y(x) = 0
 4.1674     (     2)  ′′     ( 2    ) ′
x  1− x   y (x)−  x  + 7 y (x)+ 4xy(x) = 0
 4.1675   x (x2 + 1) y′′(x)− 2 (x2 + 1) y′(x)+ 2xy(x) = 0
 4.1676   x (x2 + 1) y′′(x)− 2 (1− x2) y′(x)− 2xy(x) = 0
 4.1677   x (1− x2) y′′(x)− 2 (1− x2) y′(x)−  2xy(x) = 0
 4.1678   (     2)  ′               ( 2   )  ′′
 a+ bx   y(x)+  cxy (x)+ x  x + 1  y (x ) = 0
 4.1679   (      2) ′                           (     2) ′′
 a + bx  y (x)+ (a − 1)x(a+ b)y(x)+ x  1−  x  y (x) = 0
 4.1680   (     2)  ′                     (     2)  ′′
 a+ bx  y (x)+ 2 (1 − b)xy(x)+ x  1− x   y (x) = 0
 4.1681   (            )                 (      )
 a− (a + 1)x2 y′(x )+ cxy(x)+ x  1−  x2 y′′(x) = 0
 4.1682   (      )                  (     )
 a+ bx2  y′(x)+  cxy (x)+ x  1− x2  y′′(x ) = 0
 4.1683     (     )
x  x2 + 2 y′′(x)− y′(x)− 6xy (x ) = 0
 4.1684   x (2− x2) y′′(x)− (x2 + 4x+  2)y(x)− (− x3 − 3x2 + 2x + 2)y ′(x) = 0
 4.1685     (      2) ′′     (        2)  ′
x  a0+ x   y (x)+  a1 + b1x  y (x)+ a2xy (x) = 0
 4.1686    2        ′′        ′           3
x (x + 1)y (x)+ xy (x)+ (x + 1) (− y(x)) = 0
 4.1687   (1 − x)x2y′′(x)− (x + 1)xy′(x) + y(x) = 0
 4.1688   (x + 1)x2y′′(x)− (2x + 1)xy′(x )+ (2x+ 1 )y(x) = 0
 4.1689   (1 − x)x2y′′(x)+ 2(2 − x)xy′(x )+ 2(x+ 1 )y(x) = 0
 4.1690   (1 − x)x2y′′(x)− (4 − 5x)xy′(x )+ (6− 9x )y(x) = 0
 4.1691   (x + 1)x2y′′(x)+ 2(3x + 2)xy′(x )+ 2(3x+  1)y(x) = 0
 4.1692               ′                             2 ′′
x(a1 + b1x)y (x )+ y(x)(a2+ b2x )+ (1− x )x  y (x ) = 0
 4.1693    2         ′′                 ′
x (a0 + x)y (x)+ x(a1 + b1x)y (x )+ y(x)(a2+ b2x ) = 0
 4.1694          2  ′′
(1 − x) xy (x)− 2y(x) = 0
 4.1695     (     )
x  x2 + 1 y′′(x)+ x(x + 1)y′(x) + y(x) = 0
 4.1696   x(2 − x)2y′′(x)+ 2(2 − x)y′(x) + 2y(x) = 0
 4.1697                             (                                                  )
(1− x)x(a − x)y′′(x)+ y′(x) x2(a0 + a1+ 1) + a0a2− a0 + a1+  x(a2+ a3) − a3+ 1  + a0a1(x−  k)y (x ) = 0
 4.1698                           ′′      ′   (              2)
(a1 − x)(a2 − x)(a3−  x)y (x )+ y (x ) b0+ b1x + b2x   + y(x)(c0+ c1x ) = 0
 4.1699   (     3)  ′′       2 ′
 1− 2x   y (x)+ 6x y (x)− 6xy (x ) = 0
 4.1700   2(1 − x)x2y′′(x)+ (3 − 5x)xy′(x )− (x+ 1 )y(x) = 0
 4.1701   2(2 − x)x2y′′(x)− (4 − x)xy′(x )+ (3− x)y(x) = 0
 4.1702     (       )
x  3x2 + 1 y′′(x)+ 2y′(x)− 6xy (x ) = 0
 4.1703   4(x + 1)x2y′′(x)− 4x2y ′(x)+ (3x + 1)y(x) = 0
 4.1704   x2(a + bx)y′′(x)− 2x(2a + bx)y′(x) + 2y(x)(3a + bx) = 0
 4.1705                           ′′                     ′
y(x)(a+  bx )+ 4(1− x )xy  (x )+ 2(1− 3x )(1 − x)y (x ) = 0
 4.1706                    ′′      ′   (             2)       (        2)
4(1 − x)x(1− ax )y (x) + y(x)  a0+ a1x + a2x   + y(x) b0 + b1x  =  0
 4.1707   a2y(x) + x4y′′(x) = 0
 4.1708             (       )
x4y′′(x) +  1− 2x2  y(x) = 0
 4.1709             (       )
x4y′′(x) −  2x2 + 1 y(x) = 0
 4.1710   (e2∕x − a2)y(x)+  x4y′′(x ) = 0
 4.1711    4 ′′       ′
x y (x) + xy (x )− 2y(x) = 0
 4.1712    4 ′′       2 ′
x y (x) − 2x y (x )+ (2x+  1)y (x ) = 0
 4.1713    4 ′′      3 ′
x y (x) + x y (x )+ y(x) = 0
 4.1714   x4y′′(x) + x3y′(x )− (x+ 1 )y(x) = 0
 4.1715        (            )
y(x)  a+ bx2 + cx4 + x4y′′(x) + x3y′(x) = 0
 4.1716             (     )
x4y′′(x) +  x2 + 1 xy′(x )+ y(x) = 0
 4.1717   x4y′′(x) − (1− x2) xy′(x )+ (1−  x2)y(x) = 0
 4.1718   a2y(x) + x4y′′(x)+ 2x3y ′(x) = 0
 4.1719    4 ′′     (  2    )  ′
x y (x) +  2x + 1  xy (x )− y(x) = 0
 4.1720      2       ′             4 ′′
2x (a + x)y (x )+ by(x)+ x y (x) = 0
 4.1721   (x3 + 1) xy′′(x)− (1 − x3)y′(x)+ x2(− y(x)) = 0
 4.1722   x3 (− y′(x))+ (1− x2)x2y′′(x)− 2y (x ) = 0
 4.1723   (1− x2) x2y′′(x)− (2 − x2)xy′(x)+ (2 − x2)y(x) = 0
 4.1724                   3 ′     (     2) 2 ′′
a(a + 1)y(x)− 2x y (x)+  1 − x  x y (x) = 0
 4.1725   (     )2           (     )
 x2 + 1 y ′′(x )+ 2x x2 + 1 y ′(x)+ y (x ) = 0
 4.1726   (     )            (     )
 x2 + 1 2y ′′(x )+ 2x x2 + 1 y ′(x)+ 4y (x ) = 0
 4.1727    2          ( 2    )2 ′′        (     2) ′
a (− y(x))+  x  + 1  y (x)− 2x  1− x   y(x) = 0
 4.1728         (  2          (     2))  (     2)2 ′′       (     2) ′
− y(x) m  − n (n + 1) 1 − x   +  1 − x   y (x) − 2x 1 − x  y (x) = 0
 4.1729         (            (      ))  (      )           (      )
− y(x) k2 − p(p + 1) 1 − x2  +  1 − x2 2y′′(x) − 2x 1 − x2 y′(x) = 0
 4.1730   − y (x )(a2 − k (1− x2)) + (1− x2)2 y′′(x)− 2x (1− x2) y′(x) = 0
 4.1731       (        2      4)  (     2)2 ′′       (     2) ′
y(x) a0 + a2x + a4x   +  1 − x   y (x)− 2x  1−  x  y (x ) = 0
 4.1732        ∑                (     )            (     )
y(x)(  nm=0 a(m )xm) +  1− x2 2 y′′(x)− 2x  1− x2  y′(x) = 0
 4.1733   ax (1− x2 )y′(x )+ by(x)+ (x2 + 1)2y′′(x) = 0
 4.1734       (    2)  ′         (             2)   (     2)2  ′′
a1x 1 − x  y (x)+ y (x ) a2+ b2x + c2x   +  1− x    y (x) = 0
 4.1735      (      )           (       )
x2  a2 + x2 2 y′′(x )+ x a2 + 2x2 y ′(x)+ b2y(x) = 0
 4.1736   (a2 + x2)2y′′(x)+ 2x (a2 + 2x2)y′(x )− y(x)(a0 + a2x2 + a4x4) = 0
 4.1737   ( 2    2)2 ′′       ( 2    2) ′         (        2      4)
 a  − x   y (x)− 2x  a −  x  y (x )+ y(x) a0 + a2x +  a4x  = 0
 4.1738   (      ) (       )         (         )            (        )
 a2 + x2 2 b2 + x2 y′′(x)+ x a0 + b0x2 y′(x)+ y(x) a1 + b1x2  = 0
 4.1739   (a2 − x2)2(b2 − x2)y′′(x)+ x(a0 + b0x2)y′(x)+ y(x) (a1 + b1x2 + c1x4) = 0
 4.1740   (1− x )x (x + 1)2y′′(x)+ 2(3 − x)x(x+ 1)y′(x)− 2(1 − x)y(x) = 0
 4.1741        (          2)         2 2 ′′
y(x)  a+ bx + cx  + (1 − x) x y (x) = 0
 4.1742          2 2 ′′                     ′
(1 − x) x y (x)+ (1−  2x)(1 − x)xy (x)− y(x) = 0
 4.1743                                (              )
(1 − x)x(a1+ b2x )y ′(x)+ y (x ) a2+ b2x + c2x2  + (1− x)2x2y′′(x) = 0
 4.1744        ∑
y(x)(  nm=0 a(m )xm) + (1− x)2x2y′′(x)+  (1 − 2x)(1− x )xy ′(x) = 0
 4.1745   x2(a − x)2y′′(x)+ by(x) = 0
 4.1746   (a − x)2(b − x)2y′′(x) = k2y (x )
 4.1747   (a− x )(A + 2x )(b − x)y′(x) + (a − x)2(b− x)2y′′(x) + By(x) = 0
 4.1748          4 ′′             3 ′
(a − x) y (x)− 2(a − x) y(x)−  y(x ) = 0
 4.1749                  2 ′′               ′
(1− 2x )(1 − x)x y (x)+ 2(2 − 3x)xy (x)+ 2(3x + 1)y(x ) = 0
 4.1750   (1− 2x)(1 − x)x2y′′(x)+ 2(1 − 2x)(2− x)xy′(x)+ 2(1 − x)y(x) = 0
 4.1751         (      (      ) (     ))    (      )           (      )
− y(x) 4k2 +  4p2 + 1 1 − x2   + 4 1−  x2 2y′′(x)− 8x  1− x2  y′(x) = 0
 4.1752   − y(x)(4k2 + (1− 4p2) (1 − x2)) + 4(1−  x2)2y′′(x)− 8x (1− x2 )y′(x) = 0
 4.1753         (                 2 )    (     2) 2 ′′                      ′
− y(x) a(a+  1)(1 − x)+ b x  + 4 1 − x  x y (x) + 2(1− 3x)(1 − x)xy(x) = 0
 4.1754           4 ′′
(a + bx) y (x )+ y(x) = 0
 4.1755         (           )
y′′(x) a + bx + cx2 2 + Ay (x) = 0
 4.1756   x5y′′(x) + xy′(x )− y(x) = 0
 4.1757   x5y′′(x) − (1− 2x3 )xy′(x )+ (1−  2x3)y(x) = 0
 4.1758     ( 2   2) ( 2   2)  ′′     (        4) ′      3    (        2      4)
x  a − x    b − x  y (x) +  a0+ b0x   y (x )+ x y(x) a1 + b1x  + c1x  =  0
 4.1759   ay(x) + x6y′′(x)− x5y ′(x) = 0
 4.1760   x6y′′(x) + 3x5y′(x )+ y(x) = 0
 4.1761   x3 (a+ 3x2) y′(x) + x6y′′(x)+ y(x) = 0
 4.1762                       ′   (             2)         2      2      2 ′′         (              2)
(a− x)(b−  x)(c − x)y(x)  a1+ b1x + c1x   + (a − x) (b− x )(c− x )y (x) + y(x) a2 + b2x + c2x  = 0
 4.1763      6 ′′     (      2)        (  2   )  3 ′
4x y  (x )+  1− 2x   y(x)+ 4  2x + 1  x y(x) = 0
 4.1764              (             )        (       )
4x6y′′(x) +  8x4 + 10x2 + 1 y (x )− 4 2x2 + 1 x3y′(x) = 0
 4.1765   (      )
 4b− a2 y(x) + 4x6y′′(x)+ 12x5y ′(x) = 0
 4.1766   x2ay′′(x) + ax2a−1y′(x) + (1− a)2y(x) = 0
 4.1767   a2xa− 1y(x )+ xa+1y′′(x) + (1 − 2a)xay′(x) = 0
 4.1768   x2y′′(x)(a0 + b0xk) + xy′(x )(a1+ b1xk )+ y(x)(a2 + b2xk) = 0
 4.1769   (    2   2   ) ′′      2             ′         (         2   )
 1− a  cos (x) y (x) + a sin(x )cos(x )y (x)+ y (x ) a0+ a1 cos(x)  = 0
 4.1770         (      (     )        )
− y(x) 4k2 −  1− p2 sinh2(x) + 4 sinh2(x)y′′(x) + 4sinh (x)cosh(x)y′(x) = 0
 4.1771   y′′(x) = 0
 4.1772   y′′(x) = ay(x)
 4.1773   y′′(x) = 6y(x)2
 4.1774    ′′          2
y (x) = 6y(x) + x
 4.1775    ′′                  2
y (x) = a + bx+ cy(x)
 4.1776    ′′          3
y (x) = 2y(x)
 4.1777   y′′(x) = a + by(x )+ 2y(x)3
 4.1778   y′′(x) = a + 2y(x)3 + xy(x)
 4.1779   y′′(x) = f(x)+  g(x )y (x )+ 2y(x)3
 4.1780   y′′(x) = − 2abxy(x)+ a + 2b2y(x)3
 4.1781   y′′(x) = a0+  a1xy(x)+ a2y(x) + a3y(x)3
 4.1782    ′′                        2         3
y (x) = a0+  a1y(x)+ a2y(x) +  a3y(x)
 4.1783      r   s    ′′
ax y (x ) + y (x) = 0
 4.1784   a sin(y(x))+ y′′(x) = 0
 4.1785   aey(x) + y′′(x) = 0
 4.1786   y′′(x) = f(y(x))
 4.1787   y(x) (f′(x) + 2f(x)2)+ 3f(x)y′(x)+ y′′(x) = 2y(x)3
 4.1788   y′′(x) + y(x)y′(x) = 0
 4.1789    ′′          ′         3
y (x) + y(x)y(x) = y(x)
 4.1790            ′′          ′         3
ay(x) + y (x)+ y(x)y (x) = y (x )
 4.1791    ′′          ′        ′                      3
y (x) + y(x)y(x) = 12f (x)− 12f(x)y(x) + y(x)
 4.1792   2a2y(x) + (3a+ y(x))y′(x )+ ay(x)2 + y ′′(x ) = y(x)3
 4.1793               (             )
y′′(x) = y(x) f′(x)− 2f (x )2  + (3f(x)− y(x))y′(x )+ f(x)y(x)2 + y(x)3
 4.1794   y′′(x) = (3f1(x) − y(x))y′(x)+  f1(x)y(x)2 + f2(x)+ f3(x)y(x)+ y(x)3
 4.1795    ′′      ′                               3           2
y (x) = y (x )(f0(x)y(x)+ f1(x))+ g0(x)y(x) + g1(x)y(x) + g2 (x )y(x) + g3(x)
 4.1796   y′′(x) = y(x)f′(x )+ (f(x)− 2y (x ))y′(x)
 4.1797   y′′(x) = (f(x)− 2y (x ))y′(x)+ f(x)y(x)2 + g(x )
 4.1798   y′′(x) = (f1(x) − 2y(x))y′(x)+  f2(x)y(x)2 + f3(x)
 4.1799   y′′(x) = (f1(x)−  2y(x))y′(x)+ f2(x)y(x)2 + f3(x)y(x)+ f4(x)
 4.1800   y′′(x) = a + 4b2y (x )+ 3by(x)2 + 3y(x)y′(x)
 4.1801    ′′           ′                          3
y (x) + 3y(x)y(x) = f(x)+  g(x )y(x )− y(x)
 4.1802    ′′                   ′             2      3
y (x) = (f(x)− 3y (x ))y (x)+ f(x)y(x) −  y(x )
 4.1803   y′′(x) = a(2y(x)y′(x)+ 1)
 4.1804   a (y(x)2 − 1)y ′(x)+ by(x) + y′′(x) = 0
 4.1805   y′(x)f(x,y(x))+ g(x,y(x))+ y ′′(x ) = 0
 4.1806    ′′     ( 2    ′  )2
y (x) =  x −  y(x)  + 2x
 4.1807    ′′       ′  2                    ′
y (x) + 2y(x) tan(y(x))+  2cot(x)y(x) = 0
 4.1808   y′′(x) = ay′(x)2
 4.1809   y′′(x) = a2 + b2y′(x)2
 4.1810   ay′(x)2 + by(x) + y′′(x) = 0
 4.1811   ay′(x)2 + b sin(y(x))+ y′′(x) = 0
 4.1812   ay′(x)2 + by′(x)+ cy(x)+ y′′(x ) = 0
 4.1813    ′′      x ′  2
y (x) = e y (x )
 4.1814        ′          ′   2   ′′
f (x )y (x)+ g(x)y (x) + y (x) = 0
 4.1815         ′   2           ′′
ay(x)y (x) + by(x)+  y (x ) = 0
 4.1816   f (y(x))y′(x)2 + g(y(x))+ y′′(x) = 0
 4.1817   f (x )y′(x)+ g(y(x))y′(x )2 + y′′(x) = 0
 4.1818   f (y(x))y′(x)+ g(y(x))y′(x)2 + y′′(x ) = 0
 4.1819   f (y(x))y′(x)+ g(y(x))y′(x)2 + h(y(x )) + y′′(x) = 0
 4.1820   y′′(x) + y′(x)3 + y′(x) = 0
 4.1821    ′′             ′  3
y (x) = (a− x )y (x)
 4.1822    ′′     ( 2y(x)   )  ′  3
y (x) +  e    + x  y(x)  = 0
 4.1823   y′′(x) + 4y′(x)3 + 2y′(x) = 0
 4.1824   ay′(x)3 + y′′(x) = 0
 4.1825   y′′(x) = xy′(x)3
 4.1826   y′(x)3(ax + by(x))+ y ′′(x ) = 0
 4.1827         ( ′  2    )2   ′′
ay(x)  y(x) +  1  + y (x) = 0
 4.1828                          k
y′′(x) = a(xy ′(x)− y (x ))
 4.1829   f (x )y′(x)k + g(x)y′(x)+ y′′(x) = 0
 4.1830   y′′(x) = Axay (x)by′(x)c
 4.1831            ∘ ---------
y′′(x) = a  y′(x)2 + 1
 4.1832   y′′(x) = a∘by-(x)2 +-y-′(x)2
 4.1833    ′′      ( ′   2   )3∕2
y (x) = a y (x) + 1
 4.1834              (         )
y′′(x) = ax  y′(x)2 + 1 3∕2
 4.1835    ′′          ( ′   2   )3∕2
y (x) = ay(x) y (x) + 1
 4.1836                (               )3∕2
y′′(x) = ay(x)  (b − y′(x))2 + 1
 4.1837            (         )3∕2
y′′(x) = a y′(x)2 + 1   (b+ cx + y(x))
 4.1838                               ∘ -------------
y′′(x) + y(x)3y′(x) = y(x)y′(x)  4y′(x)+ y(x)4
 4.1839   y′′(x) = f (y′(x))
 4.1840   y′′(x) = f (ax + by(x),y′(x ))
 4.1841                (   y′(x))
y′′(x) = y(x)f  x,y(x)
 4.1842   y′′(x) = xn−2f (x−ny(x),x1−ny ′(x))
 4.1843   2y′′(x) = 12y(x)2 + 1
 4.1844     ′′         (        2)
2y (x) = y(x) a − y(x)
 4.1845     ′′       ′  4
8y (x) + 9y(x)  = 0
 4.1846   axey(x) + xy ′′(x) + y′(x) = 0
 4.1847   xy ′′(x )+ 2y′(x )+ xy(x)5 = 0
 4.1848   xy (x )n + xy′′(x) + 2y′(x) = 0
 4.1849   xmy (x)n + xy′′(x)+ 2y′(x) = 0
 4.1850   axmy (x)n + xy′′(x)+ 2y′(x) = 0
 4.1851   ay′(x)+ bxey(x) + xy′′(x) = 0
 4.1852   (2− ax2) y′(x) + xy′′(x) = 0
 4.1853      ′′               ′
xy  (x ) = (1− y(x ))y (x)
 4.1854      ′′       ′   2   ′
xy  (x )+ xy (x) = y (x)
 4.1855      ′′       ′   2   ′
xy  (x ) = xy (x) + y (x)
 4.1856   xy ′′(x )+ 2xy′(x)2 − 2y′(x) = 0
 4.1857   xy ′′(x ) = x2y′(x)2 − 2y′(x)− y(x)2
 4.1858   ax2y ′(x)2 + xy ′′(x )+ 2y′(x) = b
 4.1859   (axy ′(x) − y(x))2 + xy′′(x) = b
 4.1860   xy ′′(x ) = y′(x)3 + y′(x )
 4.1861      ′′       ′       2k ′   k
xy  (x )+ 2y (x ) = ax y (x)
 4.1862       ′′      ′  3    ′
2xy  (x )+ y (x ) + y(x) = 0
 4.1863   ay(x) (1 − y(x)n)+ x2y ′′(x ) = 0
 4.1864   aey(x)− 1 + x2y′′(x) = 0
 4.1865                               (                    )
(a + 1)xy′(x) + x2y′′(x) = xkf xky(x),ky(x) + xy′(x )
 4.1866   x2y′′(x) + y′(x)2 = 0
 4.1867   x2y′′(x) = (3x − 2y′(x))y′(x)
 4.1868    2 ′′      2 ′  2      ′
x y (x) + x y (x ) + 4xy (x )+ 2 = 0
 4.1869    2 ′′      4 ′   2    2    2
x y (x) = x y (x) − 4x y(x) +  6y(x)
 4.1870        ′         2    2 ′′       2
a (xy(x) − y(x)) + x y (x) = bx
 4.1871   ax4y ′(x)2 + x2y ′′(x )+ 2xy(x) = b
 4.1872   ay(x)y′(x)2 + bx + x2y′′(x) = 0
 4.1873             ∘ -----------------
x2y′′(x) =   ax2y′(x)2 + by(x)2
 4.1874    2 ′′           (xy′(x))
x y (x) = y(x)f   y(x)
 4.1875   (     )
 x2 + 1 y′′(x )+ y′(x )2 + 1 = 0
 4.1876   ay(x)3 + 9x2y′′(x)+ 2y(x) = 0
 4.1877   x3y′′(x) − x2y′(x ) = 3 − x2
 4.1878   x3 (y′′(x)+ y(x)y′(x )− y(x)3)+ 12xy (x)+ 24 = 0
 4.1879    3 ′′          ′         2
x y (x) = a (xy (x)−  y(x ))
 4.1880                                        (                       )
2x3y′′(x) + x2(2xy(x)+ 9)y′(x)+ xy (x ) − 2x2y (x )2 + 3xy(x)+ 12 − 6 = 0
 4.1881               (         )
x4y′′(x) = x  x2 + 2y(x) y ′(x)− 4y (x )2
 4.1882   x4y′′(x) = x2y′(x)(y′(x) + x)− 4y(x)2
 4.1883   x4y′′(x) + (xy′(x) − y(x))3 = 0
 4.1884    a  ′′         b
x y  (x )+ y(x) = 0
 4.1885   (       2)(  ′         2)     (      2)( ′′          ′        3)
 1 − 12x   3y (x)+ y(x)  +  2x 1 − 4x   y (x)+ y (x )y (x)− y(x )  − 48xy(x)+ 24 = 0
 4.1886               (            )(             )    (        )(                       )
axy(x) + b−  kxk−1 − 12x2  3y ′(x)+ y (x )2  + 2 xk − 4x3  y′′(x) + y(x)y′(x)−  y(x )3  = 0
 4.1887     --
√ xy′′(x) = y(x)3∕2
 4.1888    3∕2 ′′      ( y√(x))
x   y (x) = f   x
 4.1889                              (          )
y′′(x) (a + 2bx + cx2)3∕2 = f  √---x-----
                             a+2bx+cx2
 4.1890         ′   ′         2 ′′                  ′
f (x )f (x)y (x )+ f(x) y (x) = g(y(x),f(x)y(x))
 4.1891       2 ′′       ′  (       ′         2           3)         4
f(x) y (x) = y(x) 3f (x )f (x)−  f(x) y(x )+ 3f(x)  −  24f(x)
 4.1892   f(x)2y′′(x) = − af (x)5 + 3f(x)f′(x)− f (x )2y(x) + 3f(x)3
 4.1893                                                                   (                           )
2f(x)2y′′(x) = f(x)y′(x)(3f′(x)− 2f (x )y (x )) + f(x)y(x)2f′(x) + y(x) f(x)f′′(x) − 2f′(x)2 − 2f(x)3 + 2f &
 4.1894   y(x)y′′(x) = a
 4.1895   y(x)y′′(x) = y′(x )2
 4.1896   y(x)y′′(x) + y′(x)2 = 0
 4.1897        ′′      ′  2    2
y(x)y (x) = y (x ) − a
 4.1898        ′′      ′  2    2
y(x)y (x) + y(x)  = a
 4.1899        ′′      ′  2      2
y(x)y (x) + y(x) +  y(x ) = 0
 4.1900   2a2y(x)2 + y(x)y′′(x)+ y′(x )2 = 0
 4.1901   y(x)y′′(x) = a0+ a1y (x )+ y(x)3(a2+ a3y(x))+  y′(x)2
 4.1902   y(x)y′′(x) = a0+ a1y(x) + a2y(x)2 + a3y(x)3 + a4y(x)4 + y′(x )2
 4.1903   y(x)y′′(x) = y′(x )2 + y(x)y′(x)
 4.1904   y(x)y′′(x) = exy(x )(a0+ a1y (x )2) + e2x(a2+ a3y (x )4) + y′(x)2
 4.1905        ′′      ′  2       2
y(x)y (x) = y (x ) + y(x) log(y (x ))
 4.1906        ′′        2    2   ′  2       2
y(x)y (x) = − x y(x ) + y (x ) + y(x) log(y(x ))
 4.1907   y(x)y′′(x) + y′(x)2 = y′(x)
 4.1908   y(x)y′′(x) = y′(x )2 − y′(x)
 4.1909   y(x)y′′(x) = y(x)2(f(x)y(x)+ g′(x))+ y′(x)2 + y′(x )
 4.1910   y(x)y′′(x) = y′(x )2 − 2y′(x)
 4.1911   y(x)y′′(x) + y′(x)2 − xy′(x) + y(x) = 0
 4.1912       ′          ′′      ′  2
axy (x)+  y(x )y  (x )+ y (x ) = 0
 4.1913        ′′            ′          ′      ′  2      3
y(x)y (x) = − y (x )f (x)+ f(x)y(x) + y(x) + y (x )
 4.1914        ′′           ′′          ′             3    ′  2      4
y(x)y (x) = y(x )f  (x )− f(x)y (x)− f(x)y(x) +  y(x) + y (x )
 4.1915   y(x)y′′(x) = − ay (x)y′(x)− by(x)2 + y′(x)2
 4.1916   y(x)y′′(x) = ay(x)y′(x )+ by(x)2 + y′(x)2 + y(x)3
 4.1917   y(x)y′′(x) = y(x)2y′(x) + y′(x)2
 4.1918   y(x)y′′(x) = f(x)y(x)y′(x) + g(x)y(x)2 + y′(x)2
 4.1919        ′′           ( ′         2 ′  )    ′  (               2)   ′   2
y(x)y (x) = − y (x ) f(x) − y(x) g(x) + y(x) f (x )+ g(x)y(x)  + y (x)
 4.1920        ′′       ′  2       2
y(x)y (x) = 2y (x ) + y(x)
 4.1921                (             )
y(x)y′′(x) = 2 y′(x)2 − y(x)2
 4.1922   y(x)y′′(x) = − 3y (x )y ′(x)+ 3y ′(x)2 + y(x)2
 4.1923   y(x)y′′(x) = ay′(x )2
 4.1924   y(x)y′′(x) = ay′(x )2 + b
 4.1925   y(x)y′′(x) = ay′(x )2 + by(x)3
 4.1926        ′′       ′  2                      2        3         2        4
y(x)y (x) = ay (x ) + a0+ a1y (x )+ a2y(x) + a3y (x ) + a3y(x) + a4y(x)
 4.1927        ′′       ′  2         ′         2
y(x)y (x) = ay (x ) + by(x)y(x) + cy(x )
 4.1928   y(x)y′′(x) = a3y(x)a+1 + ay′(x)2 + a1y(x)y′(x)+ a2y(x)2
 4.1929   ay′(x)2 + f (x )y(x)y′(x)+ g(x)y(x)2 + y(x )y ′′(x) = 0
 4.1930   y(x)y′′(x) + y′(x)3 = 0
 4.1931   y(x)y′′(x) + y′(x)3 − y′(x)2 = 0
 4.1932   y(x)y′′(x) = y′(x)2(y′(x )(− sin(y(x))) − y(x)y′(x)cos(y(x))+ 1)
 4.1933              ′′       ′  2
(1 − y(x))y (x) + 2y (x ) = 0
 4.1934              ′′      ′   2
(a + y(x))y  (x ) = y (x)
 4.1935   (a + y(x))y ′′(x )+ y′(x )2 = b
 4.1936   (a + y(x))y ′′(x )+ by′(x)2 = 0
 4.1937   (y(x)+  x)y′′(x )+ y′(x )2 − y′(x ) = 0
 4.1938   (x − y(x))y′′(x )+ 2y′(x )(y′(x) + 1) = 0
 4.1939   (x − y(x))y′′(x ) = (y′(x )+ 1)(y′(x)2 + 1)
 4.1940              ′′         ′
(x − y(x))y (x ) = f (y(x))
 4.1941         ′′      ′  2
2y(x)y (x) = y (x )
 4.1942   2y(x)y′′(x) + y′(x)2 + 1 = 0
 4.1943   2y(x)y′′(x) = a+  y′(x)2
 4.1944   2y(x)y′′(x) = y′(x )2 + 8y(x)3
 4.1945         ′′      ′  2        3       2
2y(x)y (x) = y (x ) + 8y(x) + 4y(x)
 4.1946         ′′      ′  2                  2
2y(x)y (x) = y (x ) + 4(2y(x)+ x)y(x)
 4.1947         ′′         2             ′   2
2y(x)y (x) = y(x) (a+ by(x))+ y (x)
 4.1948   2y(x)y′′(x) = ay(x)3 + y′(x)2 − 2xy (x )2 − 1
 4.1949   2y(x)y′′(x) = y(x)2(ax+ by(x))+  y′(x)2
 4.1950   2y(x)y′′(x) = y′(x )2 + 3y(x)4
 4.1951   2y(x)y′′(x) = − a2 − 4(b− x2)y(x)2 + y′(x)2 + 3y(x)4 + 8xy(x)3
 4.1952         ′′           2 ( ′         2)            ′      ′   2       3
2y(x)y (x) = − 2y (x ) f(x) + f(x)  − 3f (x )y(x)y (x)+ y (x) + 8y(x)
 4.1953         ′′               2       2 ′      ′   2      4
2y(x)y (x) = 2xf(x)y(x) − 4y(x) y (x)+ y (x) − y(x) − 1
 4.1954   2y(x)y′′(x) = 3y′(x )2
 4.1955   2y(x)y′′(x) = 3y′(x )2 + 4y(x)2
 4.1956   2y(x)y′′(x) = f(x)y(x)2 + 3y′(x)2
 4.1957   2y(x)y′′(x) = 6y′(x )2 + (1− 3y (x )2) y(x )2
 4.1958   2y(x)y′′(x) = 6y′(x )2 − y(x)2(ay(x)3 + 1)
 4.1959         ′′      ′  2 ( ′  2    )
2y(x)y (x) = y (x )  y(x) + 1
 4.1960         ′′       ′  2         2
3y(x)y (x) = 2y (x ) + 36y(x)
 4.1961         ′′       ′  2
3y(x)y (x) = 5y (x )
 4.1962   4y(x)y′′(x) = 3y′(x )2 − 4y(x)
 4.1963   4y(x)y′′(x) = 3y′(x )2 + 12y(x)2
 4.1964   4y(x)y′′(x) = ay(x)+ by(x)2 + cy(x)3 + 3y′(x )2
 4.1965   5y(x)y′′(x) = y′(x )2
 4.1966   12y(x)y′′(x) = 15y′(x )2 − 8y(x)3
 4.1967         ′′            ′   2
ay(x)y (x) = (a− 1)y (x)
 4.1968          2     ′′      2          3 ′      2    2    4          2        2 ′                 2 ′   2
a(a+ 2) y(x)y (x) = a (a+ 2)y(x) f (x )− a f(x) y(x) + a(a + 2) f(x)y(x) y(x)+  (a − 1)(a+ 2) y (x)
 4.1969   xy (x )y′′(x) + xy′(x )2 + y(x)y′(x) = 0
 4.1970   xy (x )y′′(x) + xy′(x )2 = y(x)y′(x)
 4.1971   xy (x )y′′(x) = xy′(x)2 − y(x)y′(x)
 4.1972         ′′       (          4)       (          2)     ′  2        ′
xy (x )y (x) = x  a0+ a1y(x)  +  y(x ) a2+ a3y (x )  + xy (x ) − y(x)y(x)
 4.1973         ′′       ′  2         ′
xy (x )y (x) + xy (x ) + 2y(x)y (x ) = 0
 4.1974         ′′       ′  2         ′
xy (x )y (x) + xy (x ) = 3y(x)y (x)
 4.1975   ay(x)y′(x)+ f (x )+ xy(x)y′′(x)+ xy ′(x)2 = 0
 4.1976   xy (x )y′′(x) = ay(x)y′(x)+ xy′(x)2 + xy(x)3
 4.1977   xy (x )y′′(x) = ay(x)y′(x)+ b2xy(x)3 + xy ′(x)2
 4.1978   xy (x )y′′(x) + 2xy′(x)2 + y(x)y′(x ) = 0
 4.1979   xy (x )y′′(x) − 2xy′(x)2 + y(x)y′(x ) = 0
 4.1980         ′′        ′   2       ′
xy (x )y (x) − 2xy (x) − y(x)y (x ) = 0
 4.1981         ′′        ′   2            ′
xy (x )y (x) = 2xy (x) − (y(x)+ 1 )y (x)
 4.1982   ay(x)y′(x)+ xy (x )y ′′(x) + 2xy′(x)2 = 0
 4.1983   ay(x)y′(x)+ xy (x )y ′′(x) − 2xy′(x)2 = 0
 4.1984   xy (x )y′′(x) − 4xy′(x)2 + 4y(x)y′(x ) = 0
 4.1985   ay′(x)(xy′(x)− y(x))+ xy (x)y′′(x ) = 0
 4.1986   x(y(x) + x)y′′(x)+ xy′(x)2 + (x − y(x))y′(x) = y(x)
 4.1987           ′′       ′   2       ′
2xy (x )y  (x ) = xy (x) − y(x)y (x)
 4.1988    2               ′′        ′  2              ′
x +  x(2y(x)+ x)y (x)+  2xy(x)  + 4(y (x )+ x)y (x )+ 2y(x) = 0
 4.1989   x2y(x)y′′(x) + (xy′(x) − y(x))2 = 0
 4.1990   x2y(x)y′′(x) + (xy′(x) − y(x))2 = 3y(x)2
 4.1991   x2y(x)y′′(x) = axy(x)y′(x)+ ay(x)2 + 2x2y′(x)2
 4.1992      2 ′  2         ′          2    2     ′′
ax y (x) + bxy (x )y (x)+ cy(x)  + x y(x)y (x)
 4.1993    2           ′′       2 ′  2               ′               2
x (1 − y(x))y  (x )+ 2x y (x ) − 2x(1−  y(x ))y (x)+ 2(1 − y(x))y(x) = 0
 4.1994                                    2
x2(x − y(x))y′′(x) = (xy′(x)− y(x))
 4.1995   x2(x − y(x))y′′(x)+ (xy′(x)− y(x))2 = 0
 4.1996   x2(x − y(x))y′′(x) = a (xy′(x) − y(x))2
 4.1997   2x2y (x )y′′(x) = x2y′(x)2 − y(x)2
 4.1998   2x2y (x )y′′(x) = x2y′(x)2 + 2xy(x)y′(x)− 4y(x)2
 4.1999    3     ′′      3 ′  2     2     ′           2
x y(x)y (x) + x y(x)  + 6x y(x)y(x) + 3xy(x) =  a
 4.2000                                        (     )
x(x + 1)2y(x)y′′(x) = a(x + 2)y(x)2 − 2 x2 + 1 y(x )y ′(x)+ x (x + 1)2y′(x )2
 4.2001     (     )             (      )
8  1− x3  y(x )y′′(x )+ 4 1 − x3 y′(x)2 − 12x2y(x)y′(x)+ 3xy (x )2 = 0
 4.2002   √a2--+-x2(by′(x )2 + y(x)y′′(x)) = y(x)y′(x)
 4.2003   √ -2----2(      ′′       ′  2        ′  )      ′  2
  a − x   xy(x)y (x)− xy (x) −  y(x )y (x) =  bxy(x)
 4.2004             ′′          ′   2            ′             2
f0(x )y (x )y (x) + f1 (x )y (x) + f2(x)y(x)y(x) + f3(x)y(x) =  0
 4.2005                        (                  )
4f(x)y(x)y′′(x) = y′(x) 2f ′(x)− 6f (x)y(x )2  + 2y(x)3f′(x )+ 3f(x)g(x)y(x)2 + 3f (x )y′(x)2 − f(x)y(x)4 + 4f(x�
 4.2006   y(x)2y′′(x) = a
 4.2007   ax + y(x)2y′′(x)+ y(x)y′(x)2 = 0
 4.2008   y(x)2y′′(x)+  y(x )y ′(x)2 = a+ bx
 4.2009   (    2   ) ′′                ′  2
 y(x) + 1 y (x) + (1− 2y(x))y (x ) = 0
 4.2010   (    2   ) ′′           ′  2
 y(x) + 1 y (x) = 3y(x)y (x )
 4.2011   (    2   ) ′′                ′   2
 y(x) + 1 y (x) = (a+ 3y (x ))y (x)
 4.2012   (        )             (        )
 y(x)2 + 1 y′′(x) + y′(x) y ′(x)2 + 1 = 0
 4.2013   (        )
 y(x)2 + x y ′′(x )+ 2y(x)y′(x)2 + 2y′(x) = a
 4.2014   (         )         (         )
 y(x)2 + x y′′(x) = 2 x − y(x)2 y′(x )3 − y′(x )(4y(x )y ′(x)+ 1)
 4.2015   (x2 + y(x)2)y′′(x) = (y(x)2 + 1) (xy′(x) − y(x))
 4.2016   (x2 + y(x)2)y′′(x) = 2(y(x)2 + 1)(xy′(x)− y(x))
 4.2017                  ′′                ′   2
2(1 − y(x))y (x )y (x) = (1− 2y (x ))y (x)
 4.2018                  ′′                        ′                ′  2
2(1− y(x))y(x)y (x) = f(x)(1 − y(x))y(x)y(x) + (1− 2y(x))y (x )
 4.2019   2(1 − y(x))y (x )y′′(x) = (1− 3y (x ))y′(x)2
 4.2020   2(1 − y(x))y (x )y′′(x) = 4y(x)y′(x )(f(x) + g(x)y(x))+ (1− 3y (x ))y′(x)2
 4.2021   2(1− y(x))y(x)y′′(x) = − 4y(x)2(1− y (x )) (f ′(x) + f(x)2 + g′(x)− g(x)2)− 4y (x )y′(x)(f(x)+ g(x)y(x&
 4.2022                  ′′                 ′   2
3(1 − y(x))y (x )y (x) = 2(1− 2y (x ))y (x)
 4.2023                  ′′                 ′   2
4(1 − y(x))y (x )y (x) = 3(1− 2y (x ))y (x)
 4.2024        2 ′′
xy (x )y (x) = a
 4.2025                 (        )
xy (x )2y′′(x) =  a−  y(x )2  y′(x) + xy(x)y′(x)2
 4.2026                  (         )
x2y(x)2y′′(x) =  x2 + y (x )2 (xy ′(x) − y(x))
 4.2027   (a2 − x2) (a2 − y(x)2)y′′(x)+ (a2 − x2)y (x )y′(x)2 = x(a2 − y(x )2) y′(x)
 4.2028   (1− y(x))3(a0 + a1y(x)2)+ a2x(1 − y(x))y(x )2 + a3x3y(x)2(y(x)+ 1) + 2x2(1− y(x))y(x)y′′(x)− x2(1 − 3y(x))y ′(x&
 4.2029    3    2 ′′                 ′          3
x y(x) y (x) + (y (x )+ x)(xy (x)− y(x)) =  0
 4.2030   y(x)3y′′(x) = a2
 4.2031   y(x) (y(x )2 + 1)y′′(x) + (1− 3y(x)2)y′(x)2 = 0
 4.2032   2y(x)3y′′(x) + y(x )2y ′(x)2 = 2
 4.2033                           ′′                                                          ′   2        2 (       2)          2        2         2              2         2             2    2
2(1− y(x))y(x)(a− y(x))y (x)+  (− (1− y(x))(a−  y(x )) + y(x)(a− y(x))+ (1 − y(x))y(x))y (x) = a0y (x &#
 4.2034                                ′′      ′  2                                                                           2         2         2              2         2              2         2             2         2
2(a− y(x))(b− y(x))(c− y(x))y (x)+ y (x) ((a − y(x))(b− y(x))+ (a − y(x))(c − y(x))+ (b− y(x))�
 4.2035                                                    (                 )                   (                          )        (         )
2(1− x)x(1 − y(x))(x − y(x))y(x)y′′(x) = 2(1 − y(x)) x2 − 2xy(x)+ y(x) y (x )y′(x)+ (1 − x)x 3y(x)2 − 2xy(x)&#x
 4.2036   2(1− x)x(1 − y(x))(x − y(x))y(x)y′′(x) = f (x)((1 − y(x))(x − y(x))y(x))3∕2 + 2(1 − y(x))(x2 − 2xy(x&#x
 4.2037          2 2                        ′′                  2         2                    2    2                     2         2              2    2         2               ( 2               )          2 ′           2 2(     2                    )  ′  2
2(1− x) x (1 − y(x))(x − y(x))y(x)y (x ) = a0x (x − y(x)) (1− y(x)) + a1(1 − x)(x− y(x)) y(x) + (a2 − 1)(1 − x
 4.2038    ∘ -----------------------       (        ) (          )            (                  )
b  (1− y(x)2)(1 − a2y(x)2)y′(x)2 + 1 − y(x)2  1− a2y(x)2 y ′′(x )+ y(x) − 2a2y(x)2 + a2 + 1 = 0
 4.2039            (         )
a2y(x) +  x2 + y(x)2 2y′′(x) = 0
 4.2040   y′′(x) (a + 2bx + cx2 + y(x)2)2 + Ay(x) = 0
 4.2041           ′′             ′  2           ′
f0(y(x ))y (x) + f1(y(x))y (x ) + f2(y(x))y (x )+ f3(y (x )) = 0
 4.2042   ∘ ---- ′′
  y(x)y (x) = a
 4.2043   ∘ ----
  y(x)y′′(x) = 2(a+ bx )
 4.2044   y′′(x)X (x,y(x))3 = 1
 4.2045        (                )         (                )
y′′(x) a0 + a1sin2 (y(x)) +  a2y(x) a1sin2(y(x )) + a3 + a1y ′(x)2 + a1y ′(x)2sin(y(x )) cos(y(x)) = 0
 4.2046   y(x)y′′(x)(1− log(y(x)))+ y′(x)2(log(y(x))+ 1) = 0
 4.2047           ′′      ′   2 ′                   ′                2
f(y(x))y (x) = y (x) f (y(x)) − g(x )f (y(x ))y (x)− h(x)f (y(x))
 4.2048                            (    y′(x) )
f(y(x))y′′(x) = f(y(x))2F0  x, f(y(x))- + y′(x )2f ′(y(x))
 4.2049   ay′(x)2f′(y(x)) + f(y(x))y′′(x) + g(y(x)) = 0
 4.2050   y′(x)y′′(x) = a2x
 4.2051    ′    ′′       2     ′          2
y (x)y (x) = x y(x )y (x)+ xy (x)
 4.2052   (  2 ′         2)  ′′                ′   2    ′
 2x y (x)+ y(x)  y  (x )+ 2(y(x)+ x)y (x) + xy (x)+ y(x) = 0
 4.2053   ay(x)2 + x3y′(x)y′′(x) = 0
 4.2054   f1y′(x)y′′(x)+ f2y(x)y′′(x)+  f3y′(x)2 + f4y(x)y′(x) + f5y(x)2 = 0
 4.2055   3y(x)y′(x)y′′(x) = y′(x)3 − 1
 4.2056   (x2 + 2y(x)2y′(x )) y′′(x)+ 2y(x)y′(x )3 + 3xy′(x)+ y(x) = 0
 4.2057   (x− y′(x)2)y′′(x) = x2 − y′(x)
 4.2058   ( ′  2      2)  ′′         3
 y(x) + y (x )  y (x )+ y(x) =  0
 4.2059    ′′   (    ′             ′   2)
y (x) a (xy (x )− y(x))+ y (x)  = b
 4.2060         ′   2 ′′      ′   4
4y(x)y (x) y (x ) = y (x) + 3
 4.2061   y′′(x)f (y′(x))+ g(y(x))y′(x )+ h(x) = 0
 4.2062   y′′(x)2 = a+  by (x )
 4.2063   y′′(x)2 = a+  by ′(x)2
 4.2064   y′′(x)2 − xy′′(x)+ y′(x) = 0
 4.2065    2 ′′  2   ( ′  2    )3
a y (x)  =  y(x)  + 1
 4.2066          ′′  2     ′   ′′
ax + xy (x)  − 2y(x)y (x) = 0
 4.2067                 2
(xy ′′(x)− y′(x)) = y′′(x)2 + 1
 4.2068    (      )
2 x2 + 1 y′′(x)2 + 2(x− y ′(x))y′(x)− x (4y ′(x)+ x )y′′(x) = 2y (x )
 4.2069   3x2y ′′(x )2 + 4y′(x)2 − 2(3xy′(x)+ y(x))y′′(x) = 0
 4.2070   (2− 9x )x2y ′′(x )2 + 6y(x)y′′(x)− 6(1 − 6x)xy′(x)y′′(x) = 36xy′(x)2
 4.2071   f0y′′(x)2 + f1y′(x )y′′(x) + f2y (x )y′′(x) + g0y′(x )2 + g1y(x)y′(x) + hy(x)2 = 0
 4.2072   y(x)y′′(x) + 4y(x)y′(x)3 − y ′(x)2 = 0
 4.2073   (     ′′      ′  2    )2   ( ′  2   )3
 y(x)y (x )+ y (x ) + 1  =  y (x) + 1
 4.2074         (            )        (          )
y′′(x)2 a2 − b2y(x)2 + y′(x )2  1− b2y′(x)2 + 2b2y(x)y′(x)2y′′(x) = 0
 4.2075   (x2y(x)y′′(x) + x2(− y′(x)2)+  y(x )2)2 = 4xy(x) (xy ′(x)−  y(x ))3
 4.2076   y′′(x)3 = 12y′(x )(xy′′(x)− 2y′(x))
 4.2077      ′′      ′′      ′   3   (      ′′      ′  2)3
32y (x) (xy  (x )− y (x ))  +  2y(x)y (x )− y (x )   = 0
 4.2078       ′′        ′′      ′
f (y (x))+ xy (x) = y(x)
 4.2079         (  ′′  )
y′(x)f  yy′((xx))  = y′(x )2 − y(x)y′′(x)
 4.2080     ( ′′     ′       ′′    1 2 ′′        ′        )
f  y (x),y(x)−  xy (x),2x y (x)− xy (x)+  y(x ) = 0
 4.2081         ′′
f (x,y (x)) = 0
 4.2082   f (y(x),y′′(x )) = 0
 4.2083   f (y′(x),y′′(x)) = 0
 4.2084   f (x,y′(x),y′′(x)) = 0
 4.2085   f (y(x),y′(x),y′′(x)) = 0
 4.2086   y′′′(x) = 0
 4.2087    ′′′
y  (x) = cos(x)+ 1
 4.2088    ′′′
y  (x)+ sin(x) = 0
 4.2089    ′′′        3
y  (x) = sin (x)
 4.2090   y′′′(x) = y(x)
 4.2091   y′′′(x) = x2 + y(x)
 4.2092   y′′′(x) = y(x) + exx+ cos2(x)
 4.2093   y′′′(x)+ ay(x) = 0
 4.2094   y′′′(x) = xy (x )
 4.2095    ′′′      ′
y  (x)+ y (x) = 0
 4.2096    ′′′      ′
y  (x) = y (x)
 4.2097   y′′′(x)+ y′(x) = x3 + cos(x)
 4.2098   y′′′(x)− 2y′(x)+ 4y(x) = 0
 4.2099   y′′′(x)− 2y′(x)+ 4y(x) = excos(x)
 4.2100   y′′′(x)− 3y′(x)+ 2y(x) = 0
 4.2101   y′′′(x)− 3y′(x)+ 2y(x) = 3ex
 4.2102    ′′′       ′              x 2
y  (x)− 3y (x)+ 2y(x) = e x
 4.2103    ′′′       ′       2    2x
y  (x)− 4y (x) = x − 3e
 4.2104    ′′′       ′
y  (x)− 7y (x)+ 6y(x) = 0
 4.2105   y′′′(x) = a2y (x )
 4.2106   y′′′(x)+ 2xy′(x)+ y(x) = 0
 4.2107   y′′′(x)+ 2axy ′(x)+ ay (x) = 0
 4.2108   y′′′(x)+ y(x)f′(x)+ 2f (x )y′(x) = 0
 4.2109   y′′′(x)− y′′(x) + y′(x) = 0
 4.2110    ′′′      ′′      ′
y  (x)− y (x) + y(x) + y(x) = 0
 4.2111    ′′′      ′′      ′
y  (x)+ y (x) + y(x) − 3y(x) = 0
 4.2112   y′′′(x)− y′′(x) − 2y′(x) = 0
 4.2113   y′′′(x)− y′′(x) − 2y′(x) = e−x
 4.2114   y′′′(x)+ y′′(x) + 4y′(x) + 4y(x) = 0
 4.2115   y′′′(x)+ y′′(x) + 2y′(x) + 4y(x) = sin(2x)
 4.2116   y′′′(x)+ y′′(x) − 7y′(x) − 15y(x) = 0
 4.2117    ′′′       ′′      ′
y  (x)+ 2y (x) + y(x) = 0
 4.2118   y′′′(x)+ 2y′′(x) + y′(x) = (x− 1)x
 4.2119   y′′′(x)− 2y′′(x) + y′(x) = ex
 4.2120   y′′′(x)− 2y′′(x) − y′(x) + 2y(x) = sinh(x)
 4.2121    ′′′       ′′       ′
y  (x)− 2y (x) − 3y(x) = 0
 4.2122    ′′′       ′′       ′       2
y  (x)− 2y (x) − 3y(x) = 3x  + sin(x)
 4.2123    ′′′       ′′       ′       2    −x
y  (x)− 2y (x) − 3y(x) = 3x  + e
 4.2124   y′′′(x)− 2y′′(x) + 3y′(x) + 10y(x) = 0
 4.2125   y′′′(x)− a2y′(x)+ 2a2y(x) − 2y′′(x) = 0
 4.2126   y′′′(x)− a2y′(x)+ 2a2y(x) − 2y′′(x) = sinh(x)
 4.2127   y′′′(x)− 3y′′(x) + 4y(x) = 0
 4.2128   y′′′(x)+ 3y′′(x) − y′(x) − 3y(x) = 0
 4.2129    ′′′       ′′      ′              2
y  (x)− 3y (x) − y(x) + 3y(x) = x
 4.2130    ′′′       ′′      ′
y  (x)+ 3y (x) − y(x) − 3y(x) = cosh(x)
 4.2131   y′′′(x)+ 3y′′(x) + 3y′(x) + y(x) = 0
 4.2132   y′′′(x)+ 3y′′(x) + 3y′(x) + y(x) = e− xx
 4.2133                                    (        )
y′′′(x)− 3y′′(x) + 3y′(x) − y(x) = x 1− exx2
 4.2134   y′′′(x)+ 3y′′(x) + 3y′(x) + y(x) = e− x(2− x2)
 4.2135   y′′′(x)− 3y′′(x) + 4y′(x) − 2y(x) = 0
 4.2136    ′′′       ′′       ′             x
y  (x)− 3y (x) + 4y(x) − 2y(x) = e + cos(x)
 4.2137    ′′′       ′′       ′
y  (x)− 4y (x) + 5y(x) − 2y(x) = 0
 4.2138    ′′′       ′′       ′
y  (x)− 4y (x) + 5y(x) − 2y(x) = x
 4.2139   y′′′(x)− 4y′′(x) + 6y′(x) − 4y(x) = 0
 4.2140   y′′′(x)− 6y′′(x) + 9y′(x) = 0
 4.2141   y′′′(x)− 6y′′(x) + 12y′(x) − 8y(x) = e2xx2
 4.2142   y′′′(x)+ a3(− y(x))+ 3a2y′(x )− 3ay′′(x) = 0
 4.2143   y′′′(x)+ a3(− y(x))+ 3a2y′(x )− 3ay′′(x) = eax
 4.2144    ′′′        ′′
y  (x) = ay (x )
 4.2145    ′′′        ′′        ′
y  (x)+ a1y (x) + a2y(x) + a3y(x) = 0
 4.2146            (             )
y′′′(x )− 2 − 2a − 4x2 + 1 y′(x) − 8axy(x)−  6xy′′(x) = 0
 4.2147   y′′′(x)+ a3x3y(x) + 3a2x2y′(x )+ 3axy′′(x) = 0
 4.2148   y′′′(x)+ x2 (− y′′(x))+ 2xy ′(x)−  2y(x ) = 0
 4.2149   y′′′(x)+ y′′(x)(2cot(x)+ csc(x))− y′(x) = cot(x)
 4.2150   y′′′(x )− sin (x )y′′(x) − 2cos(x)y′(x) + y(x)sin (x ) = log(x)
 4.2151    ′′′      ′  (   ′  2    ′          )                     ′            ′′
y (x) + y(x) 2f (x) +  f(x) + 4g(x) + 2y(x)(2f (x )g(x) + g(x))+  3f(x)y (x) = 0
 4.2152    ′′′          ′′                ′
y  (x)+ f(x)y (x) + f(x)y(x)+ y (x) = 0
 4.2153   4y′′′(x)− 3y′(x)+ y(x) = 0
 4.2154   4y′′′(x)− 8y′′(x) − 11y′(x) − 3y(x) = 0
 4.2155   4y′′′(x)− 8y′′(x) − 11y′(x) − 3y(x)+ 18ex = 0
 4.2156   xy ′′′(x) = 2
 4.2157   xy ′′′(x)+ 3y′(x)+ xy (x ) = 0
 4.2158      ′′′      ′′       ′
xy  (x)− y (x) + xy (x )− y(x) = 0
 4.2159      ′′′      ′′       ′
xy  (x)− y (x) − xy (x )+ y(x) = 0
 4.2160   xy ′′′(x)− y′′(x) − xy′(x )+ y(x) = 1− x2
 4.2161   xy ′′′(x)+ x2(− y(x))+ 3y′′(x) = 0
 4.2162            (      )
xy ′′′(x)−  3 − x2 y′′(x) + 4xy′(x )+ 2y(x) = 0
 4.2163   (1 − 2x)y′′′(x) − (x + 4)y′′(x)− 2y ′(x) = 0
 4.2164   x2y′′′(x)+ ax2y (x )− 6y′(x ) = 0
 4.2165    2 ′′′         ′′
x y  (x)+ 2xy  (x ) = a
 4.2166    2 ′′′     ( 2    ) ′         ′′
x y  (x)+  x  + 2 y (x)+ 4xy  (x )+ 3xy(x) = f(x)
 4.2167   x2y′′′(x)+ 5xy ′′(x )+ 4y′(x ) = log(x)
 4.2168   x2y′′′(x)+ 6xy ′′(x )+ 6y′(x ) = 0
 4.2169   x2y′′′(x)+ ax2y (x )+ 6xy′′(x) + 6y′(x) = 0
 4.2170    2 ′′′               ′′        ′
x y  (x)− 2(n + 1)xy (x)+ 6ny (x) = 0
 4.2171    2 ′′′     (      3) ′       3       (    2)   ′′
x y  (x)+  6 − 2x  y (x)+ 2x y(x) +  6− x   xy (x) = 0
 4.2172   ( 2   )  ′′′        ′        ′
 x + 1  y (x)+ 8xy (x) + 10y(x) = 0
 4.2173   (     )        (      )
 x2 + 2 y ′′′(x)+ x2 + 2 y′(x)− 2xy ′′(x )− 2xy(x) = 0
 4.2174   (          )
x2 − 2x + 2 y′′′(x)+ x2 (− y′′(x))+ 2xy ′(x)− 2y (x ) = 0
 4.2175   (x + 2)2y′′′(x)+ (x + 2)y′′(x)+ y′(x) = 0
 4.2176   4x2y ′′′(x)+ 8xy ′′(x )+ y′(x ) = 0
 4.2177   x(a0 + b0x)y′′′(x) + (a1+ b1x )y′′(x) + xy′(x )+ y(x) = f(x)
 4.2178    3 ′′′
x y  (x) = a
 4.2179    3 ′′′        ′
x y  (x)+ xy (x)− y (x ) = 0
 4.2180   x3y′′′(x)+ xy ′(x)− y (x ) = x log(x)
 4.2181   x3y′′′(x)− x2y′′(x) + 2xy′(x )− 2y(x) = 0
 4.2182                                         (      )
x3y′′′(x)− x2y′′(x) + 2xy′(x )− 2y(x) = x x2 + 3
 4.2183   x3y′′′(x)+ x2y′′(x) + 3xy′(x )− 8y(x) = 0
 4.2184   x3y′′′(x)− x2y′′(x) + xy′(x ) = 0
 4.2185    3 ′′′       2  ′′
x y  (x)+ 2x y (x) + 2y(x) = 0
 4.2186    3 ′′′       2  ′′       ′
x y  (x)+ 2x y (x) − xy (x )+ y(x) = 0
 4.2187    3 ′′′       2  ′′
x y  (x)+ 3x y (x) = a
 4.2188   x3y′′′(x)+ 3x2y ′′(x) − 2xy′(x)+ 2y(x) = 0
 4.2189   x3y′′′(x)− 3x2y ′′(x) + 7xy′(x)− 8y(x) = 0
 4.2190             (      )
x3y′′′(x)+  1 − a2 xy′(x)+ 3x2y ′′(x ) = 0
 4.2191   x3y′′′(x)+ 4x2y ′′(x) − 8xy′(x)+ 8y(x) = 0
 4.2192   x3y′′′(x)− 4x2y′′(x) + (x2 + 8) xy′(x) − 2(x2 + 4)y(x) = 0
 4.2193    3 ′′′     (       3)         2 ′′
x y  (x)−  12 − ax  y(x) + 6x y (x) = 0
 4.2194    3 ′′′      2       ′′        ′              3
x y  (x)+ x  log(x)y (x) + 2xy (x )− y(x) = 2x
 4.2195   (     )
 x3 + 1 y′′′(x)+ 9x2y ′′(x )+ 18xy′(x)+ 6y(x) = 0
 4.2196     (     )          (       )
x  x2 + 1 y′′′(x) + 3 2x2 + 1 y′′(x)− 12y (x ) = 0
 4.2197     (     )        (       )
x  1− x2  y′′′(x)+  3 − 8x2 y ′′(x )− 14xy′(x)− 4y(x) = 0
 4.2198   x (x2 − 2x + 2)y ′′′(x)+ (− x3 + 3x2 − 6x + 6)y′′(x) = 0
 4.2199   (x + 1)3y ′′′(x)+ (x + 1)2y′′(x)+ 3(x + 1)y′(x)− 8y (x ) = 0
 4.2200    2          ′′′               ′′              ′
x (y(x)+ 3)y  (x)− 3(x + 2)xy (x)+ 6(x + 1)y(x)−  6y(x) = 0
 4.2201      3 ′′′        ′
4x y  (x)+ xy (x)− y (x ) = 0
 4.2202           3 ′′′             ′
(1 − 2x) y (x)+  (1 − 2x)y (x )+ 2y(x) = 0
 4.2203                                   (      )
x4y′′′(x)+ 2x3y ′′(x) + 2xy(x) = 10 x2 + 1
 4.2204   x4y′′′(x)+ 2x3y ′′(x) − x2y′(x )+ xy(x) = 1
 4.2205   (      )
 x2 + 1 x2y′′′(x)+ 8x3y′′(x) + 10x2y′(x ) = 3x2 + 2x2log(x)− 1
 4.2206   (x + 1)x3y′′′(x) − 2(2x + 1)x2y′′(x) + 2(5x+ 2)xy′(x)− 4(3x + 1)y(x) = 0
 4.2207      4 ′′′        3 ′′       2 ′
4x y  (x)− 4x y  (x )+ 4x y (x ) = 1
 4.2208   ( 2    ) 3 ′′′      (   2   )  2 ′′      (   2   )   ′      (  2    )
 x  + 1 x y  (x)− 2 2x  + 1 x  y (x )+ 2 5x  + 2 xy (x)−  4 3x  + 1 y(x) = 0
 4.2209   (a − x)3(b − x)3y′′′(x) = cy(x)
 4.2210   (x + sin(x))y′′′(x )+ 3(cos(x )+ 1)y′′(x)− 3 sin(x)y′(x )− y(x)cos(x)+ sin(x) = 0
 4.2211   y′′′′(x) = 0
 4.2212    ′′′′
y  (x) = xcos(x)
 4.2213    ′′′′       −x
y  (x) + 4e  cos(x) = 0
 4.2214   y′′′′(x) = y(x)+ cos(x)
 4.2215   y′′′′(x) = y(x)+ ex cos(x)
 4.2216   y′′′′(x) + ay(x) = 0
 4.2217    ′′′′      4       3
y  (x) = a y(x)+ x
 4.2218    ′′′′      ′′
y  (x) + y (x )+ y(x) = 0
 4.2219    ′′′′       ′′
y  (x) + 2y (x)+ y(x) = 0
 4.2220   y′′′′(x) − 2y′′(x)+ y(x) = 0
 4.2221   y′′′′(x) + 2y′′(x)+ y(x) = cos(x )
 4.2222   y′′′′(x) − 2y′′(x)+ y(x) = cos(x )
 4.2223   y′′′′(x) + 2y′′(x)+ y(x) = 24xsin(x)
 4.2224   y′′′′(x) − 2y′′(x)+ y(x) = ex + 4
 4.2225    ′′′′       ′′
y  (x) − 2y (x)− 8y(x) = 0
 4.2226    ′′′′       ′′
y  (x) + 3y (x)− 4y(x) = 0
 4.2227   y′′′′(x) + 5y′′(x)+ 6y(x) = 0
 4.2228   y′′′′(x) − 12y′′(x)+ 27y(x) = 0
 4.2229   y′′′′(x) + a2y′′(x ) = 0
 4.2230   y′′′′(x) + a4y(x)+ 2a2y′′(x) = 0
 4.2231   y′′′′(x) + a4y(x)+ 2a2y′′(x) = cosh(ax)
 4.2232    ′′′′     ( 2   2) ′′      2 2
y  (x) +  a + b  y (x) + a by (x ) = 0
 4.2233    ′′′′           ( ′′          2)      ′   ′            ′′
y  (x)+ 3y(x) f  (x)+ 3f(x)  +  10f(x)y (x)+ 10f (x )y (x) = 0
 4.2234      ′′′      ′′′′       ′′       ′
− y (x) + y  (x )− 3y (x)+  5y(x)−  2y(x ) = 0
 4.2235   − y′′′(x) + y′′′′(x )− 3y′′(x)+  5y′(x)−  2y(x ) = e3x
 4.2236   − 2y′′′(x) + y′′′′(x)+ y(x)2 = 0
 4.2237   − 2y′′′(x) + y′′′′(x)+ y(x)2 = x3
 4.2238   2y′′′(x)+ y′′′′(x)−  2y′(x)−  y(x ) = 0
 4.2239   − 2y′′′(x) + y′′′′(x)+ 2y′′(x)−  2y′(x) + y(x ) = 0
 4.2240     ′′′      ′′′′       ′′       ′
2y  (x)+ y  (x)+  3y (x )+ 2y (x )+ y(x) = 0
 4.2241     ′′′      ′′′′       ′′       ′
2y  (x)+ y  (x)−  3y (x )− 4y (x )+ 4y(x) = 0
 4.2242   − 3y′′′(x) + y′′′′(x)+ y′′(x)−  y′(x) = 0
 4.2243   − 4y′′′(x) + y′′′′(x)+ 6y′′(x)−  4y′(x) + y(x ) = 0
 4.2244   − 4y′′′(x) + y′′′′(x)+ 12y′′(x)−  16y′(x) + 16y(x) = 0
 4.2245   4axy′′′(x)+ y′′′′(x)+  a4x4y(x)+ 4a3x3y′(x)+ 6a2x2y ′′(x) = 0
 4.2246   2(y ′′′′(x) + (a2 + b2)y′′(x) + a2b2y(x)) = cos(ax)+ cos(bx)
 4.2247        ′′′       ′′′′        ′′       ′
− 12y (x) + 4y  (x)+ 11y (x) − 3y(x) = 0
 4.2248     ′′′       ′′′′
3y  (x)+ xy  (x) = 0
 4.2249     ′′′       ′′′′
5y  (x)+ xy  (x) = 0
 4.2250   x2y′′′′(x) = 2y′′′(x)
 4.2251   x2y′′′′(x) = 2y′′(x)
 4.2252   x2y′′′′(x) + 4xy′′′(x) + 2y′′(x) = 0
 4.2253   x2y′′′′(x) + 6xy′′′(x) + 6y′′(x) = 0
 4.2254   x2y′′′′(x) + 8xy′′′(x) + 12y′′(x) = 0
 4.2255    2 ′′′′        ′′′      2             ′′
x y  (x) + 8xy (x) + a (− y(x))+ 12y (x ) = 0
 4.2256          2 ′′′′
(a + x) y  (x ) = 1
 4.2257   32x(a−  b+ 2)y′′′(x) + 16x2y′′′′(x)+ 16(a − b+ 1)(a−  b+ 2)y′′(x)+ c4(− y(x )) = 0
 4.2258                           (    )
x3y′′′′(x) + 2x2y′′′(x) + a4 − x3 y(x)−  xy′′(x) = 0
 4.2259   x3y′′′′(x) + 6x2y′′′(x) + 6xy′′(x) = 0
 4.2260   x2(a+ b + c+ 3)y′′′(x )+ x3y′′′′(x)+ x (ab + ac+  a+ bc+  b+ c+ 1)y′′(x) − y′(x)(x − abc)− ky(x) = 0
 4.2261    4 ′′′′       3 ′′′       2 ′′         ′
x y  (x) + 6x y (x) + 4x y (x)− 2xy (x)− 4y (x ) = 0
 4.2262   x4y′′′′(x) + 6x3y′′′(x) + 9x2y′′(x)+ 3xy ′(x)+ y (x ) = 0
 4.2263   x4y′′′′(x) + 8x3y′′′(x) + 12x2y′′(x) = 0
 4.2264   x4y′′′′(x) + 8x3y′′′(x) + ay(x)+ 12x2y ′′(x ) = 0
 4.2265       3 ′′′      4 ′′′′         2 ′′          ′
A1x  y (x) + x y  (x )+ A2x  y (x)+ A3xy  (x )+ A4y (x ) = 0
 4.2266      4 4 ′′′′                2 3 ′′′                      2 2 ′′      4 2∕a
16a x y  (x)− 32(1 − 2a)a x y (x) + 16(1− 2a)(1 − a)a x y (x)− b x   y(x) = 0
 4.2267   4 (ex + 2)y′′′(x)+ (2x + ex)y′′′′(x)+ 6exy′′(x)+  4exy′(x)+  exy(x ) = 0
 4.2268   − y′′′(x) + y′′′′′(x)− 2y ′′(x )+ 2y′(x ) = 0
 4.2269   2y′′′(x)+ y′′′′′(x )+ y′(x ) = 0
 4.2270   2y′′′(x)+ y′′′′′(x )+ y′(x ) = ax + bcos(x)+ c sin(x)
 4.2271    ′′′′′′
y   (x) = 0
 4.2272    ′′′′′′
y   (x)+ ay (x ) = 0
 4.2273   2y′′′(x)+ y′′′′′′(x)+ y(x ) = 0
 4.2274   y′′′′′′′′(x) = y(x)
 4.2275   y′′′′′′′′(x)− 2y ′′′′ + y(x) = 0
 4.2276   y(2n)(x ) = a2ny(x)
 4.2277   xny (2n)(x) = y(x )
 4.2278    n+ 1 (2n+1 )
x   2y     (x) = y(x)
 4.2279    (n)      x
y   (x) = e x
 4.2280   amxm  −1y(x)+ axmy ′(x)+ y(n)(x) = 0
 4.2281   (a − x)n(b− x)ny(n)(x ) = cy(x)
 4.2282   y′′′(x) = y′(x)(y′(x )+ 1)
 4.2283   y′′′(x)+ y′(x)2 − y(x)y′(x ) = 0
 4.2284   y′′′(x)+ ay(x)y′′(x) = 0
 4.2285    2 ′′′        ′′                 ′         2
x y  (x)+ xy  (x )− (1− 2xy (x ))y (x)+ y(x)  = f(x)
 4.2286    2 ′′′                ′′       ′  2             ′
x y  (x)− x(1 − y(x))y (x) + xy (x ) + (1− y(x))y (x ) = 0
 4.2287   y(x)y′′′(x)+ y(x)3y′(x )− y′(x )y′′(x) = 0
 4.2288   (a + y(x))y ′′′(x)+ 3y′(x)y′′(x) = 0
 4.2289   x3y(x)y′′′(x) + 3x3y′(x)y′′(x) + 9x2y(x)y′′(x)+ 9x2y′(x)2 + 18xy (x )y ′(x)+ 3y (x )2 = 0
 4.2290   (y(x)2 + x)y′′′(x)+ 3y′′(x) + 2y′(x)3 + 6y(x )y ′(x)y′′(x) = 0
 4.2291        2 ′′′        ′   3         ′    ′′
4y(x) y  (x )+ 15y (x) − 18y(x)y (x )y  (x ) = 0
 4.2292        2 ′′′        ′   3         ′    ′′
9y(x) y  (x )+ 40y (x) − 45y(x)y (x )y  (x ) = 0
 4.2293    ′    ′′′      ′  2     ′′   2
y (x)y (x) + y(x)  = 2y (x)
 4.2294   y′(x)y′′(x) = axy ′(x)5 + 3y′′(x)2
 4.2295   2y′(x)y′′′(x) = 2y′′(x)2
 4.2296   (         )
 y′(x)2 + 1 y′′′(x) = 3y′(x )y ′′(x )2
 4.2297   (y′(x)2 + 1 )y′′′(x) = y′′(x)2(a+ 3y′(x))
 4.2298   y′(x)3y′′′(x) = 1
 4.2299    ′′   ′′′
y (x)y  (x ) = 2
 4.2300    ′′   ′′′      ∘ -2-′′--2----
y (x)y  (x ) = a b y (x)  + 1
 4.2301   2xy ′′(x )y ′′′(x) = y′′(x )2 − a2
 4.2302   (     )
 1− x2  (y′′′(x))2 + 2xy′′(x)y′′′(x )− y′′(x)2 + 1 = 0
 4.2303   ∘ ----------
  y′′(x)2 + 1(1−  y′′′(x)) = y ′′(x )y′′′(x)
 4.2304   3y′′(x)y′′′′(x) = 5(y′′′(x))2
 4.2305       ′     ′′′    2    ′    ′′    ′′′′       ′′   3
− 4y (x )(y  (x)) + 3y (x)y (x )y  (x) − 3y (x) = 0
 4.2306        ′′    ′′′   ′′′′       ′′  2 ′′′′′         ′′′   3
− 45y (x)y (x)y  (x)+ 9y (x) y   (x )+ 40(y  (x ))  = 0
 4.2307   y(n)(x) = f (x )
 4.2308   y(n)(x) = f (y(n−2)(x))
 4.2309   y(n)(x) = f (y(n−1)(x))
 4.2310    (n−2)    (n)       (n− 1)   2
y     (x)y  (x) = y    (x)
 4.2311     (   (n)  )
f  x,y  (x)  = 0
 4.2312     (                 )
f  x,y(n−1)(x),y(n)(x ) = 0
 4.2313     (                         )
f  y(n−2)(x),y(n−1)(x ),y(n)(x) = 0