2.3.14 first order ode isobaric

Table 2.421: first order ode isobaric

#

ODE

ODE classification

Solved?

27

\[ {}y^{\prime } = 2 x^{2} y^{2} \]
i.c.

[_separable]

42

\[ {}y^{\prime }+2 x y^{2} = 0 \]

[_separable]

46

\[ {}y^{\prime } = 3 \sqrt {x y} \]

[[_homogeneous, ‘class G‘]]

47

\[ {}y^{\prime } = 64^{{1}/{3}} \left (x y\right )^{{1}/{3}} \]

[[_homogeneous, ‘class G‘]]

51

\[ {}y^{\prime } = x y^{3} \]

[_separable]

77

\[ {}y^{\prime } x +2 y = 3 x \]
i.c.

[_linear]

78

\[ {}y^{\prime } x +5 y = 7 x^{2} \]
i.c.

[_linear]

79

\[ {}2 y^{\prime } x +y = 10 \sqrt {x} \]

[_linear]

80

\[ {}3 y^{\prime } x +y = 12 x \]

[_linear]

82

\[ {}2 y^{\prime } x -3 y = 9 x^{3} \]

[_linear]

84

\[ {}y^{\prime } x +3 y = 2 x^{5} \]
i.c.

[_linear]

98

\[ {}\frac {1-4 x y^{2}}{x^{\prime }} = y^{3} \]

[_linear]

105

\[ {}\left (x +y\right ) y^{\prime } = x -y \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

106

\[ {}2 x y^{\prime } y = x^{2}+2 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

107

\[ {}y^{\prime } x = y+2 \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _dAlembert]

108

\[ {}\left (x -y\right ) y^{\prime } = x +y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

109

\[ {}x \left (x +y\right ) y^{\prime } = y \left (x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

110

\[ {}\left (x +2 y\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

111

\[ {}x y^{2} y^{\prime } = x^{3}+y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

112

\[ {}x^{2} y^{\prime } = x y+x^{2} {\mathrm e}^{\frac {y}{x}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

113

\[ {}x^{2} y^{\prime } = x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

114

\[ {}x y^{\prime } y = x^{2}+3 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

115

\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

116

\[ {}x y^{\prime } y = y^{2}+x \sqrt {4 x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

117

\[ {}y^{\prime } x = y+\sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

118

\[ {}y y^{\prime }+x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

119

\[ {}x \left (x +y\right ) y^{\prime }+y \left (3 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

123

\[ {}x^{2} y^{\prime }+2 x y = 5 y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

126

\[ {}x^{2} y^{\prime }+2 x y = 5 y^{4} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

127

\[ {}y^{\prime } x +6 y = 3 x y^{{4}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

131

\[ {}3 x y^{2} y^{\prime } = 3 x^{4}+y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

135

\[ {}2 x +3 y+\left (3 x +2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

136

\[ {}4 x -y+\left (6 y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

137

\[ {}3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

166

\[ {}y^{\prime } = -\frac {y \left (2 x^{3}-y^{3}\right )}{x \left (2 y^{3}-x^{3}\right )} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

181

\[ {}x y+y^{2}-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

186

\[ {}x^{2} y^{\prime }+2 x y = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

187

\[ {}y^{\prime } x +2 y = 6 x^{2} \sqrt {y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

189

\[ {}x^{2} y^{\prime } = x y+3 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

190

\[ {}6 x y^{3}+2 y^{4}+\left (9 x^{2} y^{2}+8 x y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

192

\[ {}x^{3} y^{\prime } = x^{2} y-y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

196

\[ {}2 x^{2} y-x^{3} y^{\prime } = y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

198

\[ {}y^{\prime } x +3 y = \frac {3}{x^{{3}/{2}}} \]

[_linear]

200

\[ {}y^{\prime } x = 6 y+12 x^{4} y^{{2}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

204

\[ {}9 \sqrt {x}\, y^{{4}/{3}}-12 x^{{1}/{5}} y^{{3}/{2}}+\left (8 x^{{3}/{2}} y^{{1}/{3}}-15 x^{{6}/{5}} \sqrt {y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

205

\[ {}3 y+y^{4} x^{3}+3 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

211

\[ {}y^{\prime } = -\frac {3 x^{2}+2 y^{2}}{4 x y} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

212

\[ {}y^{\prime } = \frac {x +3 y}{y-3 x} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

669

\[ {}y^{\prime } = 2 x^{2} y^{2} \]
i.c.

[_separable]

678

\[ {}y^{\prime }+2 x y^{2} = 0 \]

[_separable]

682

\[ {}y^{\prime } = 3 \sqrt {x y} \]

[[_homogeneous, ‘class G‘]]

683

\[ {}y^{\prime } = 4 \left (x y\right )^{{1}/{3}} \]

[[_homogeneous, ‘class G‘]]

687

\[ {}y^{\prime } = x y^{3} \]

[_separable]

708

\[ {}y^{\prime } x +2 y = 3 x \]
i.c.

[_linear]

709

\[ {}2 y^{\prime } x +y = 10 \sqrt {x} \]
i.c.

[_linear]

710

\[ {}2 y^{\prime } x +y = 10 \sqrt {x} \]

[_linear]

711

\[ {}3 y^{\prime } x +y = 12 x \]

[_linear]

713

\[ {}2 y^{\prime } x -3 y = 9 x^{3} \]

[_linear]

715

\[ {}y^{\prime } x +3 y = 2 x^{5} \]
i.c.

[_linear]

729

\[ {}\left (x +y\right ) y^{\prime } = x -y \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

730

\[ {}2 x y^{\prime } y = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

731

\[ {}y^{\prime } x = y+2 \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _dAlembert]

732

\[ {}\left (x -y\right ) y^{\prime } = x +y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

733

\[ {}x \left (x +y\right ) y^{\prime } = y \left (x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

734

\[ {}\left (x +2 y\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

735

\[ {}x y^{2} y^{\prime } = x^{3}+y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

736

\[ {}x^{2} y^{\prime } = x y+x^{2} {\mathrm e}^{\frac {y}{x}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

737

\[ {}x^{2} y^{\prime } = x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

738

\[ {}x y^{\prime } y = x^{2}+3 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

739

\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

740

\[ {}x y^{\prime } y = y^{2}+x \sqrt {4 x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

741

\[ {}y^{\prime } x = y+\sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

742

\[ {}y y^{\prime }+x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

743

\[ {}x \left (x +y\right ) y^{\prime }+y \left (3 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

747

\[ {}x^{2} y^{\prime }+2 x y = 5 y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

750

\[ {}x^{2} y^{\prime }+2 x y = 5 y^{4} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

751

\[ {}y^{\prime } x +6 y = 3 x y^{{4}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

755

\[ {}3 x y^{2} y^{\prime } = 3 x^{4}+y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

759

\[ {}2 x +3 y+\left (3 x +2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

760

\[ {}4 x -y+\left (6 y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

761

\[ {}3 x^{2}+2 y^{2}+\left (4 x y+6 y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

773

\[ {}x y+y^{2}-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

778

\[ {}x^{2} y^{\prime }+2 x y = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

779

\[ {}y^{\prime } x +2 y = 6 x^{2} \sqrt {y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

781

\[ {}x^{2} y^{\prime } = x y+3 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

782

\[ {}6 x y^{3}+2 y^{4}+\left (9 x^{2} y^{2}+8 x y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

784

\[ {}x^{3} y^{\prime } = x^{2} y-y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

788

\[ {}2 x^{2} y-x^{3} y^{\prime } = y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

790

\[ {}y^{\prime } x +3 y = \frac {3}{x^{{3}/{2}}} \]

[_linear]

792

\[ {}y^{\prime } x = 6 y+12 x^{4} y^{{2}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

797

\[ {}3 y+y^{4} x^{3}+3 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

803

\[ {}y^{\prime } = \frac {-3 x^{2}-2 y^{2}}{4 x y} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

804

\[ {}y^{\prime } = \frac {x +3 y}{y-3 x} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1129

\[ {}y^{\prime } = \frac {x^{2}}{y} \]

[_separable]

1134

\[ {}y^{\prime } x = \sqrt {1-y^{2}} \]

[_separable]

1140

\[ {}r^{\prime } = \frac {r^{2}}{x} \]
i.c.

[_separable]

1158

\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

1159

\[ {}y^{\prime } = \frac {x^{2}+3 y^{2}}{2 x y} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1160

\[ {}y^{\prime } = \frac {4 y-3 x}{2 x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1161

\[ {}y^{\prime } = -\frac {4 x +3 y}{2 x +y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1162

\[ {}y^{\prime } = \frac {x +3 y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1163

\[ {}x^{2}+3 x y+y^{2}-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

1164

\[ {}y^{\prime } = \frac {x^{2}-3 y^{2}}{2 x y} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1165

\[ {}y^{\prime } = \frac {3 y^{2}-x^{2}}{2 x y} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1174

\[ {}y^{\prime } = -\frac {4 t}{y} \]

[_separable]

1175

\[ {}y^{\prime } = 2 t y^{2} \]

[_separable]

1194

\[ {}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1197

\[ {}y^{\prime } = \frac {-a x -b y}{b x +c y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1198

\[ {}y^{\prime } = \frac {-a x +b y}{b x -c y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1204

\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}} = 0 \]

[_separable]

1205

\[ {}2 x -y+\left (-x +2 y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1217

\[ {}3 x y+y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1218

\[ {}y^{\prime } = \frac {x^{3}-2 y}{x} \]

[_linear]

1231

\[ {}x +y+\left (x +2 y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1243

\[ {}y^{\prime } x = {\mathrm e}^{\frac {y}{x}} x +y \]

[[_homogeneous, ‘class A‘], _dAlembert]

1245

\[ {}3 t +2 y = -y^{\prime } t \]

[_linear]

1246

\[ {}y^{\prime } = \frac {x +y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1247

\[ {}2 x y+3 y^{2}-\left (x^{2}+2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1248

\[ {}y^{\prime } = \frac {-3 x^{2} y-y^{2}}{2 x^{3}+3 x y} \]
i.c.

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1520

\[ {}y^{\prime } x +y = x^{2} \]

[_linear]

1533

\[ {}y^{\prime } = -\frac {y \left (1+y\right )}{x} \]
i.c.

[_separable]

1567

\[ {}y^{\prime } x +2 y = 8 x^{2} \]
i.c.

[_linear]

1573

\[ {}y^{\prime } x -2 y = -1 \]
i.c.

[_separable]

1580

\[ {}y^{\prime } x +y^{2}+y = 0 \]

[_separable]

1597

\[ {}y y^{\prime }+x = 0 \]
i.c.

[_separable]

1615

\[ {}y^{\prime } = \frac {2 x +3 y}{x -4 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1626

\[ {}y^{\prime } = \frac {y+{\mathrm e}^{-\frac {y}{x}} x}{x} \]

[[_homogeneous, ‘class A‘], _dAlembert]

1628

\[ {}x^{2} y^{\prime } = y^{2}+x y-x^{2} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Riccati]

1643

\[ {}y^{\prime } = \frac {y^{2}+2 x y}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1644

\[ {}x y^{3} y^{\prime } = y^{4}+x^{4} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1645

\[ {}y^{\prime } = \frac {y}{x}+\sec \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

1646

\[ {}x^{2} y^{\prime } = x^{2}+x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

1647

\[ {}x y^{\prime } y = x^{2}+2 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1648

\[ {}y^{\prime } = \frac {2 y^{2}+x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}}}{2 x y} \]

[[_homogeneous, ‘class A‘]]

1649

\[ {}y^{\prime } = \frac {x y+y^{2}}{x^{2}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1650

\[ {}y^{\prime } = \frac {x^{3}+y^{3}}{x y^{2}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1651

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1653

\[ {}x^{2} y^{\prime } = 2 x^{2}+y^{2}+4 x y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Riccati]

1654

\[ {}x y^{\prime } y = 3 x^{2}+4 y^{2} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

1655

\[ {}y^{\prime } = \frac {x +y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1656

\[ {}\left (y^{\prime } x -y\right ) \left (\ln \left (y\right )-\ln \left (x \right )\right ) = x \]

[[_homogeneous, ‘class A‘]]

1657

\[ {}y^{\prime } = \frac {y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x \left (x +y\right )^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

1658

\[ {}y^{\prime } = \frac {x +2 y}{2 x +y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1659

\[ {}y^{\prime } = \frac {y}{y-2 x} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1660

\[ {}y^{\prime } = \frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

1661

\[ {}y^{\prime } = \frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

1663

\[ {}x y^{\prime } y = x^{2}-x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1664

\[ {}y^{\prime } = \frac {2 y^{2}-x y+2 x^{2}}{x y+2 x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1665

\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1669

\[ {}3 x y^{2} y^{\prime } = y^{3}+x \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

1670

\[ {}x y^{\prime } y = 3 x^{6}+6 y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

1671

\[ {}x^{3} y^{\prime } = 2 y^{2}+2 x^{2} y-2 x^{4} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

1675

\[ {}2 x \left (y+2 \sqrt {x}\right ) y^{\prime } = \left (y+\sqrt {x}\right )^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1677

\[ {}y^{\prime }+\frac {2 y}{x} = \frac {3 x^{2} y^{2}+6 x y+2}{x^{2} \left (2 x y+3\right )} \]
i.c.

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1678

\[ {}y^{\prime }+\frac {3 y}{x} = \frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )} \]
i.c.

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1685

\[ {}4 x +7 y+\left (3 x +4 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1687

\[ {}2 x +y+\left (2 y+2 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1692

\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}} = 0 \]

[_separable]

1702

\[ {}7 x +4 y+\left (4 x +3 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1707

\[ {}x^{2}+y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

1711

\[ {}y+\left (2 x +\frac {1}{y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1712

\[ {}-y^{2}+x^{2} y^{\prime } = 0 \]

[_separable]

1718

\[ {}27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1726

\[ {}y \sin \left (y\right )+x \left (\sin \left (y\right )-y \cos \left (y\right )\right ) y^{\prime } = 0 \]

[_separable]

1733

\[ {}x^{4} y^{3}+y+\left (x^{5} y^{2}-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

1735

\[ {}12 x y+6 y^{3}+\left (9 x^{2}+10 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

1736

\[ {}3 x^{2} y^{2}+2 y+2 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

1804

\[ {}x^{2} \left (y^{\prime }+y^{2}\right )-7 x y+7 = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

2330

\[ {}y^{\prime } t = y+\sqrt {t^{2}+y^{2}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2331

\[ {}2 t y y^{\prime } = 3 y^{2}-t^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2332

\[ {}\left (t -\sqrt {t y}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2333

\[ {}y^{\prime } = \frac {t +y}{t -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2334

\[ {}{\mathrm e}^{\frac {t}{y}} \left (y-t \right ) y^{\prime }+y \left (1+{\mathrm e}^{\frac {t}{y}}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

2346

\[ {}3 t y+y^{2}+\left (t^{2}+t y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2501

\[ {}y^{\prime } = \frac {2 y}{t}+\frac {y^{2}}{t^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2502

\[ {}y^{\prime } t = y+\sqrt {t^{2}+y^{2}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2503

\[ {}2 t y y^{\prime } = 3 y^{2}-t^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2504

\[ {}\left (t -\sqrt {t y}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2505

\[ {}y^{\prime } = \frac {t +y}{t -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2506

\[ {}{\mathrm e}^{\frac {t}{y}} \left (y-t \right ) y^{\prime }+y \left (1+{\mathrm e}^{\frac {t}{y}}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

2518

\[ {}3 t y+y^{2}+\left (t^{2}+t y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2851

\[ {}y^{\prime } = \frac {x}{y} \]

[_separable]

2853

\[ {}y^{\prime } x +y = y^{2} \]

[_separable]

2864

\[ {}x^{2} y^{\prime }+y^{2} = 0 \]
i.c.

[_separable]

2872

\[ {}\left (x +y\right ) y^{\prime }+x = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2873

\[ {}y^{\prime } x -y = \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2874

\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2875

\[ {}y^{\prime } x -y = \sqrt {x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2876

\[ {}y y^{\prime }+x = 2 y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2877

\[ {}y^{\prime } x -y+\sqrt {y^{2}-x^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

2878

\[ {}x^{2}+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2879

\[ {}\left (x y-x^{2}\right ) y^{\prime }-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2880

\[ {}y^{\prime } x +y = 2 \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _dAlembert]

2881

\[ {}x +y+\left (x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2882

\[ {}y \left (x^{2}-x y+y^{2}\right )+x y^{\prime } \left (x^{2}+x y+y^{2}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2883

\[ {}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

2884

\[ {}y^{\prime } = \frac {y}{x}+\cosh \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

2885

\[ {}x^{2}+y^{2} = 2 x y^{\prime } y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2886

\[ {}\left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }+1 = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2887

\[ {}{\mathrm e}^{\frac {y}{x}} x +y = y^{\prime } x \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

2888

\[ {}y^{\prime } = \frac {x +y}{x -y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2889

\[ {}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right ) \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

2890

\[ {}\left (3 x y-2 x^{2}\right ) y^{\prime } = 2 y^{2}-x y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2891

\[ {}y^{\prime } = \frac {y}{x -k \sqrt {x^{2}+y^{2}}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

2892

\[ {}y^{2} \left (y y^{\prime }-x \right )+x^{3} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2893

\[ {}y^{\prime } = \frac {y}{x}+\tanh \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

2914

\[ {}x +y+\left (x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2915

\[ {}3 x +y+\left (x +3 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2919

\[ {}2 x y-\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

2927

\[ {}\frac {y \left (2+x^{3} y\right )}{x^{3}} = \frac {\left (1-2 x^{3} y\right ) y^{\prime }}{x^{2}} \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2929

\[ {}\frac {2 y}{x^{3}}+\frac {2 x}{y^{2}} = \left (\frac {1}{x^{2}}+\frac {2 x^{2}}{y^{3}}\right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _exact, _rational]

2934

\[ {}\frac {x^{2}+3 y^{2}}{x \left (3 x^{2}+4 y^{2}\right )}+\frac {\left (2 x^{2}+y^{2}\right ) y^{\prime }}{y \left (3 x^{2}+4 y^{2}\right )} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

2935

\[ {}\frac {x^{2}-y^{2}}{x \left (2 x^{2}+y^{2}\right )}+\frac {\left (x^{2}+2 y^{2}\right ) y^{\prime }}{y \left (2 x^{2}+y^{2}\right )} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

2938

\[ {}x y+\left (y+x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2939

\[ {}\left (x -2 x y\right ) y^{\prime }+2 y = 0 \]

[_separable]

2940

\[ {}x^{2} y+y^{2}+x^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

2941

\[ {}x y^{3}-1+x^{2} y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

2942

\[ {}\left (x^{3} y^{3}-1\right ) y^{\prime }+x^{2} y^{4} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

2943

\[ {}y \left (y-x^{2}\right )+x^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

2944

\[ {}y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2946

\[ {}2 x y+\left (y-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2947

\[ {}y = x \left (x^{2} y-1\right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2950

\[ {}\left (2 x +3 x^{2} y\right ) y^{\prime }+y+2 x y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2952

\[ {}y \left (1-x^{4} y^{2}\right )+y^{\prime } x = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

2954

\[ {}x^{2} y^{2}-y+\left (2 x^{3} y+x \right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2957

\[ {}y \left (x +y^{2}\right )+x \left (x -y^{2}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational]

2958

\[ {}y^{\prime } x +2 y = x^{2} \]

[_linear]

2964

\[ {}y+\left (2 x -3 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2965

\[ {}y^{\prime } x -2 x^{4}-2 y = 0 \]

[_linear]

2972

\[ {}2 y = \left (y^{4}+x \right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational]

2980

\[ {}y+2 \left (x -2 y^{2}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational]

2986

\[ {}x y^{\prime } y = x^{2}-y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2988

\[ {}x^{\prime } t +x \left (1-x^{2} t^{4}\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

2989

\[ {}x^{2} y^{\prime }+y^{2} = x y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2994

\[ {}y^{\prime } x +2 y = 3 x^{3} y^{{4}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3001

\[ {}2 y = \left (x^{2} y^{4}+x \right ) y^{\prime } \]
i.c.

[[_homogeneous, ‘class G‘], _rational]

3005

\[ {}y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3006

\[ {}2 x +y-\left (x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3011

\[ {}6+2 y = x y^{\prime } y \]

[_separable]

3014

\[ {}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3018

\[ {}2 x y+y^{4}+\left (x y^{3}-2 x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

3019

\[ {}y+\left (3 x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3021

\[ {}\left (3 x +4 y\right ) y^{\prime }+2 x +y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3023

\[ {}y^{\prime } x -y-\sqrt {x^{2}+y^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3026

\[ {}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _dAlembert]

3030

\[ {}2 y^{\prime } x -y+\frac {x^{2}}{y^{2}} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3031

\[ {}y^{\prime } x +y \left (1+y^{2}\right ) = 0 \]

[_separable]

3032

\[ {}y \sqrt {x^{2}+y^{2}}+x y = x^{2} y^{\prime } \]

[[_homogeneous, ‘class A‘], _dAlembert]

3036

\[ {}y \cos \left (\frac {x}{y}\right )-\left (y+x \cos \left (\frac {x}{y}\right )\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

3037

\[ {}y \left (3 x^{2}+y\right )-x \left (x^{2}-y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3039

\[ {}y^{\prime } x -5 y-x \sqrt {y} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3041

\[ {}x y-y^{2}-x^{2} y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

3044

\[ {}y^{\prime } x -2 y-2 x^{4} y^{3} = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3045

\[ {}\left (-2 x^{2}-3 x y\right ) y^{\prime }+y^{2} = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3047

\[ {}y^{\prime } x +y = x^{3} y^{6} \]
i.c.

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3049

\[ {}x^{2}+y^{2} = 2 x y^{\prime } y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

3050

\[ {}3 x y+\left (3 x^{2}+y^{2}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3054

\[ {}y^{2}+\left (x^{3}-2 x y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3056

\[ {}y^{3}+2 x^{2} y+\left (-3 x^{3}-2 x y^{2}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3302

\[ {}2 x^{2} y+{y^{\prime }}^{2} = x^{3} y^{\prime } \]

[[_1st_order, _with_linear_symmetries]]

3307

\[ {}x^{2}-3 y y^{\prime }+x {y^{\prime }}^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

3320

\[ {}{y^{\prime }}^{3}+x y^{\prime } y = 2 y^{2} \]

[[_1st_order, _with_linear_symmetries]]

3410

\[ {}y^{\prime } = x^{2} y^{2} \]

[_separable]

3413

\[ {}y^{\prime } x = \sqrt {1-y^{2}} \]

[_separable]

3432

\[ {}y^{\prime } = -\frac {t}{y} \]

[_separable]

3449

\[ {}y^{\prime } t = y+t^{3} \]
i.c.

[_linear]

3452

\[ {}y^{\prime } t = -y+t^{3} \]
i.c.

[_linear]

3457

\[ {}y^{\prime }-x y^{3} = 0 \]

[_separable]

3461

\[ {}2 y^{\prime } x +3 x +y = 0 \]

[_linear]

3465

\[ {}\left (y^{3}+x \right ) y^{\prime } = y \]

[[_homogeneous, ‘class G‘], _rational]

3467

\[ {}\left (y-x \right ) y^{\prime }+2 x +3 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3471

\[ {}x \left (1-2 x^{2} y\right ) y^{\prime }+y = 3 x^{2} y^{2} \]
i.c.

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3476

\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3477

\[ {}y^{\prime }-\frac {y^{2}}{x^{2}} = {\frac {1}{4}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3544

\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3545

\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3546

\[ {}\sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right ) = x \cos \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

3547

\[ {}y^{\prime } x = \sqrt {16 x^{2}-y^{2}}+y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3548

\[ {}y^{\prime } x -y = \sqrt {9 x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3549

\[ {}x \left (x^{2}-y^{2}\right )-x \left (x^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3550

\[ {}y^{\prime } x +y \ln \left (x \right ) = y \ln \left (y\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

3551

\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3552

\[ {}2 x y^{\prime } y-2 y^{2}-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}} = 0 \]

[[_homogeneous, ‘class A‘]]

3553

\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3554

\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3555

\[ {}2 x \left (2 x +y\right ) y^{\prime } = y \left (4 x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3556

\[ {}y^{\prime } x = x \tan \left (\frac {y}{x}\right )+y \]

[[_homogeneous, ‘class A‘], _dAlembert]

3557

\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \]

[[_homogeneous, ‘class A‘], _dAlembert]

3636

\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3637

\[ {}\left (3 x -y\right ) y^{\prime } = 3 y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3638

\[ {}y^{\prime } = \frac {\left (x +y\right )^{2}}{2 x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3639

\[ {}\sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right ) = x \cos \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

3640

\[ {}y^{\prime } x = \sqrt {16 x^{2}-y^{2}}+y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3641

\[ {}y^{\prime } x -y = \sqrt {9 x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3643

\[ {}y^{\prime } x +y \ln \left (x \right ) = y \ln \left (y\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

3644

\[ {}y^{\prime } = \frac {y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3645

\[ {}2 x y^{\prime } y-2 y^{2}-x^{2} {\mathrm e}^{-\frac {y^{2}}{x^{2}}} = 0 \]

[[_homogeneous, ‘class A‘]]

3646

\[ {}x^{2} y^{\prime } = y^{2}+3 x y+x^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

3647

\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3648

\[ {}2 x \left (2 x +y\right ) y^{\prime } = y \left (4 x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3649

\[ {}y^{\prime } x = x \tan \left (\frac {y}{x}\right )+y \]

[[_homogeneous, ‘class A‘], _dAlembert]

3650

\[ {}y^{\prime } = \frac {x \sqrt {x^{2}+y^{2}}+y^{2}}{x y} \]

[[_homogeneous, ‘class A‘], _dAlembert]

3651

\[ {}y^{\prime } = \frac {-2 x +4 y}{x +y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3652

\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3653

\[ {}y^{\prime } = \frac {y-\sqrt {x^{2}+y^{2}}}{x} \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

3654

\[ {}y^{\prime } x -y = \sqrt {4 x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

3655

\[ {}y^{\prime } = \frac {x +a y}{a x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3656

\[ {}y^{\prime } = \frac {x +\frac {y}{2}}{\frac {x}{2}-y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3661

\[ {}y^{\prime }+\frac {2 y}{x} = 6 x^{4} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3662

\[ {}2 x \left (y^{\prime }+x^{2} y^{3}\right )+y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

3675

\[ {}y^{\prime } = \frac {y \left (\ln \left (x y\right )-1\right )}{x} \]

[[_homogeneous, ‘class G‘]]

3679

\[ {}y^{\prime }+\frac {2 y}{x}-y^{2} = -\frac {2}{x^{2}} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

3680

\[ {}y^{\prime }+\frac {7 y}{x}-3 y^{2} = \frac {3}{x^{2}} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

3682

\[ {}\frac {y^{\prime }}{y}-\frac {2 \ln \left (y\right )}{x} = \frac {1-2 \ln \left (x \right )}{x} \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

4080

\[ {}4 x y^{2}+6 y+\left (5 x^{2} y+8 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4098

\[ {}x^{3}+y^{3}-x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4103

\[ {}y^{\prime } = \frac {x^{2}+y^{2}}{2 x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

4112

\[ {}y^{\prime } = \frac {2 x -y}{2 x +y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

4190

\[ {}y y^{\prime } = x \]

[_separable]

4196

\[ {}y^{\prime } x +y = x \]

[_linear]

4197

\[ {}y^{\prime } x -y = x^{3} \]

[_linear]

4214

\[ {}y^{\prime } = 6 x y^{2} \]

[_separable]

4223

\[ {}-y^{2}+x^{2} y^{\prime } = 0 \]
i.c.

[_separable]

4230

\[ {}x \cos \left (y\right ) y^{\prime } = 1+\sin \left (y\right ) \]
i.c.

[_separable]

4231

\[ {}y^{\prime } x = 2 y \left (-1+y\right ) \]
i.c.

[_separable]

4232

\[ {}2 y^{\prime } x = 1-y^{2} \]
i.c.

[_separable]

4238

\[ {}x y^{\prime } y = \sqrt {y^{2}-9} \]
i.c.

[_separable]

4240

\[ {}x y^{\prime } y = 2 x^{2}-y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4241

\[ {}x^{2}-y^{2}+x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4242

\[ {}x^{2} y^{\prime }-2 x y-2 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4243

\[ {}x^{2} y^{\prime } = 3 \left (x^{2}+y^{2}\right ) \arctan \left (\frac {y}{x}\right )+x y \]

[[_homogeneous, ‘class A‘], _dAlembert]

4244

\[ {}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )+x \]

[[_homogeneous, ‘class A‘], _dAlembert]

4250

\[ {}\left (x +\frac {2}{y}\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4261

\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4263

\[ {}\left (x +3 y^{4} x^{3}\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

4265

\[ {}y-\left (x +x y^{3}\right ) y^{\prime } = 0 \]

[_separable]

4267

\[ {}\left (x +y\right ) y^{\prime } = y-x \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

4269

\[ {}y^{\prime } x -3 y = x^{4} \]

[_linear]

4274

\[ {}2 y-x^{3} = y^{\prime } x \]

[_linear]

4275

\[ {}\left (1-x y\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4277

\[ {}y^{\prime } x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4278

\[ {}y^{2} = \left (x^{3}-x y\right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4279

\[ {}x^{2} y^{3}+y = \left (x^{3} y^{2}-x \right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational]

4281

\[ {}\left (x y-x^{2}\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4283

\[ {}y+x^{2} = y^{\prime } x \]

[_linear]

4290

\[ {}y^{2}-3 x y-2 x^{2} = \left (x^{2}-x y\right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4300

\[ {}\frac {x}{x^{2}+y^{2}}+\frac {y}{x^{2}}+\left (\frac {y}{x^{2}+y^{2}}-\frac {1}{x}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

4314

\[ {}y^{\prime }+\frac {x}{y}+2 = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

4315

\[ {}y^{\prime } x -y = x \cot \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4316

\[ {}x \cos \left (\frac {y}{x}\right )^{2}-y+y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4317

\[ {}y^{\prime } x = y \left (1+\ln \left (y\right )-\ln \left (x \right )\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4318

\[ {}x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4319

\[ {}\left (1-{\mathrm e}^{-\frac {y}{x}}\right ) y^{\prime }+1-\frac {y}{x} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4320

\[ {}x^{2}-x y+y^{2}-x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4333

\[ {}2 x y+\left (x^{2}+2 x y+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4347

\[ {}x -\sqrt {x^{2}+y^{2}}+\left (y-\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _dAlembert]

4349

\[ {}y^{2}-\left (x y+x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4351

\[ {}2 x^{2} y^{2}+y+\left (x^{3} y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4352

\[ {}y^{2}+\left (x y+\tan \left (x y\right )\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘]]

4353

\[ {}2 x^{2} y^{4}-y+\left (4 x^{3} y^{3}-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

4357

\[ {}x^{2} y^{2}-2 y+\left (x^{3} y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4358

\[ {}2 x^{3} y+y^{3}-\left (x^{4}+2 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

4363

\[ {}\left (y^{3}+\frac {x}{y}\right ) y^{\prime } = 1 \]

[[_homogeneous, ‘class G‘], _rational]

4376

\[ {}y^{\prime } = \frac {4 x^{3} y^{2}}{y x^{4}+2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4382

\[ {}6 y^{2}-x \left (2 x^{3}+y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4389

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

4390

\[ {}{y^{\prime }}^{3}+y^{2} = x y^{\prime } y \]

[[_1st_order, _with_linear_symmetries]]

4392

\[ {}y = y^{\prime } x -x^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

4393

\[ {}y \left (y-2 y^{\prime } x \right )^{3} = {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘]]

4396

\[ {}x y^{2} \left (y^{\prime } x +y\right ) = 1 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4397

\[ {}5 y+{y^{\prime }}^{2} = x \left (x +y^{\prime }\right ) \]

[[_homogeneous, ‘class G‘]]

4399

\[ {}y^{\prime } x = y-{\mathrm e}^{\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

4401

\[ {}2 \sqrt {x y}-y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4402

\[ {}y^{\prime } = {\mathrm e}^{\frac {x y^{\prime }}{y}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

4405

\[ {}y^{\prime } x -y = x \tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4408

\[ {}2 y-x \left (\ln \left (x^{2} y\right )-1\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘]]

4416

\[ {}y^{\prime } x = y+\sqrt {x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4419

\[ {}y^{3}+\left (3 x^{2}-2 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

4421

\[ {}2 x^{3} y y^{\prime }+3 x^{2} y^{2}+7 = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

4422

\[ {}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4424

\[ {}y^{4}+x y+\left (x y^{3}-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

4433

\[ {}2 y^{\prime }+x = 4 \sqrt {y} \]

[[_1st_order, _with_linear_symmetries], _Chini]

4441

\[ {}x +\sin \left (\frac {y}{x}\right )^{2} \left (y-y^{\prime } x \right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4443

\[ {}x y^{3}-1+x^{2} y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4675

\[ {}y^{\prime } = a x y^{2} \]

[_separable]

4690

\[ {}y^{\prime } = x y^{3} \]

[_separable]

4692

\[ {}y^{\prime } = \left (a +b x y\right ) y^{2} \]

[[_homogeneous, ‘class G‘], _Abel]

4702

\[ {}y^{\prime } = a x +b \sqrt {y} \]

[[_homogeneous, ‘class G‘], _Chini]

4703

\[ {}y^{\prime }+x^{3} = x \sqrt {x^{4}+4 y} \]

[[_1st_order, _with_linear_symmetries]]

4740

\[ {}2 y^{\prime }+a x = \sqrt {a^{2} x^{2}-4 b \,x^{2}-4 c y} \]

[[_homogeneous, ‘class G‘]]

4741

\[ {}3 y^{\prime } = x +\sqrt {x^{2}-3 y} \]

[[_1st_order, _with_linear_symmetries], _dAlembert]

4743

\[ {}y^{\prime } x +x +y = 0 \]

[_linear]

4744

\[ {}y^{\prime } x +x^{2}-y = 0 \]

[_linear]

4745

\[ {}y^{\prime } x = x^{3}-y \]

[_linear]

4754

\[ {}y^{\prime } x = a x +b y \]

[_linear]

4755

\[ {}y^{\prime } x = a \,x^{2}+b y \]

[_linear]

4766

\[ {}y^{\prime } x = a +b y^{2} \]

[_separable]

4772

\[ {}y^{\prime } x +\left (1-x y\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4773

\[ {}y^{\prime } x = \left (1-x y\right ) y \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

4774

\[ {}y^{\prime } x = \left (x y+1\right ) y \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

4777

\[ {}y^{\prime } x = y \left (1+2 x y\right ) \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

4787

\[ {}y^{\prime } x = y \left (1+y^{2}\right ) \]

[_separable]

4788

\[ {}y^{\prime } x +y \left (1-x y^{2}\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4792

\[ {}y^{\prime } x = 4 y-4 \sqrt {y} \]

[_separable]

4793

\[ {}y^{\prime } x +2 y = \sqrt {1+y^{2}} \]

[_separable]

4794

\[ {}y^{\prime } x = y+\sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4795

\[ {}y^{\prime } x = y+\sqrt {x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

4798

\[ {}y^{\prime } x = y+a \sqrt {y^{2}+b^{2} x^{2}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

4800

\[ {}y^{\prime } x +x -y+x \cos \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4801

\[ {}y^{\prime } x = y-x \cos \left (\frac {y}{x}\right )^{2} \]

[[_homogeneous, ‘class A‘], _dAlembert]

4803

\[ {}y^{\prime } x = y-\cot \left (y\right )^{2} \]

[_separable]

4805

\[ {}y^{\prime } x -y+x \sec \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

4806

\[ {}y^{\prime } x = y+x \sec \left (\frac {y}{x}\right )^{2} \]

[[_homogeneous, ‘class A‘], _dAlembert]

4808

\[ {}y^{\prime } x = y+x \sin \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4809

\[ {}y^{\prime } x +\tan \left (y\right ) = 0 \]

[_separable]

4811

\[ {}y^{\prime } x = y-x \tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4813

\[ {}y^{\prime } x = {\mathrm e}^{\frac {y}{x}} x +y \]

[[_homogeneous, ‘class A‘], _dAlembert]

4814

\[ {}y^{\prime } x = x +y+{\mathrm e}^{\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

4815

\[ {}y^{\prime } x = y \ln \left (y\right ) \]

[_separable]

4816

\[ {}y^{\prime } x = \left (1+\ln \left (x \right )-\ln \left (y\right )\right ) y \]

[[_homogeneous, ‘class A‘], _dAlembert]

4817

\[ {}y^{\prime } x +\left (1-\ln \left (x \right )-\ln \left (y\right )\right ) y = 0 \]

[[_homogeneous, ‘class G‘]]

4818

\[ {}y^{\prime } x = y-2 x \tanh \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

4836

\[ {}2 y^{\prime } x = 2 x^{3}-y \]

[_linear]

4838

\[ {}2 y^{\prime } x = y \left (1+y^{2}\right ) \]

[_separable]

4839

\[ {}2 y^{\prime } x +y \left (1+y^{2}\right ) = 0 \]

[_separable]

4841

\[ {}2 y^{\prime } x +4 y+a +\sqrt {a^{2}-4 b -4 c y} = 0 \]

[_separable]

4847

\[ {}3 y^{\prime } x = \left (2+x y^{3}\right ) y \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4853

\[ {}x^{2} y^{\prime } = a +b x y \]

[_linear]

4857

\[ {}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

4860

\[ {}x^{2} y^{\prime } = \left (x +a y\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4861

\[ {}x^{2} y^{\prime } = \left (a x +b y\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4862

\[ {}x^{2} y^{\prime }+a \,x^{2}+b x y+c y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

4864

\[ {}x^{2} y^{\prime }+2+x y \left (4+x y\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

4866

\[ {}x^{2} y^{\prime } = a +b \,x^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]]

4868

\[ {}x^{2} y^{\prime } = a +b x y+c \,x^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

4871

\[ {}x^{2} y^{\prime } = 2 y \left (x -y^{2}\right ) \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4874

\[ {}x^{2} y^{\prime } = \left (a x +b y^{3}\right ) y \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4875

\[ {}x^{2} y^{\prime }+x y+\sqrt {y} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

4929

\[ {}2 x^{2} y^{\prime }+1+2 x y-x^{2} y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

4938

\[ {}a \,x^{2} y^{\prime } = x^{2}+y a x +b^{2} y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

4943

\[ {}x^{3} y^{\prime } = a +b \,x^{2} y \]

[_linear]

4945

\[ {}x^{3} y^{\prime } = x^{4}+y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

4946

\[ {}x^{3} y^{\prime } = y \left (y+x^{2}\right ) \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

4949

\[ {}x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

4951

\[ {}x^{3} y^{\prime } = \left (2 x^{2}+y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4965

\[ {}2 x^{3} y^{\prime } = y \left (x^{2}-y^{2}\right ) \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4966

\[ {}2 x^{3} y^{\prime } = \left (3 x^{2}+a y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4969

\[ {}x^{4} y^{\prime } = \left (x^{3}+y\right ) y \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

4978

\[ {}x^{5} y^{\prime } = 1-3 y x^{4} \]

[_linear]

5015

\[ {}y y^{\prime }+x = 0 \]

[_separable]

5018

\[ {}y y^{\prime }+a x +b y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5032

\[ {}\left (x +y\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5033

\[ {}\left (x -y\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5034

\[ {}\left (x +y\right ) y^{\prime }+x -y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5035

\[ {}\left (x +y\right ) y^{\prime } = x -y \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5039

\[ {}\left (x -y\right ) y^{\prime } = \left ({\mathrm e}^{-\frac {x}{y}}+1\right ) y \]

[[_homogeneous, ‘class A‘], _dAlembert]

5044

\[ {}\left (2 x +y\right ) y^{\prime }+x -2 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5051

\[ {}\left (4 x -y\right ) y^{\prime }+2 x -5 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5056

\[ {}\left (x^{2}-y\right ) y^{\prime } = 4 x y \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5060

\[ {}\left (x -2 y\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5061

\[ {}\left (x +2 y\right ) y^{\prime }+2 x -y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5062

\[ {}\left (x -2 y\right ) y^{\prime }+2 x +y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5081

\[ {}\left (x +4 y\right ) y^{\prime }+4 x -y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5088

\[ {}\left (7 x +5 y\right ) y^{\prime }+10 x +8 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5097

\[ {}\left (a x +b y\right ) y^{\prime }+x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

5098

\[ {}\left (a x +b y\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5099

\[ {}\left (a x +b y\right ) y^{\prime }+b x +a y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5100

\[ {}\left (a x +b y\right ) y^{\prime } = b x +a y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5101

\[ {}x y^{\prime } y+1+y^{2} = 0 \]

[_separable]

5102

\[ {}x y^{\prime } y = x +y^{2} \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

5103

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5104

\[ {}x y^{\prime } y+x^{4}-y^{2} = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

5106

\[ {}x y^{\prime } y = x^{2}-x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5107

\[ {}x y^{\prime } y+2 x^{2}-2 x y-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5108

\[ {}x y^{\prime } y = a +b y^{2} \]

[_separable]

5111

\[ {}x y^{\prime } y+x^{2} \operatorname {arccot}\left (\frac {y}{x}\right )-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5112

\[ {}x y^{\prime } y+x^{2} {\mathrm e}^{-\frac {2 y}{x}}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5113

\[ {}\left (x y+1\right ) y^{\prime }+y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5122

\[ {}x \left (x +y\right ) y^{\prime }+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5123

\[ {}x \left (x -y\right ) y^{\prime }+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5124

\[ {}x \left (x +y\right ) y^{\prime } = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5125

\[ {}x \left (x -y\right ) y^{\prime }+2 x^{2}+3 x y-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5126

\[ {}x \left (x +y\right ) y^{\prime }-y \left (x +y\right )+x \sqrt {x^{2}-y^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5128

\[ {}x \left (2 x +y\right ) y^{\prime } = x^{2}+x y-y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5129

\[ {}x \left (4 x -y\right ) y^{\prime }+4 x^{2}-6 x y-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5130

\[ {}x \left (x^{3}+y\right ) y^{\prime } = \left (x^{3}-y\right ) y \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5131

\[ {}x \left (2 x^{3}+y\right ) y^{\prime } = \left (2 x^{3}-y\right ) y \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5132

\[ {}x \left (2 x^{3}+y\right ) y^{\prime } = 6 y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5136

\[ {}2 x y^{\prime } y+a +y^{2} = 0 \]

[_separable]

5137

\[ {}2 x y^{\prime } y = a x +y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

5138

\[ {}x^{2}+y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

5139

\[ {}2 x y^{\prime } y = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5143

\[ {}x \left (x -2 y\right ) y^{\prime }+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5144

\[ {}x \left (x +2 y\right ) y^{\prime }+\left (2 x -y\right ) y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5145

\[ {}x \left (x -2 y\right ) y^{\prime }+\left (2 x -y\right ) y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5148

\[ {}2 x \left (2 x^{2}+y\right ) y^{\prime }+\left (12 x^{2}+y\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5150

\[ {}x \left (2 x +3 y\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5151

\[ {}x \left (2 x +3 y\right ) y^{\prime }+3 \left (x +y\right )^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5154

\[ {}a x y y^{\prime } = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5155

\[ {}a x y y^{\prime }+x^{2}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5156

\[ {}x \left (a +b y\right ) y^{\prime } = c y \]

[_separable]

5157

\[ {}x \left (x -a y\right ) y^{\prime } = y \left (y-a x \right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5161

\[ {}x \left (1-x y\right ) y^{\prime }+\left (x y+1\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5163

\[ {}x \left (2-x y\right ) y^{\prime }+2 y-x y^{2} \left (x y+1\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

5164

\[ {}x \left (3-x y\right ) y^{\prime } = y \left (x y-1\right ) \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5170

\[ {}x \left (1-2 x y\right ) y^{\prime }+y \left (1+2 x y\right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5171

\[ {}x \left (1+2 x y\right ) y^{\prime }+\left (2+3 x y\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5172

\[ {}x \left (1+2 x y\right ) y^{\prime }+\left (1+2 x y-x^{2} y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

5173

\[ {}x^{2} \left (x -2 y\right ) y^{\prime } = 2 x^{3}-4 x y^{2}+y^{3} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

5175

\[ {}3 x^{2} y y^{\prime }+1+2 x y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

5176

\[ {}x^{2} \left (4 x -3 y\right ) y^{\prime } = \left (6 x^{2}-3 x y+2 y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

5177

\[ {}\left (1-x^{3} y\right ) y^{\prime } = x^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5178

\[ {}2 x^{3} y y^{\prime }+a +3 x^{2} y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

5180

\[ {}x \left (3+2 x^{2} y\right ) y^{\prime }+\left (4+3 x^{2} y\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5181

\[ {}8 x^{3} y y^{\prime }+3 x^{4}-6 x^{2} y^{2}-y^{4} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5183

\[ {}3 x^{4} y y^{\prime } = 1-2 x^{3} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

5192

\[ {}x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5193

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5194

\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5195

\[ {}\left (x^{2}-y^{2}\right ) y^{\prime }+x \left (x +2 y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5196

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5201

\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5202

\[ {}\left (x^{4}+y^{2}\right ) y^{\prime } = 4 x^{3} y \]

[[_homogeneous, ‘class G‘], _rational]

5210

\[ {}\left (x^{2}+2 x y-y^{2}\right ) y^{\prime }+x^{2}-2 x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5211

\[ {}\left (x +y\right )^{2} y^{\prime } = x^{2}-2 x y+5 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5213

\[ {}\left (2 x^{2}+4 x y-y^{2}\right ) y^{\prime } = x^{2}-4 x y-2 y^{2} \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5214

\[ {}\left (3 x +y\right )^{2} y^{\prime } = 4 \left (3 x +2 y\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5219

\[ {}\left (2 x^{2}+3 y^{2}\right ) y^{\prime }+x \left (3 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5221

\[ {}\left (3 x^{2}+2 x y+4 y^{2}\right ) y^{\prime }+2 x^{2}+6 x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5225

\[ {}\left (x^{2}+a y^{2}\right ) y^{\prime } = x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5226

\[ {}\left (x^{2}+x y+a y^{2}\right ) y^{\prime } = a \,x^{2}+x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5227

\[ {}\left (a \,x^{2}+2 x y-a y^{2}\right ) y^{\prime }+x^{2}-2 y a x -y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5228

\[ {}\left (a \,x^{2}+2 b x y+c y^{2}\right ) y^{\prime }+k \,x^{2}+2 y a x +b y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5230

\[ {}x \left (3 x -y^{2}\right ) y^{\prime }+\left (5 x -2 y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5234

\[ {}x \left (2 x^{2}+y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5236

\[ {}x \left (a +y\right )^{2} y^{\prime } = b y^{2} \]

[_separable]

5237

\[ {}x \left (x^{2}-x y+y^{2}\right ) y^{\prime }+\left (x^{2}+x y+y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5238

\[ {}x \left (x^{2}-x y-y^{2}\right ) y^{\prime } = \left (x^{2}+x y-y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5239

\[ {}x \left (x^{2}+y a x +y^{2}\right ) y^{\prime } = \left (x^{2}+b x y+y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5240

\[ {}x \left (x^{2}-2 y^{2}\right ) y^{\prime } = \left (2 x^{2}-y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5241

\[ {}x \left (x^{2}+2 y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5242

\[ {}2 x \left (5 x^{2}+y^{2}\right ) y^{\prime } = x^{2} y-y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5243

\[ {}x \left (x^{2}+y a x +2 y^{2}\right ) y^{\prime } = \left (a x +2 y\right ) y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5244

\[ {}3 x y^{2} y^{\prime } = 2 x -y^{3} \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

5246

\[ {}x \left (x -3 y^{2}\right ) y^{\prime }+\left (2 x -y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

5249

\[ {}6 x y^{2} y^{\prime }+x +2 y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

5250

\[ {}x \left (6 y^{2}+x \right ) y^{\prime }+x y-3 y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5251

\[ {}x \left (x^{2}-6 y^{2}\right ) y^{\prime } = 4 \left (x^{2}+3 y^{2}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5252

\[ {}x \left (3 x -7 y^{2}\right ) y^{\prime }+\left (5 x -3 y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5254

\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = x y^{3} \]

[[_homogeneous, ‘class G‘], _rational]

5255

\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = \left (x y+1\right ) y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5256

\[ {}x \left (x y^{2}+1\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5257

\[ {}x \left (x y^{2}+1\right ) y^{\prime } = \left (2-3 x y^{2}\right ) y \]

[[_homogeneous, ‘class G‘], _rational]

5263

\[ {}x^{3} \left (1+y^{2}\right ) y^{\prime }+3 x^{2} y = 0 \]

[_separable]

5264

\[ {}x \left (1-x y\right )^{2} y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5265

\[ {}\left (1-x^{4} y^{2}\right ) y^{\prime } = x^{3} y^{3} \]

[[_homogeneous, ‘class G‘], _rational]

5267

\[ {}\left (x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5268

\[ {}\left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (a x +3 y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5272

\[ {}\left (3 x^{2}+y^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5274

\[ {}2 y^{3} y^{\prime } = x^{3}-x y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5276

\[ {}\left (3 x^{2}+2 y^{2}\right ) y y^{\prime }+x^{3} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5277

\[ {}\left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5279

\[ {}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5280

\[ {}\left (x^{3}+a y^{3}\right ) y^{\prime } = x^{2} y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5282

\[ {}x \left (x -y^{3}\right ) y^{\prime } = \left (3 x +y^{3}\right ) y \]

[[_homogeneous, ‘class G‘], _rational]

5283

\[ {}x \left (2 x^{3}+y^{3}\right ) y^{\prime } = \left (2 x^{3}-x^{2} y+y^{3}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5284

\[ {}x \left (2 x^{3}-y^{3}\right ) y^{\prime } = \left (x^{3}-2 y^{3}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5285

\[ {}x \left (x^{3}+3 x^{2} y+y^{3}\right ) y^{\prime } = \left (3 x^{2}+y^{2}\right ) y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5286

\[ {}x \left (x^{3}-2 y^{3}\right ) y^{\prime } = \left (2 x^{3}-y^{3}\right ) y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5287

\[ {}x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5295

\[ {}x \left (1-x y\right ) \left (1-x^{2} y^{2}\right ) y^{\prime }+\left (x y+1\right ) \left (1+x^{2} y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5296

\[ {}\left (x^{2}-y^{4}\right ) y^{\prime } = x y \]

[[_homogeneous, ‘class G‘], _rational]

5297

\[ {}\left (x^{3}-y^{4}\right ) y^{\prime } = 3 x^{2} y \]

[[_homogeneous, ‘class G‘], _rational]

5299

\[ {}2 \left (x -y^{4}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class G‘], _rational]

5301

\[ {}\left (a \,x^{3}+\left (a x +b y\right )^{3}\right ) y y^{\prime }+x \left (\left (a x +b y\right )^{3}+b y^{3}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5303

\[ {}2 x \left (x^{3}+y^{4}\right ) y^{\prime } = \left (x^{3}+2 y^{4}\right ) y \]

[[_homogeneous, ‘class G‘], _rational]

5304

\[ {}x \left (1-x^{2} y^{4}\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5305

\[ {}\left (x^{2}-y^{5}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class G‘], _rational]

5306

\[ {}x \left (x^{3}+y^{5}\right ) y^{\prime } = \left (x^{3}-y^{5}\right ) y \]

[[_homogeneous, ‘class G‘], _rational]

5313

\[ {}y^{\prime } \sqrt {y} = \sqrt {x} \]

[_separable]

5315

\[ {}y^{\prime } \sqrt {x y}+x -y = \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5316

\[ {}\left (x -2 \sqrt {x y}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _dAlembert]

5319

\[ {}\left (x -\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5321

\[ {}x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}} = 0 \]

[[_homogeneous, ‘class G‘], _dAlembert]

5322

\[ {}x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = x y^{2}-\left (x^{2}-y^{2}\right )^{{3}/{2}} \]

[[_1st_order, _with_linear_symmetries], _dAlembert]

5328

\[ {}x \left (x -y \tan \left (\frac {y}{x}\right )\right ) y^{\prime }+\left (x +y \tan \left (\frac {y}{x}\right )\right ) y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5336

\[ {}{y^{\prime }}^{2} = y+x^{2} \]

[[_homogeneous, ‘class G‘]]

5337

\[ {}{y^{\prime }}^{2}+x^{2} = 4 y \]

[[_homogeneous, ‘class G‘]]

5338

\[ {}{y^{\prime }}^{2}+3 x^{2} = 8 y \]

[[_homogeneous, ‘class G‘]]

5339

\[ {}{y^{\prime }}^{2}+a \,x^{2}+b y = 0 \]

[[_homogeneous, ‘class G‘]]

5384

\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b \,x^{2}+c y = 0 \]

[[_homogeneous, ‘class G‘]]

5387

\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5388

\[ {}{y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5389

\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 y x^{4} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5407

\[ {}{y^{\prime }}^{2}+x y^{2} y^{\prime }+y^{3} = 0 \]

[[_homogeneous, ‘class G‘]]

5408

\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5412

\[ {}{y^{\prime }}^{2}-3 x y^{{2}/{3}} y^{\prime }+9 y^{{5}/{3}} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5416

\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \]

[[_homogeneous, ‘class G‘]]

5419

\[ {}3 {y^{\prime }}^{2}+4 y^{\prime } x +x^{2}-y = 0 \]

[[_homogeneous, ‘class G‘]]

5425

\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5439

\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5441

\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \]

[[_homogeneous, ‘class G‘]]

5450

\[ {}x {y^{\prime }}^{2}-3 y y^{\prime }+9 x^{2} = 0 \]

[[_homogeneous, ‘class G‘]]

5468

\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime }-y^{4} = 0 \]

[[_homogeneous, ‘class G‘]]

5470

\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \]

[[_homogeneous, ‘class G‘]]

5484

\[ {}x^{2} {y^{\prime }}^{2}+x \left (x^{3}-2 y\right ) y^{\prime }-\left (2 x^{3}-y\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5486

\[ {}x^{2} {y^{\prime }}^{2}-3 x y^{\prime } y+x^{3}+2 y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5508

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \]

[[_homogeneous, ‘class G‘]]

5511

\[ {}x^{4} {y^{\prime }}^{2}-y^{\prime } x -y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5513

\[ {}x^{4} {y^{\prime }}^{2}+x y^{2} y^{\prime }-y^{3} = 0 \]

[[_homogeneous, ‘class G‘]]

5516

\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \]

[[_homogeneous, ‘class G‘]]

5517

\[ {}x^{6} {y^{\prime }}^{2}-2 y^{\prime } x -4 y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5518

\[ {}x^{8} {y^{\prime }}^{2}+3 y^{\prime } x +9 y = 0 \]

[[_homogeneous, ‘class G‘]]

5526

\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5535

\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5549

\[ {}y^{2} {y^{\prime }}^{2}-3 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

5550

\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5571

\[ {}9 y^{2} {y^{\prime }}^{2}-3 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

5576

\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+a^{2} x = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5578

\[ {}2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-a = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5580

\[ {}4 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

5581

\[ {}3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }+1 = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5582

\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5602

\[ {}{y^{\prime }}^{3}-a x y y^{\prime }+2 a y^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5603

\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5620

\[ {}3 {y^{\prime }}^{3}-x^{4} y^{\prime }+2 x^{3} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5625

\[ {}x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

5632

\[ {}2 x^{3} {y^{\prime }}^{3}+6 x^{2} y {y^{\prime }}^{2}-\left (1-6 x y\right ) y y^{\prime }+2 y^{3} = 0 \]

[[_homogeneous, ‘class G‘]]

5633

\[ {}x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3} = 1 \]

[[_1st_order, _with_linear_symmetries]]

5634

\[ {}x^{6} {y^{\prime }}^{3}-y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5638

\[ {}y^{2} {y^{\prime }}^{3}-y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5639

\[ {}y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5640

\[ {}4 y^{2} {y^{\prime }}^{3}-2 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5641

\[ {}16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5644

\[ {}y^{4} {y^{\prime }}^{3}-6 y^{\prime } x +2 y = 0 \]

[[_1st_order, _with_linear_symmetries]]

5662

\[ {}2 \left (1+y\right )^{{3}/{2}}+3 y^{\prime } x -3 y = 0 \]

[_separable]

5689

\[ {}y^{\prime } = \frac {x y}{x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5694

\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5695

\[ {}y+x y^{2}-y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

5705

\[ {}\left (y-x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5706

\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5707

\[ {}y^{\prime } x -y-\sqrt {x^{2}+y^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5708

\[ {}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5709

\[ {}\left (7 x +5 y\right ) y^{\prime }+10 x +8 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5733

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5734

\[ {}\left (7 x +5 y\right ) y^{\prime }+10 x +8 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5735

\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5736

\[ {}y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5738

\[ {}\left (x^{2} y^{2}+x y\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5739

\[ {}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5763

\[ {}y = y^{\prime } x +x \sqrt {1+{y^{\prime }}^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5771

\[ {}2 x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5772

\[ {}\left (x +\sqrt {y^{2}-x y}\right ) y^{\prime }-y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5773

\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5774

\[ {}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5775

\[ {}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5776

\[ {}y^{2}+\left (x \sqrt {y^{2}-x^{2}}-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _dAlembert]

5777

\[ {}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5778

\[ {}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5779

\[ {}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5780

\[ {}{\mathrm e}^{\frac {y}{x}} x -y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5781

\[ {}x^{2}+y^{2} = 2 x y^{\prime } y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5782

\[ {}{\mathrm e}^{\frac {y}{x}} x +y = y^{\prime } x \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

5783

\[ {}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

5784

\[ {}x y-y^{2}-x^{2} y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5819

\[ {}y \left (2 x^{2} y^{3}+3\right )+x \left (x^{2} y^{3}-1\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5839

\[ {}y^{\prime } x +y = x^{3} \]

[_linear]

5855

\[ {}y^{\prime } x +x y^{2}-y = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

5859

\[ {}y^{\prime }+\frac {y}{x} = \frac {y^{2}}{x} \]
i.c.

[_separable]

5864

\[ {}y^{\prime } = \frac {1}{x^{2}}-\frac {y}{x}-y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

5865

\[ {}y^{\prime } = 1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

5872

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5874

\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5876

\[ {}x^{2} y+y^{2}+x^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

5886

\[ {}y^{\prime } x -y^{2}+1 = 0 \]

[_separable]

5888

\[ {}y^{\prime } x = x +y+{\mathrm e}^{\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

5890

\[ {}y^{\prime } x -y \left (\ln \left (x y\right )-1\right ) = 0 \]

[[_homogeneous, ‘class G‘]]

5891

\[ {}x^{3} y^{\prime }-y^{2}-x^{2} y = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

5893

\[ {}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

5894

\[ {}y^{2}-3 x y-2 x^{2}+\left (x y-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5896

\[ {}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

5903

\[ {}y^{2}+12 x^{2} y+\left (2 x y+4 x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5905

\[ {}\left (x^{2}-y\right ) y^{\prime }-4 x y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5906

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5907

\[ {}2 x y^{\prime } y+3 x^{2}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

5908

\[ {}\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5909

\[ {}\left (x y-1\right )^{2} x y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

5910

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

5911

\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

5912

\[ {}2 y^{3} y^{\prime }+x y^{2}-x^{3} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6019

\[ {}-a y^{3}-\frac {b}{x^{{3}/{2}}}+y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Abel]

6020

\[ {}a x y^{3}+b y^{2}+y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _Abel]

6032

\[ {}y^{\prime } = a x y^{2} \]

[_separable]

6039

\[ {}a x y^{\prime }+2 y = x y^{\prime } y \]

[_separable]

6096

\[ {}x y^{\prime } y+1+y^{2} = 0 \]
i.c.

[_separable]

6100

\[ {}y^{\prime }+2 x y^{2} = 0 \]
i.c.

[_separable]

6104

\[ {}\left (x +x y\right ) y^{\prime }+y = 0 \]
i.c.

[_separable]

6120

\[ {}y^{\prime }+\frac {y}{x} = 2 x^{{3}/{2}} \sqrt {y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6121

\[ {}3 x y^{2} y^{\prime }+3 y^{3} = 1 \]

[_separable]

6125

\[ {}x^{2} y^{\prime }+y^{2}-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6126

\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6127

\[ {}x y+\left (y^{2}-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6128

\[ {}y^{2}-x y+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6130

\[ {}y^{\prime } = \frac {y}{x}-\tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

6132

\[ {}y^{\prime } = x y^{2}-\frac {2 y}{x}-\frac {1}{x^{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

6208

\[ {}x^{2} y^{\prime }-x y = \frac {1}{x} \]

[_linear]

6214

\[ {}3 x^{3} y^{2} y^{\prime }-x^{2} y^{3} = 1 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6224

\[ {}\left (2 x +y\right ) y^{\prime }-x +2 y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6233

\[ {}y^{\prime } x -y = x^{2} \]
i.c.

[_linear]

6262

\[ {}y^{\prime } x = \frac {1}{y^{3}} \]

[_separable]

6266

\[ {}x v^{\prime } = \frac {1-4 v^{2}}{3 v} \]

[_separable]

6277

\[ {}x^{2}+2 y y^{\prime } = 0 \]
i.c.

[_separable]

6289

\[ {}y^{\prime } = x y^{3} \]

[_separable]

6303

\[ {}y^{\prime } x +2 y = \frac {1}{x^{3}} \]

[_linear]

6306

\[ {}y x^{\prime }+2 x = 5 y^{3} \]

[_linear]

6319

\[ {}y^{\prime }+\frac {3 y}{x} = x^{2} \]

[_linear]

6323

\[ {}x^{{10}/{3}}-2 y+y^{\prime } x = 0 \]

[_linear]

6344

\[ {}t^{3} y^{2}+\frac {t^{4} y^{\prime }}{y^{6}} = 0 \]

[_separable]

6406

\[ {}3 y^{\prime } x +y+x^{2} y^{4} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6415

\[ {}y^{\prime }-\frac {2 y}{x}-x^{2} = 0 \]

[_linear]

6416

\[ {}y^{\prime }+\frac {2 y}{x}-x^{3} = 0 \]

[_linear]

6421

\[ {}y^{\prime } x -y = x^{2} \]

[_linear]

6423

\[ {}x \cos \left (y\right ) y^{\prime }-\sin \left (y\right ) = 0 \]

[_separable]

6424

\[ {}\left (x y^{2}+x^{3}\right ) y^{\prime } = 2 y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6428

\[ {}y^{\prime }+\frac {y}{x} = y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6429

\[ {}y^{\prime } x +3 y = x^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6430

\[ {}x \left (y-3\right ) y^{\prime } = 4 y \]

[_separable]

6435

\[ {}\left (-x +2 y\right ) y^{\prime } = 2 x +y \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6436

\[ {}x y+y^{2}+\left (x^{2}-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6437

\[ {}x^{3}+y^{3} = 3 x y^{2} y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6438

\[ {}y-3 x +\left (3 x +4 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6439

\[ {}\left (x^{3}+3 x y^{2}\right ) y^{\prime } = y^{3}+3 x^{2} y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6449

\[ {}\left (x y+1\right ) y+x \left (1+x y+x^{2} y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

6457

\[ {}x^{2}-2 x y+5 y^{2} = \left (x^{2}+2 x y+y^{2}\right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6461

\[ {}y^{\prime } = \frac {y^{2}+2 x y}{x^{2}+2 x y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6464

\[ {}x^{2} y^{\prime } = y^{2}-x y^{\prime } y \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6467

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6468

\[ {}2 x y^{\prime } y = x^{2}-y^{2} \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

6478

\[ {}y^{\prime }+\frac {y}{x} = x y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6533

\[ {}y^{\prime }+\frac {4 y}{x} = x^{4} \]

[_linear]

6542

\[ {}y^{\prime }-\frac {y}{x} = x^{2} \]

[_linear]

6570

\[ {}y y^{\prime }+x = 0 \]

[_separable]

6572

\[ {}2 x^{3} y^{\prime } = y \left (3 x^{2}+y^{2}\right ) \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6581

\[ {}y^{2}-x^{2} y^{\prime } = 0 \]

[_separable]

6583

\[ {}x y^{2}+y+\left (x^{2} y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6584

\[ {}x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

6586

\[ {}y \sqrt {x^{2}+y^{2}}-x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _dAlembert]

6590

\[ {}x +2 y+\left (2 x +3 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6591

\[ {}2 y^{\prime } x -2 y = \sqrt {x^{2}+4 y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6594

\[ {}y^{2}-x^{2}+x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6595

\[ {}y \left (1+2 x y\right )+x \left (1-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

6597

\[ {}x^{3}+y^{3}+3 x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

6600

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6615

\[ {}y \left (x -2 y\right )-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6616

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

6617

\[ {}x^{2}+y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

6648

\[ {}y^{\prime } x +y-x^{3} y^{6} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6657

\[ {}2 y^{5} x -y+2 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

6669

\[ {}3 x^{4} {y^{\prime }}^{2}-y^{\prime } x -y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

6671

\[ {}y^{2} {y^{\prime }}^{2}+3 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

6673

\[ {}16 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

6676

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

6682

\[ {}y^{2} {y^{\prime }}^{2}+3 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

6686

\[ {}y = -y^{\prime } x +x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

6689

\[ {}{y^{\prime }}^{3}-4 x^{4} y^{\prime }+8 x^{3} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

6885

\[ {}y^{2}-1+y^{\prime } x = 0 \]

[_separable]

6892

\[ {}y^{\prime } = 2 x y^{2} \]

[_separable]

6895

\[ {}2 x y+\left (x^{2}-y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6905

\[ {}y^{\prime } = -\frac {x}{y} \]

[_separable]

6914

\[ {}3 y^{\prime } x +5 y = 10 \]

[_separable]

6935

\[ {}y^{\prime }+2 x y^{2} = 0 \]
i.c.

[_separable]

6936

\[ {}y^{\prime }+2 x y^{2} = 0 \]
i.c.

[_separable]

6937

\[ {}y^{\prime }+2 x y^{2} = 0 \]
i.c.

[_separable]

6938

\[ {}y^{\prime }+2 x y^{2} = 0 \]
i.c.

[_separable]

6950

\[ {}y^{\prime } = \sqrt {x y} \]

[[_homogeneous, ‘class G‘]]

6955

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

6956

\[ {}\left (y-x \right ) y^{\prime } = x +y \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

6968

\[ {}y y^{\prime } = 3 x \]
i.c.

[_separable]

6969

\[ {}y y^{\prime } = 3 x \]
i.c.

[_separable]

6970

\[ {}y y^{\prime } = 3 x \]
i.c.

[_separable]

6981

\[ {}y^{\prime } = x \sqrt {y} \]
i.c.

[_separable]

6992

\[ {}y^{\prime } x +y = 2 x \]
i.c.

[_linear]

7004

\[ {}y^{\prime } x +y = \frac {1}{y^{2}} \]

[_separable]

7007

\[ {}\left (1-x y\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7034

\[ {}y y^{\prime } = -x \]
i.c.

[_separable]

7035

\[ {}y y^{\prime } = -x \]
i.c.

[_separable]

7044

\[ {}y^{\prime } = 1-\frac {y}{x} \]
i.c.

[_linear]

7045

\[ {}y^{\prime } = 1-\frac {y}{x} \]
i.c.

[_linear]

7068

\[ {}y^{\prime }+2 x y^{2} = 0 \]

[_separable]

7092

\[ {}x \sinh \left (y\right ) y^{\prime } = \cosh \left (y\right ) \]
i.c.

[_separable]

7104

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

7105

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

7106

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

7107

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

7127

\[ {}y^{\prime } = -\frac {x}{y} \]
i.c.

[_separable]

7128

\[ {}y^{\prime } = x \sqrt {y} \]

[_separable]

7154

\[ {}y^{\prime } x +2 y = 3 \]

[_separable]

7159

\[ {}y-4 \left (x +y^{6}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

7382

\[ {}y^{\prime } = \frac {x^{2}}{y} \]

[_separable]

7385

\[ {}y^{\prime } x = \sqrt {1-y^{2}} \]

[_separable]

7387

\[ {}x y^{\prime } y = \sqrt {1+y^{2}} \]

[_separable]

7390

\[ {}y^{\prime } x +y = y^{2} \]
i.c.

[_separable]

7398

\[ {}\frac {1}{\sqrt {x}}+\frac {y^{\prime }}{\sqrt {y}} = 0 \]

[_separable]

7403

\[ {}y^{\prime } = \frac {\sqrt {y}}{\sqrt {x}} \]

[_separable]

7404

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]

[_separable]

7417

\[ {}\left (x +y\right ) y^{\prime }+x -y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7420

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7421

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7422

\[ {}y^{\prime } x -y = x \tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

7423

\[ {}y^{\prime } x = y-{\mathrm e}^{\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

7424

\[ {}y^{\prime } x -y = \left (x +y\right ) \ln \left (\frac {x +y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

7425

\[ {}y^{\prime } x = y \cos \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

7426

\[ {}y+\sqrt {x y}-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7427

\[ {}y^{\prime } x -\sqrt {x^{2}-y^{2}}-y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7428

\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7429

\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7430

\[ {}y^{\prime } x -y = y y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7431

\[ {}y^{2}+\left (x^{2}-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7432

\[ {}x^{2}+x y+y^{2} = x^{2} y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

7433

\[ {}\frac {1}{x^{2}-x y+y^{2}} = \frac {y^{\prime }}{2 y^{2}-x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7434

\[ {}y^{\prime } = \frac {2 x y}{3 x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7435

\[ {}y^{\prime } = \frac {x}{y}+\frac {y}{x} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7436

\[ {}y^{\prime } x = y+\sqrt {y^{2}-x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7437

\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

7438

\[ {}y^{\prime } x = y \ln \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

7442

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7444

\[ {}y^{\prime }+\frac {x +2 y}{x} = 0 \]

[_linear]

7445

\[ {}y^{\prime } = \frac {y}{x +y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7446

\[ {}y^{\prime } x = x +\frac {y}{2} \]
i.c.

[_linear]

7460

\[ {}2 y^{\prime } x +\left (x^{2} y^{4}+1\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7461

\[ {}2 x \left (x -y^{2}\right ) y^{\prime }+y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

7462

\[ {}x^{3} \left (y^{\prime }-x \right ) = y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

7463

\[ {}2 x^{2} y^{\prime } = y^{3}+x y \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7464

\[ {}y+x \left (1+2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7465

\[ {}2 y^{\prime }+x = 4 \sqrt {y} \]

[[_1st_order, _with_linear_symmetries], _Chini]

7466

\[ {}y^{\prime } = y^{2}-\frac {2}{x^{2}} \]

[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]]

7467

\[ {}2 y^{\prime } x +y = y^{2} \sqrt {x -x^{2} y^{2}} \]

[[_homogeneous, ‘class G‘]]

7468

\[ {}\frac {2 x y^{\prime } y}{3} = \sqrt {x^{6}-y^{4}}+y^{2} \]

[[_homogeneous, ‘class G‘]]

7469

\[ {}2 y+\left (x^{2} y+1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7470

\[ {}x \left (1-x y\right ) y^{\prime }+\left (x y+1\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7471

\[ {}\left (1+x^{2} y^{2}\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

7472

\[ {}\left (x^{2}-y^{4}\right ) y^{\prime }-x y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

7473

\[ {}y \left (1+\sqrt {x^{2} y^{4}-1}\right )+2 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘]]

7477

\[ {}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7505

\[ {}y^{\prime } x -2 \sqrt {x y} = y \]

[[_homogeneous, ‘class A‘], _dAlembert]

7509

\[ {}x^{2} y^{\prime }+y^{2}-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7510

\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7511

\[ {}y^{\prime } = \frac {y}{2 x}+\frac {x^{2}}{2 y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7512

\[ {}y^{\prime } = -\frac {2}{t}+\frac {y}{t}+\frac {y^{2}}{t} \]

[_separable]

7543

\[ {}y+\sqrt {x^{2}+y^{2}}-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7547

\[ {}\left (1+x^{2} y^{2}\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

7560

\[ {}x^{2}+y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7561

\[ {}x^{2}-y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7564

\[ {}x +y y^{\prime }+y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7607

\[ {}x^{2} y^{\prime }+2 x y = 1 \]

[_linear]

7732

\[ {}y y^{\prime } = x \]

[_separable]

7739

\[ {}y^{\prime } = \frac {x +y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7740

\[ {}y^{\prime } = \frac {y^{2}}{x^{2}+x y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7741

\[ {}y^{\prime } = \frac {x^{2}+x y+y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

7742

\[ {}y^{\prime } = \frac {y+x \,{\mathrm e}^{-\frac {2 y}{x}}}{x} \]

[[_homogeneous, ‘class A‘], _dAlembert]

7781

\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7782

\[ {}2 x y^{\prime } y = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7783

\[ {}y^{\prime } x +y = x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

7784

\[ {}y^{\prime } = \frac {y^{2}}{x y-x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7805

\[ {}y^{\prime } = \frac {2 x y^{2}}{1-x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7807

\[ {}x^{5} y^{\prime }+y^{5} = 0 \]

[_separable]

7811

\[ {}y \ln \left (y\right )-y^{\prime } x = 0 \]

[_separable]

7815

\[ {}x y^{\prime } y = -1+y \]

[_separable]

7816

\[ {}x y^{2}-x^{2} y^{\prime } = 0 \]

[_separable]

7821

\[ {}y^{\prime } = x^{2} y^{2} \]
i.c.

[_separable]

7841

\[ {}y^{\prime } x +y = x^{4} y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7843

\[ {}y^{\prime } x +y = x y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7850

\[ {}\left (x +\frac {2}{y}\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7871

\[ {}x^{2}-2 y^{2}+x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7872

\[ {}x^{2} y^{\prime }-3 x y-2 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7873

\[ {}x^{2} y^{\prime } = 3 \left (x^{2}+y^{2}\right ) \arctan \left (\frac {y}{x}\right )+x y \]

[[_homogeneous, ‘class A‘], _dAlembert]

7874

\[ {}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )+x \]

[[_homogeneous, ‘class A‘], _dAlembert]

7875

\[ {}y^{\prime } x = y+2 \,{\mathrm e}^{-\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

7876

\[ {}x -y-\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7877

\[ {}y^{\prime } x = 2 x -6 y \]

[_linear]

7878

\[ {}y^{\prime } x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7879

\[ {}x^{2} y^{\prime } = y^{2}+2 x y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7880

\[ {}x^{3}+y^{3}-x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7886

\[ {}y^{\prime } = \frac {1-x y^{2}}{2 x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7887

\[ {}y^{\prime } = \frac {2+3 x y^{2}}{4 x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

7888

\[ {}y^{\prime } = \frac {y-x y^{2}}{x +x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

7889

\[ {}y^{\prime } = \sin \left (\frac {y}{x}\right )-\cos \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

7890

\[ {}{\mathrm e}^{\frac {x}{y}}-\frac {y y^{\prime }}{x} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

7891

\[ {}y^{\prime } = \frac {x^{2}-x y}{y^{2} \cos \left (\frac {x}{y}\right )} \]

[[_homogeneous, ‘class A‘], _dAlembert]

7892

\[ {}y^{\prime } = \frac {y \tan \left (\frac {y}{x}\right )}{x} \]

[[_homogeneous, ‘class A‘], _dAlembert]

7904

\[ {}y^{\prime } = \frac {2 y}{x}+\frac {x^{3}}{y}+x \tan \left (\frac {y}{x^{2}}\right ) \]

[[_homogeneous, ‘class G‘]]

7917

\[ {}y^{\prime } x +y = x \]

[_linear]

7921

\[ {}y^{\prime } = \frac {x^{2}+y^{2}}{x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7922

\[ {}y^{\prime } = \frac {x +2 y}{2 x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7927

\[ {}y^{2} y^{\prime } = x \]
i.c.

[_separable]

7929

\[ {}y^{\prime } = \frac {x +y}{x -y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

7930

\[ {}y^{\prime } = \frac {x^{2}+2 y^{2}}{x^{2}-2 y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

7931

\[ {}2 x \cos \left (y\right )-x^{2} \sin \left (y\right ) y^{\prime } = 0 \]
i.c.

[_separable]

8089

\[ {}y^{\prime }-\frac {y}{x} = x^{2} \]

[_linear]

8090

\[ {}y^{\prime }+\frac {y}{x} = x \]

[_linear]

8455

\[ {}3 x^{4} {y^{\prime }}^{2}-y^{\prime } x -y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

8458

\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 y x^{4} = 0 \]

[[_1st_order, _with_linear_symmetries]]

8459

\[ {}4 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

8460

\[ {}4 y^{3} {y^{\prime }}^{2}+4 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

8462

\[ {}y^{4} {y^{\prime }}^{3}-6 y^{\prime } x +2 y = 0 \]

[[_1st_order, _with_linear_symmetries]]

8463

\[ {}{y^{\prime }}^{2}+x^{3} y^{\prime }-2 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

8464

\[ {}{y^{\prime }}^{2}+4 x^{5} y^{\prime }-12 y x^{4} = 0 \]

[[_1st_order, _with_linear_symmetries]]

8465

\[ {}2 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+x^{4} = 0 \]

[[_1st_order, _with_linear_symmetries]]

8468

\[ {}x^{8} {y^{\prime }}^{2}+3 y^{\prime } x +9 y = 0 \]

[[_homogeneous, ‘class G‘]]

8471

\[ {}3 x^{4} {y^{\prime }}^{2}-y^{\prime } x -y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

8474

\[ {}x^{6} {y^{\prime }}^{3}-3 y^{\prime } x -3 y = 0 \]

[[_1st_order, _with_linear_symmetries]]

8475

\[ {}y = x^{6} {y^{\prime }}^{3}-y^{\prime } x \]

[[_1st_order, _with_linear_symmetries]]

8533

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+4 = 0 \]

[[_homogeneous, ‘class G‘], _rational]

8535

\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

8536

\[ {}4 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries], _rational]

8537

\[ {}x^{6} {y^{\prime }}^{2}-2 y^{\prime } x -4 y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

8540

\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \]

[[_homogeneous, ‘class G‘]]

8541

\[ {}4 y^{2} {y^{\prime }}^{3}-2 y^{\prime } x +y = 0 \]

[[_1st_order, _with_linear_symmetries]]

8544

\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \]

[[_homogeneous, ‘class G‘]]

8547

\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-1 = 0 \]

[[_homogeneous, ‘class G‘], _rational]

8549

\[ {}x^{6} {y^{\prime }}^{2} = 8 y^{\prime } x +16 y \]

[[_homogeneous, ‘class G‘]]

8556

\[ {}x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}+4 x^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

8699

\[ {}y^{\prime }+\frac {2 y}{x} = 5 x^{2} \]

[_linear]

8701

\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

8702

\[ {}y^{\prime }+\frac {2 y}{x} = 6 x^{4} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

8728

\[ {}y^{\prime } = \frac {-x y-1}{4 x^{3} y-2 x^{2}} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

8733

\[ {}y^{\prime } = \sqrt {y}+x \]

[[_1st_order, _with_linear_symmetries], _Chini]

8734

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

8742

\[ {}y^{\prime } = \frac {5 x^{2}-x y+y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

8746

\[ {}y^{2}+\frac {2}{x}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

8751

\[ {}y y^{\prime }-y = x \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

8886

\[ {}y^{\prime } = {\mathrm e}^{-\frac {y}{x}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

10052

\[ {}-a y^{3}-\frac {b}{x^{{3}/{2}}}+y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Abel]

10055

\[ {}a x y^{3}+b y^{2}+y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _Abel]

10071

\[ {}y^{\prime }-a \sqrt {y}-b x = 0 \]

[[_homogeneous, ‘class G‘], _Chini]

10108

\[ {}y^{\prime } x -y^{2}+1 = 0 \]

[_separable]

10113

\[ {}y^{\prime } x +x y^{2}-y = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

10124

\[ {}y^{\prime } x -y-\sqrt {x^{2}+y^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10125

\[ {}y^{\prime } x +a \sqrt {x^{2}+y^{2}}-y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10128

\[ {}y^{\prime } x -{\mathrm e}^{\frac {y}{x}} x -y-x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10129

\[ {}y^{\prime } x -y \ln \left (y\right ) = 0 \]

[_separable]

10130

\[ {}y^{\prime } x -y \left (\ln \left (x y\right )-1\right ) = 0 \]

[[_homogeneous, ‘class G‘]]

10134

\[ {}y^{\prime } x -y-x \sin \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10135

\[ {}y^{\prime } x +x -y+x \cos \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10136

\[ {}y^{\prime } x +x \tan \left (\frac {y}{x}\right )-y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10137

\[ {}y^{\prime } x -y f \left (x y\right ) = 0 \]

[[_homogeneous, ‘class G‘]]

10141

\[ {}2 y^{\prime } x -y-2 x^{3} = 0 \]

[_linear]

10147

\[ {}x^{2} y^{\prime }+x^{2}+x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

10148

\[ {}x^{2} y^{\prime }-y^{2}-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

10149

\[ {}x^{2} y^{\prime }-y^{2}-x y-x^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

10151

\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+4 x y+2 = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

10152

\[ {}x^{2} \left (y^{\prime }+y^{2}\right )+y a x +b = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

10154

\[ {}x^{2} \left (y^{\prime }+a y^{2}\right )-b = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]]

10178

\[ {}3 x^{2} y^{\prime }-7 y^{2}-3 x y-x^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

10181

\[ {}x^{3} y^{\prime }-y^{2}-x^{4} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

10182

\[ {}x^{3} y^{\prime }-y^{2}-x^{2} y = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

10183

\[ {}x^{3} y^{\prime }-x^{4} y^{2}+x^{2} y+20 = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

10214

\[ {}y y^{\prime }+a y+x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

10221

\[ {}y y^{\prime }-x \,{\mathrm e}^{\frac {x}{y}} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10228

\[ {}\left (y-x^{2}\right ) y^{\prime }+4 x y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

10232

\[ {}\left (-x +2 y\right ) y^{\prime }-y-2 x = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

10241

\[ {}x y^{\prime } y+x^{2}+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

10248

\[ {}y^{2}-3 x y-2 x^{2}+\left (x y-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10249

\[ {}2 x y^{\prime } y-y^{2}+a x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

10250

\[ {}2 x y^{\prime } y-y^{2}+a \,x^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

10251

\[ {}2 x y^{\prime } y+2 y^{2}+1 = 0 \]

[_separable]

10254

\[ {}\left (2 x y+4 x^{3}\right ) y^{\prime }+y^{2}+112 x^{2} y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10255

\[ {}x \left (2 x +3 y\right ) y^{\prime }+3 \left (x +y\right )^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10263

\[ {}x \left (x y-2\right ) y^{\prime }+x^{2} y^{3}+x y^{2}-2 y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

10264

\[ {}x \left (x y-3\right ) y^{\prime }+x y^{2}-y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10269

\[ {}\left (x +2 x^{2} y\right ) y^{\prime }-x^{2} y^{3}+2 x y^{2}+y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

10270

\[ {}\left (2 x^{2} y-x \right ) y^{\prime }-2 x y^{2}-y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10271

\[ {}\left (2 x^{2} y-x^{3}\right ) y^{\prime }+y^{3}-4 x y^{2}+2 x^{3} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

10273

\[ {}2 x \left (x^{3} y+1\right ) y^{\prime }+\left (3 x^{3} y-1\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

10279

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

10280

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10284

\[ {}\left (y^{2}-x^{2}\right ) y^{\prime }+2 x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10285

\[ {}\left (x^{4}+y^{2}\right ) y^{\prime }-4 x^{3} y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10289

\[ {}x^{2}+2 x y-y^{2}+\left (y^{2}+2 x y-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10292

\[ {}\left (x^{2}+4 y^{2}\right ) y^{\prime }-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10293

\[ {}\left (3 x^{2}+2 x y+4 y^{2}\right ) y^{\prime }+2 x^{2}+6 x y+y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

10298

\[ {}\left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

10301

\[ {}x \left (y^{2}-3 x \right ) y^{\prime }+2 y^{3}-5 x y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10303

\[ {}x \left (y^{2}+x y-x^{2}\right ) y^{\prime }-y^{3}+x y^{2}+x^{2} y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10305

\[ {}2 x \left (5 x^{2}+y^{2}\right ) y^{\prime }+y^{3}-x^{2} y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10306

\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

10307

\[ {}\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 x y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

10308

\[ {}6 x y^{2} y^{\prime }+x +2 y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

10309

\[ {}\left (x^{2}+6 x y^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10310

\[ {}\left (x^{2} y^{2}+x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10311

\[ {}\left (x y-1\right )^{2} x y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10312

\[ {}\left (10 x^{3} y^{2}+x^{2} y+2 x \right ) y^{\prime }+5 x^{2} y^{3}+x y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10314

\[ {}\left (y^{3}-x^{3}\right ) y^{\prime }-x^{2} y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10316

\[ {}2 y^{3} y^{\prime }+x y^{2} = 0 \]

[_separable]

10318

\[ {}\left (2 y^{3}+5 x^{2} y\right ) y^{\prime }+5 x y^{2}+x^{3} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

10319

\[ {}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

10323

\[ {}\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10333

\[ {}y \left (y^{3}-2 x^{3}\right ) y^{\prime }+\left (2 y^{3}-x^{3}\right ) x = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10334

\[ {}y \left (\left (b x +a y\right )^{3}+b \,x^{3}\right ) y^{\prime }+x \left (\left (b x +a y\right )^{3}+a y^{3}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10339

\[ {}\left (\sqrt {x y}-1\right ) x y^{\prime }-\left (\sqrt {x y}+1\right ) y = 0 \]

[[_homogeneous, ‘class G‘]]

10340

\[ {}\left (2 x^{{5}/{2}} y^{{3}/{2}}+x^{2} y-x \right ) y^{\prime }-x^{{3}/{2}} y^{{5}/{2}}+x y^{2}-y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10344

\[ {}\left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }-y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

10345

\[ {}\left (y \sqrt {x^{2}+y^{2}}+\left (y^{2}-x^{2}\right ) \sin \left (\alpha \right )-2 x y \cos \left (\alpha \right )\right ) y^{\prime }+x \sqrt {x^{2}+y^{2}}+2 x y \sin \left (\alpha \right )+\left (y^{2}-x^{2}\right ) \cos \left (\alpha \right ) = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10349

\[ {}x \left (3 \,{\mathrm e}^{x y}+2 \,{\mathrm e}^{-x y}\right ) \left (y^{\prime } x +y\right )+1 = 0 \]

[[_homogeneous, ‘class G‘]]

10356

\[ {}x y^{\prime } \cot \left (\frac {y}{x}\right )+2 x \sin \left (\frac {y}{x}\right )-y \cot \left (\frac {y}{x}\right ) = 0 \]

[[_homogeneous, ‘class A‘]]

10360

\[ {}x \cos \left (y\right ) y^{\prime }+\sin \left (y\right ) = 0 \]

[_separable]

10369

\[ {}\left (x^{2} y \sin \left (x y\right )-4 x \right ) y^{\prime }+x y^{2} \sin \left (x y\right )-y = 0 \]

[[_homogeneous, ‘class G‘]]

10370

\[ {}\left (y^{\prime } x -y\right ) \cos \left (\frac {y}{x}\right )^{2}+x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10371

\[ {}\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }-\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

10374

\[ {}{y^{\prime }}^{2}+a y+b \,x^{2} = 0 \]

[[_homogeneous, ‘class G‘]]

10388

\[ {}{y^{\prime }}^{2}+a x y^{\prime }+b y+c \,x^{2} = 0 \]

[[_homogeneous, ‘class G‘]]

10390

\[ {}{y^{\prime }}^{2}-2 x^{2} y^{\prime }+2 x y = 0 \]

[[_homogeneous, ‘class G‘]]

10391

\[ {}{y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10400

\[ {}{y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3} = 0 \]

[[_1st_order, _with_linear_symmetries]]

10401

\[ {}{y^{\prime }}^{2}-3 x y^{{2}/{3}} y^{\prime }+9 y^{{5}/{3}} = 0 \]

[[_1st_order, _with_linear_symmetries]]

10403

\[ {}2 {y^{\prime }}^{2}-2 x^{2} y^{\prime }+3 x y = 0 \]

[[_homogeneous, ‘class G‘]]

10405

\[ {}3 {y^{\prime }}^{2}+4 y^{\prime } x +x^{2}-y = 0 \]

[[_homogeneous, ‘class G‘]]

10407

\[ {}a {y^{\prime }}^{2}+b \,x^{2} y^{\prime }+c x y = 0 \]

[[_homogeneous, ‘class G‘]]

10416

\[ {}x {y^{\prime }}^{2}+y y^{\prime }-x^{2} = 0 \]

[[_homogeneous, ‘class G‘]]

10417

\[ {}x {y^{\prime }}^{2}+y y^{\prime }+x^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10418

\[ {}x {y^{\prime }}^{2}+y y^{\prime }-y^{4} = 0 \]

[[_homogeneous, ‘class G‘]]

10457

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a = 0 \]

[[_homogeneous, ‘class G‘]]

10459

\[ {}x^{4} {y^{\prime }}^{2}-y^{\prime } x -y = 0 \]

[[_homogeneous, ‘class G‘], _rational]

10471

\[ {}y {y^{\prime }}^{2}+x^{3} y^{\prime }-x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10477

\[ {}9 y {y^{\prime }}^{2}+4 x^{3} y^{\prime }-4 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10488

\[ {}y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10528

\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

10538

\[ {}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10539

\[ {}2 \left (y^{\prime } x +y\right )^{3}-y y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘]]

10542

\[ {}y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10543

\[ {}16 y^{2} {y^{\prime }}^{3}+2 y^{\prime } x -y = 0 \]

[[_1st_order, _with_linear_symmetries]]

10545

\[ {}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0 \]

[[_homogeneous, ‘class G‘]]

10558

\[ {}x \left (\sqrt {1+{y^{\prime }}^{2}}+y^{\prime }\right )-y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

10619

\[ {}y^{\prime } = \frac {1}{y+\sqrt {x}} \]

[[_homogeneous, ‘class G‘], [_Abel, ‘2nd type‘, ‘class C‘]]

10621

\[ {}y^{\prime } = \frac {x^{2}}{y+x^{{3}/{2}}} \]

[[_1st_order, _with_linear_symmetries], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

10622

\[ {}y^{\prime } = \frac {x^{{5}/{3}}}{y+x^{{4}/{3}}} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]

11927

\[ {}y^{\prime } = f \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

11940

\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b \]

[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]]

11976

\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

12728

\[ {}\frac {y^{2}-2 x^{2}}{x y^{2}-x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

12729

\[ {}\frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

12730

\[ {}y^{\prime } x +x +y = 0 \]

[_linear]

12736

\[ {}{\mathrm e}^{\frac {y}{x}} x +y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

12737

\[ {}2 x^{2} y+3 y^{3}-\left (x^{3}+2 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12738

\[ {}x^{2} y^{\prime }+y^{2}-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

12739

\[ {}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

12740

\[ {}y^{3}+x^{3} y^{\prime } = 0 \]

[_separable]

12741

\[ {}x +y \cos \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

12745

\[ {}y+2 x y^{2}-x^{2} y^{3}+2 x^{2} y y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

12746

\[ {}2 y+3 x y^{2}+\left (x +2 x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

12747

\[ {}y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

12758

\[ {}x^{4} y \left (3 y+2 y^{\prime } x \right )+x^{2} \left (4 y+3 y^{\prime } x \right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

12760

\[ {}2 x^{3} y-y^{2}-\left (2 x^{4}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

12761

\[ {}x^{2} y^{\prime }+y^{2}-x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

12763

\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

12764

\[ {}x^{2}+y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

12765

\[ {}x -y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

12767

\[ {}3 x^{2}+6 x y+3 y^{2}+\left (2 x^{2}+3 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

12771

\[ {}y^{2}-x^{2}+2 m x y+\left (m y^{2}-m \,x^{2}-2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12774

\[ {}x +y y^{\prime }+y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

12782

\[ {}\left (y-x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

12783

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12784

\[ {}y^{\prime } x -y = \sqrt {x^{2}-y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12785

\[ {}x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

12789

\[ {}x y^{2} \left (y^{\prime } x +3 y\right )-2 y+y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational]

12791

\[ {}5 x y-3 y^{3}+\left (3 x^{2}-7 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

12793

\[ {}y+x y^{2}-y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

12795

\[ {}3 x^{2} y+\left (x^{3}+x^{3} y^{2}\right ) y^{\prime } = 0 \]

[_separable]

12798

\[ {}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12801

\[ {}1+{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

12804

\[ {}\left (2 \sqrt {x y}-x \right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

12815

\[ {}y = -y^{\prime } x +x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

12820

\[ {}{y^{\prime }}^{3}-4 x y^{\prime } y+8 y^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

12825

\[ {}x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+x = 0 \]

[[_homogeneous, ‘class G‘], _rational]

12827

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

12832

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+1 = 0 \]

[[_homogeneous, ‘class G‘], _rational]

12839

\[ {}y = y^{\prime } x +\frac {y {y^{\prime }}^{2}}{x^{2}} \]

[[_1st_order, _with_linear_symmetries]]

12955

\[ {}x^{\prime } = -\frac {t}{x} \]

[_separable]

12999

\[ {}x^{\prime } = \frac {4 t^{2}+3 x^{2}}{2 t x} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13002

\[ {}y^{\prime } = \frac {y^{2}+2 t y}{t^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13010

\[ {}x^{\prime } = -\frac {2 x}{t}+t \]

[_linear]

13013

\[ {}x^{\prime } t = -x+t^{2} \]

[_linear]

13029

\[ {}x^{\prime } = \frac {2 x}{3 t}+\frac {2 t}{x} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13031

\[ {}x^{\prime } = -\frac {x}{t}+\frac {1}{t x^{2}} \]

[_separable]

13032

\[ {}t^{2} y^{\prime }+2 t y-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13038

\[ {}x+3 t x^{2} x^{\prime } = 0 \]

[_separable]

13039

\[ {}x^{2}-t^{2} x^{\prime } = 0 \]

[_separable]

13040

\[ {}t \cot \left (x\right ) x^{\prime } = -2 \]

[_separable]

13179

\[ {}x^{2}+y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

13180

\[ {}y^{\prime } x +y = x^{3} y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

13201

\[ {}3 x +2 y+\left (2 x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13208

\[ {}\frac {\left (2 s-1\right ) s^{\prime }}{t}+\frac {s-s^{2}}{t^{2}} = 0 \]

[_separable]

13216

\[ {}4 x +3 y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

13217

\[ {}y^{2}+2 x y-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13223

\[ {}\tan \left (\theta \right )+2 r \theta ^{\prime } = 0 \]

[_separable]

13227

\[ {}2 x y+3 y^{2}-\left (x^{2}+2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13228

\[ {}v^{3}+\left (u^{3}-u v^{2}\right ) v^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

13229

\[ {}x \tan \left (\frac {y}{x}\right )+y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

13230

\[ {}\left (2 s^{2}+2 s t +t^{2}\right ) s^{\prime }+s^{2}+2 s t -t^{2} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

13231

\[ {}x^{3}+y^{2} \sqrt {x^{2}+y^{2}}-x y \sqrt {x^{2}+y^{2}}\, y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

13232

\[ {}\sqrt {x +y}+\sqrt {x -y}+\left (\sqrt {x -y}-\sqrt {x +y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class C‘], _dAlembert]

13236

\[ {}x^{2}+3 y^{2}-2 x y^{\prime } y = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13237

\[ {}\left (4 x -y\right ) y^{\prime }+2 x -5 y = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13238

\[ {}3 x^{2}+9 x y+5 y^{2}-\left (6 x^{2}+4 x y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13239

\[ {}x +2 y+\left (2 x -y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13240

\[ {}3 x -y-\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13241

\[ {}x^{2}+2 y^{2}+\left (4 x y-y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

13242

\[ {}2 x^{2}+2 x y+y^{2}+\left (x^{2}+2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13243

\[ {}y^{\prime }+\frac {3 y}{x} = 6 x^{2} \]

[_linear]

13244

\[ {}x^{4} y^{\prime }+2 x^{3} y = 1 \]

[_linear]

13257

\[ {}y^{\prime }-\frac {y}{x} = -\frac {y^{2}}{x} \]

[_separable]

13258

\[ {}y^{\prime } x +y = -2 x^{6} y^{4} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

13261

\[ {}y^{\prime } x -2 y = 2 x^{4} \]
i.c.

[_linear]

13267

\[ {}y^{\prime }+\frac {y}{2 x} = \frac {x}{y^{3}} \]
i.c.

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

13281

\[ {}\left (3 x^{2} y^{2}-x \right ) y^{\prime }+2 x y^{3}-y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

13284

\[ {}3 x -5 y+\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13287

\[ {}2 x^{2}+x y+y^{2}+2 x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

13288

\[ {}y^{\prime } = \frac {4 x^{3} y^{2}-3 x^{2} y}{x^{3}-2 y x^{4}} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13290

\[ {}y^{\prime } = \frac {2 x -7 y}{3 y-8 x} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13291

\[ {}x y+x^{2} y^{\prime } = x y^{3} \]

[_separable]

13293

\[ {}y^{\prime } = \frac {2 x^{2}+y^{2}}{2 x y-x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13294

\[ {}x^{2}+y^{2}-2 x y^{\prime } y = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13298

\[ {}4 x y^{\prime } y = 1+y^{2} \]
i.c.

[_separable]

13299

\[ {}y^{\prime } = \frac {2 x +7 y}{2 x -2 y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13308

\[ {}4 x y^{2}+6 y+\left (5 x^{2} y+8 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13309

\[ {}8 x^{2} y^{3}-2 y^{4}+\left (5 x^{3} y^{2}-8 x y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13660

\[ {}y^{\prime }+\frac {y}{x} = x^{2} \]

[_linear]

13676

\[ {}x y+y^{2}+x^{2}-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

13677

\[ {}x^{\prime } = \frac {x^{2}+t \sqrt {x^{2}+t^{2}}}{t x} \]

[[_homogeneous, ‘class A‘], _dAlembert]

13779

\[ {}y^{\prime } x = y+\sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

13780

\[ {}y^{\prime } x +y = x^{3} \]

[_linear]

13781

\[ {}y-y^{\prime } x = x^{2} y y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

13786

\[ {}x \left (\ln \left (x \right )-\ln \left (y\right )\right ) y^{\prime }-y = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

13789

\[ {}x^{\prime } = {\mathrm e}^{\frac {x}{t}}+\frac {x}{t} \]

[[_homogeneous, ‘class A‘], _dAlembert]

13791

\[ {}y = y^{\prime } x +\frac {1}{y} \]

[_separable]

13793

\[ {}y^{\prime } = \frac {y}{y^{3}+x} \]

[[_homogeneous, ‘class G‘], _rational]

13806

\[ {}y \left (x -y\right )-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13808

\[ {}x^{\prime } = \frac {x}{t}+\frac {x^{2}}{t^{3}} \]
i.c.

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

13813

\[ {}y = x^{2}+2 y^{\prime } x +\frac {{y^{\prime }}^{2}}{2} \]

[[_homogeneous, ‘class G‘]]

13814

\[ {}y^{\prime }-\frac {3 y}{x}+x^{3} y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

13817

\[ {}3 y^{2}-x +2 y \left (y^{2}-3 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

13818

\[ {}y \left (x -y\right )-x^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

13824

\[ {}\left (y^{2}-x^{2}\right ) y^{\prime }+2 x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

13825

\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

13882

\[ {}y^{\prime } x +y = x y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

14104

\[ {}y-x +\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14105

\[ {}y^{\prime } x +x +y = 0 \]

[_linear]

14106

\[ {}x +y+\left (y-x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14107

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

14108

\[ {}\left (7 x +5 y\right ) y^{\prime }+10 x +8 y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14109

\[ {}2 \sqrt {s t}-s+t s^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

14111

\[ {}x y^{2} y^{\prime } = x^{3}+y^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

14112

\[ {}x \cos \left (\frac {y}{x}\right ) \left (y^{\prime } x +y\right ) = y \sin \left (\frac {y}{x}\right ) \left (y^{\prime } x -y\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

14116

\[ {}\frac {y-y^{\prime } x}{\sqrt {x^{2}+y^{2}}} = m \]

[[_homogeneous, ‘class A‘], _dAlembert]

14117

\[ {}\frac {y y^{\prime }+x}{\sqrt {x^{2}+y^{2}}} = m \]

[[_homogeneous, ‘class A‘], _exact, _dAlembert]

14118

\[ {}y+\frac {x}{y^{\prime }} = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

14119

\[ {}y y^{\prime } = \sqrt {x^{2}+y^{2}}-x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

14137

\[ {}\left (y^{3}-x \right ) y^{\prime } = y \]

[[_homogeneous, ‘class G‘], _exact, _rational]

14140

\[ {}\frac {x}{\left (x +y\right )^{2}}+\frac {\left (2 x +y\right ) y^{\prime }}{\left (x +y\right )^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

14141

\[ {}\frac {1}{x^{2}}+\frac {3 y^{2}}{x^{4}} = \frac {2 y y^{\prime }}{x^{3}} \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

14142

\[ {}\frac {x^{2} y^{\prime }}{\left (x -y\right )^{2}}-\frac {y^{2}}{\left (x -y\right )^{2}} = 0 \]

[_separable]

14153

\[ {}y^{\prime } = \frac {2 y}{x}-\sqrt {3} \]

[_linear]

14205

\[ {}\frac {x^{2} y^{\prime }}{\left (x -y\right )^{2}}-\frac {y^{2}}{\left (x -y\right )^{2}} = 0 \]

[_separable]

14209

\[ {}x \cos \left (\frac {y}{x}\right ) y^{\prime } = y \cos \left (\frac {y}{x}\right )-x \]

[[_homogeneous, ‘class A‘], _dAlembert]

14265

\[ {}y^{\prime } = x \sqrt {y} \]

[_separable]

14292

\[ {}y^{\prime } = \frac {x}{y} \]

[_separable]

14300

\[ {}y^{\prime } = \frac {2 x -y}{x +3 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14303

\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

14304

\[ {}y^{\prime } = \frac {1}{x y} \]

[_separable]

14307

\[ {}y^{\prime } = \frac {y}{y-x} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14308

\[ {}y^{\prime } = \frac {x}{y^{2}} \]

[_separable]

14309

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]

[_separable]

14311

\[ {}y^{\prime } = \left (x y\right )^{{1}/{3}} \]

[[_homogeneous, ‘class G‘]]

14313

\[ {}y^{\prime } = -\frac {y}{x}+y^{{1}/{4}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

14325

\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \]
i.c.

[[_1st_order, _with_linear_symmetries], _Clairaut]

14341

\[ {}y^{\prime } = \frac {2 x}{y} \]
i.c.

[_separable]

14347

\[ {}2 x y^{\prime } y+y^{2} = -1 \]

[_separable]

14349

\[ {}y^{\prime } = -\frac {y \left (2 x +y\right )}{x \left (x +2 y\right )} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

14350

\[ {}y^{\prime } = \frac {y^{2}}{1-x y} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

14358

\[ {}x -y y^{\prime } = 0 \]

[_separable]

14360

\[ {}y^{\prime } x +x^{2}-y = 0 \]

[_linear]

14377

\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \]
i.c.

[_separable]

14378

\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \]
i.c.

[_separable]

14379

\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \]
i.c.

[_separable]

14380

\[ {}y^{\prime } = -\frac {3 x^{2}}{2 y} \]
i.c.

[_separable]

14381

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]
i.c.

[_separable]

14382

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]
i.c.

[_separable]

14383

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]
i.c.

[_separable]

14384

\[ {}y^{\prime } = \frac {\sqrt {y}}{x} \]
i.c.

[_separable]

14388

\[ {}y^{\prime } = 3 x y^{{1}/{3}} \]
i.c.

[_separable]

14389

\[ {}y^{\prime } = 3 x y^{{1}/{3}} \]
i.c.

[_separable]

14393

\[ {}y^{\prime } = \frac {y}{y-x} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14394

\[ {}y^{\prime } = \frac {y}{y-x} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14395

\[ {}y^{\prime } = \frac {y}{y-x} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14396

\[ {}y^{\prime } = \frac {y}{y-x} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

14398

\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

14405

\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \]
i.c.

[[_1st_order, _with_linear_symmetries], _Clairaut]

14408

\[ {}y^{\prime } = -\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \]
i.c.

[[_1st_order, _with_linear_symmetries], _Clairaut]

14531

\[ {}y^{\prime } = t^{2} y^{2} \]

[_separable]

14538

\[ {}y^{\prime } = \frac {t}{y} \]

[_separable]

14540

\[ {}y^{\prime } = t y^{{1}/{3}} \]

[_separable]

14542

\[ {}y^{\prime } = \frac {2 y+1}{t} \]

[_separable]

14671

\[ {}y^{\prime } = -\frac {y}{t}+2 \]

[_linear]

14672

\[ {}y^{\prime } = \frac {3 y}{t}+t^{5} \]

[_linear]

14679

\[ {}y^{\prime } = -\frac {y}{t}+2 \]
i.c.

[_linear]

14681

\[ {}y^{\prime }-\frac {2 y}{t} = 2 t^{2} \]
i.c.

[_linear]

14724

\[ {}y^{\prime } = \frac {2 y+1}{t} \]

[_separable]

14912

\[ {}y y^{\prime } = 2 x \]

[_separable]

14956

\[ {}x^{2} y^{\prime }+x y^{2} = x \]

[_separable]

14974

\[ {}y^{\prime } = \frac {x}{y} \]

[_separable]

14976

\[ {}x y^{\prime } y = y^{2}+9 \]

[_separable]

14980

\[ {}y^{\prime } = \frac {x}{y} \]
i.c.

[_separable]

15000

\[ {}y^{\prime } = 3 x y^{3} \]

[_separable]

15008

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

15009

\[ {}y^{\prime } x = y^{2}-y \]
i.c.

[_separable]

15010

\[ {}y^{\prime } = \frac {-1+y^{2}}{x y} \]
i.c.

[_separable]

15026

\[ {}y^{\prime } x +3 y-10 x^{2} = 0 \]

[_linear]

15028

\[ {}y^{\prime } x = \sqrt {x}+3 y \]

[_linear]

15035

\[ {}y^{\prime } x +3 y = 20 x^{2} \]
i.c.

[_linear]

15045

\[ {}x^{2} y^{\prime }-x y = y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15046

\[ {}y^{\prime } = \frac {x}{y}+\frac {y}{x} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15047

\[ {}\cos \left (\frac {y}{x}\right ) \left (y^{\prime }-\frac {y}{x}\right ) = 1+\sin \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

15048

\[ {}y^{\prime } = \frac {x -y}{x +y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15050

\[ {}y^{\prime }-\frac {3 y}{x} = \frac {y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15052

\[ {}y^{\prime }-\frac {y}{x} = \frac {1}{y} \]
i.c.

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

15053

\[ {}y^{\prime } = \frac {y}{x}+\frac {x^{2}}{y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15055

\[ {}3 y^{\prime }+\frac {2 y}{x} = 4 \sqrt {y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

15058

\[ {}\left (x +y\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15059

\[ {}\left (2 x y+2 x^{2}\right ) y^{\prime } = x^{2}+2 x y+2 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15060

\[ {}y^{\prime }+\frac {y}{x} = x^{2} y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

15063

\[ {}y^{\prime } x -y = \sqrt {x^{2}+x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

15066

\[ {}y^{\prime }+2 x = 2 \sqrt {y+x^{2}} \]

[[_1st_order, _with_linear_symmetries], _Clairaut]

15068

\[ {}y^{\prime } = x \left (1+\frac {2 y}{x^{2}}+\frac {y^{2}}{x^{4}}\right ) \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

15069

\[ {}y^{\prime } = \frac {1}{y}-\frac {y}{2 x} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

15070

\[ {}{\mathrm e}^{x y^{2}-x^{2}} \left (y^{2}-2 x \right )+2 \,{\mathrm e}^{x y^{2}-x^{2}} x y y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

15071

\[ {}2 x y+y^{2}+\left (x^{2}+2 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15072

\[ {}2 x y^{3}+4 x^{3}+3 x^{2} y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

15075

\[ {}4 x^{3} y+\left (x^{4}-y^{4}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

15076

\[ {}1+\ln \left (x y\right )+\frac {x y^{\prime }}{y} = 0 \]

[[_homogeneous, ‘class G‘], _exact]

15077

\[ {}1+{\mathrm e}^{y}+x \,{\mathrm e}^{y} y^{\prime } = 0 \]

[_separable]

15079

\[ {}1+y^{4}+x y^{3} y^{\prime } = 0 \]

[_separable]

15080

\[ {}y+\left (y^{4}-3 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

15081

\[ {}\frac {2 y}{x}+\left (4 x^{2} y-3\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15083

\[ {}3 y+3 y^{2}+\left (2 x +4 x y\right ) y^{\prime } = 0 \]

[_separable]

15085

\[ {}2 y^{3}+\left (4 x^{3} y^{3}-3 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15086

\[ {}4 x y+\left (3 x^{2}+5 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15087

\[ {}6+12 x^{2} y^{2}+\left (7 x^{3} y+\frac {x}{y}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

15088

\[ {}y^{\prime } x = 2 y-6 x^{3} \]

[_linear]

15089

\[ {}y^{\prime } x = 2 y^{2}-6 y \]

[_separable]

15093

\[ {}x y^{\prime } y-y^{2} = \sqrt {x^{2} y^{2}+x^{4}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

15095

\[ {}4 x y-6+x^{2} y^{\prime } = 0 \]

[_linear]

15096

\[ {}x y^{2}-6+x^{2} y y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]

15097

\[ {}x^{3}+y^{3}+x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15098

\[ {}3 y-x^{3}+y^{\prime } x = 0 \]

[_linear]

15100

\[ {}3 x y^{3}-y+y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

15104

\[ {}y^{\prime } = \frac {1}{x y-3 x} \]

[_separable]

15109

\[ {}x y^{\prime } y = 2 x^{2}+2 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15111

\[ {}y^{\prime } = \frac {x +2 y}{2 x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15112

\[ {}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

15118

\[ {}x y^{\prime } y = x^{2}+x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15120

\[ {}x y^{3} y^{\prime } = y^{4}-x^{2} \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

15123

\[ {}x y^{2}+\left (x^{2} y+10 y^{4}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

15718

\[ {}2 x -y-y y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15731

\[ {}y^{\prime } = -\frac {x}{y} \]

[_separable]

15732

\[ {}3 y \left (t^{2}+y\right )+t \left (t^{2}+6 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

15733

\[ {}y^{\prime } = -\frac {2 y}{x}-3 \]

[_linear]

15762

\[ {}y^{\prime } = \frac {y^{2}+2 x y}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

15779

\[ {}2 x -3 y+\left (2 y-3 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15794

\[ {}\frac {y^{\prime }}{t} = \sqrt {y} \]
i.c.

[_separable]

15809

\[ {}y^{\prime } t +y = t^{3} \]
i.c.

[_linear]

15821

\[ {}y^{\prime } = -\frac {t}{y} \]
i.c.

[_separable]

15823

\[ {}y^{\prime } = \frac {x}{y^{2}} \]

[_separable]

15824

\[ {}\frac {1}{2 \sqrt {t}}+y^{2} y^{\prime } = 0 \]

[_separable]

15825

\[ {}y^{\prime } = \frac {\sqrt {y}}{x^{2}} \]

[_separable]

15868

\[ {}y^{\prime } = \sqrt {\frac {y}{t}} \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

15899

\[ {}y^{\prime } t +y = t^{2} \]

[_linear]

15900

\[ {}y^{\prime } t +y = t \]

[_linear]

15915

\[ {}y-\left (x +3 y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

15917

\[ {}p^{\prime } = t^{3}+\frac {p}{t} \]

[_linear]

15954

\[ {}\frac {t}{\sqrt {t^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {t^{2}+y^{2}}} = 0 \]

[_separable]

15957

\[ {}3 t y^{2}+y^{3} y^{\prime } = 0 \]

[_separable]

15960

\[ {}\ln \left (t y\right )+\frac {t y^{\prime }}{y} = 0 \]

[[_homogeneous, ‘class G‘], _exact]

15967

\[ {}-\frac {1}{y}+\left (\frac {t}{y^{2}}+3 y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

15968

\[ {}2 t y+\left (t^{2}+y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

15969

\[ {}2 t y^{3}+\left (1+3 t^{2} y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational]

15970

\[ {}\sin \left (y\right )^{2}+t \sin \left (2 y\right ) y^{\prime } = 0 \]

[_separable]

15971

\[ {}3 t^{2}+3 y^{2}+6 t y y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]

15980

\[ {}-\frac {y^{2} {\mathrm e}^{\frac {y}{t}}}{t^{2}}+1+{\mathrm e}^{\frac {y}{t}} \left (1+\frac {y}{t}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _dAlembert]

15981

\[ {}2 t \sin \left (\frac {y}{t}\right )-y \cos \left (\frac {y}{t}\right )+t \cos \left (\frac {y}{t}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _dAlembert]

15983

\[ {}1+\frac {y}{t^{2}}-\frac {y^{\prime }}{t} = 0 \]
i.c.

[_linear]

15997

\[ {}2 t y+y^{2}-t^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16005

\[ {}\frac {9 t}{5}+2 y+\left (2 t +2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16006

\[ {}2 t +\frac {19 y}{10}+\left (\frac {19 t}{10}+2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16013

\[ {}y^{\prime }-\frac {y}{t} = t y^{2} \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

16014

\[ {}y^{\prime }-\frac {y}{t} = \frac {y^{2}}{t^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16015

\[ {}y^{\prime }-\frac {y}{t} = \frac {y^{2}}{t} \]

[_separable]

16016

\[ {}y^{\prime }-\frac {y}{t} = t^{2} y^{{3}/{2}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

16017

\[ {}\cos \left (\frac {t}{t +y}\right )+{\mathrm e}^{\frac {2 y}{t}} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

16018

\[ {}y \ln \left (\frac {t}{y}\right )+\frac {t^{2} y^{\prime }}{t +y} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

16020

\[ {}\frac {2}{t}+\frac {1}{y}+\frac {t y^{\prime }}{y^{2}} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16023

\[ {}2 t +\left (y-3 t \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

16024

\[ {}2 y-3 t +y^{\prime } t = 0 \]

[_linear]

16025

\[ {}t y-y^{2}+t \left (t -3 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16026

\[ {}t^{2}+t y+y^{2}-t y y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16027

\[ {}t^{3}+y^{3}-t y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16028

\[ {}y^{\prime } = \frac {t +4 y}{4 t +y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16030

\[ {}y+\left (t +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16031

\[ {}2 t^{2}-7 t y+5 y^{2}+t y y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16032

\[ {}y+2 \sqrt {t^{2}+y^{2}}-y^{\prime } t = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

16033

\[ {}y^{2} = \left (t y-4 t^{2}\right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16034

\[ {}y-\left (3 \sqrt {t y}+t \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

16035

\[ {}\left (t^{2}-y^{2}\right ) y^{\prime }+y^{2}+t y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16036

\[ {}t y y^{\prime }-t^{2} {\mathrm e}^{-\frac {y}{t}}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

16037

\[ {}y^{\prime } = \frac {1}{\frac {2 y \,{\mathrm e}^{-\frac {t}{y}}}{t}+\frac {t}{y}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

16038

\[ {}t \left (\ln \left (t \right )-\ln \left (y\right )\right ) y^{\prime } = y \]

[[_homogeneous, ‘class A‘], _dAlembert]

16043

\[ {}y^{\prime } t -y-\sqrt {t^{2}+y^{2}} = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16044

\[ {}t^{3}+y^{2} \sqrt {t^{2}+y^{2}}-t y \sqrt {t^{2}+y^{2}}\, y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

16045

\[ {}y^{3}-t^{3}-t y^{2} y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16046

\[ {}t y^{3}-\left (t^{4}+y^{4}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16047

\[ {}y^{4}+\left (t^{4}-t y^{3}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16063

\[ {}t^{{1}/{3}} y^{{2}/{3}}+t +\left (t^{{2}/{3}} y^{{1}/{3}}+y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

16064

\[ {}y^{\prime } = \frac {y^{2}-t^{2}}{t y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16065

\[ {}y \sin \left (\frac {t}{y}\right )-\left (t +t \sin \left (\frac {t}{y}\right )\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _dAlembert]

16066

\[ {}y^{\prime } = \frac {2 t^{5}}{5 y^{2}} \]

[_separable]

16068

\[ {}y^{\prime }-\frac {y}{t} = \frac {y^{2}}{t} \]

[_separable]

16069

\[ {}y^{\prime } = \frac {{\mathrm e}^{8 y}}{t} \]

[_separable]

16075

\[ {}3 t +\left (t -4 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

16076

\[ {}y-t +\left (t +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16078

\[ {}y^{2}+\left (t^{2}+t y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16079

\[ {}r^{\prime } = \frac {r^{2}+t^{2}}{r t} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16080

\[ {}x^{\prime } = \frac {5 t x}{x^{2}+t^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16087

\[ {}x^{\prime }+\frac {x}{y} = y^{2} \]

[_linear]

16103

\[ {}y^{\prime } = t y^{3} \]
i.c.

[_separable]

16104

\[ {}y^{\prime } = \frac {t}{y^{3}} \]
i.c.

[_separable]

16594

\[ {}y^{\prime } = \frac {x}{y} \]

[_separable]

16597

\[ {}y^{\prime } = \sqrt {x^{2}-y}-x \]

[[_1st_order, _with_linear_symmetries], _dAlembert]

16618

\[ {}y^{\prime } = \frac {x +y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16631

\[ {}y^{\prime } x = 2 x -y \]
i.c.

[_linear]

16633

\[ {}x y^{\prime } y+1+y^{2} = 0 \]

[_separable]

16635

\[ {}1+y^{2} = y^{\prime } x \]

[_separable]

16639

\[ {}y \ln \left (y\right )+y^{\prime } x = 1 \]
i.c.

[_separable]

16666

\[ {}y^{\prime } x = y+x \cos \left (\frac {y}{x}\right )^{2} \]

[[_homogeneous, ‘class A‘], _dAlembert]

16668

\[ {}y^{\prime } x = y \left (\ln \left (y\right )-\ln \left (x \right )\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

16669

\[ {}x^{2} y^{\prime } = x^{2}-x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

16670

\[ {}y^{\prime } x = y+\sqrt {y^{2}-x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16671

\[ {}2 x^{2} y^{\prime } = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

16672

\[ {}4 x -3 y+\left (2 y-3 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16673

\[ {}y-x +\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16682

\[ {}2 x \left (x -y^{2}\right ) y^{\prime }+y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

16683

\[ {}4 y^{6}+x^{3} = 6 x y^{5} y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

16684

\[ {}y \left (1+\sqrt {x^{2} y^{4}+1}\right )+2 y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘]]

16685

\[ {}x +y^{3}+3 \left (y^{3}-x \right ) y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

16687

\[ {}x^{2}-y^{\prime } x = y \]
i.c.

[_linear]

16694

\[ {}\left (2 x -y^{2}\right ) y^{\prime } = 2 y \]

[[_homogeneous, ‘class G‘], _rational]

16706

\[ {}y^{\prime } x +y = 2 x \]

[_linear]

16710

\[ {}3 x y^{2} y^{\prime }-2 y^{3} = x^{3} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16723

\[ {}x \left (2 x^{2}+y^{2}\right )+y \left (x^{2}+2 y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

16733

\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

16735

\[ {}3 x^{2} y+y^{3}+\left (x^{3}+3 x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

16737

\[ {}x^{2}+y-y^{\prime } x = 0 \]

[_linear]

16738

\[ {}x +y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

16743

\[ {}3 y^{2}-x +\left (2 y^{3}-6 x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

16754

\[ {}{y^{\prime }}^{2}-4 y^{\prime } x +2 y+2 x^{2} = 0 \]

[[_homogeneous, ‘class G‘]]

16780

\[ {}x^{2} y^{\prime } = 1+x y+x^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

16783

\[ {}{y^{\prime }}^{3}-4 x y^{\prime } y+8 y^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

16786

\[ {}\left (y^{\prime } x +y\right )^{2}+3 x^{5} \left (y^{\prime } x -2 y\right ) = 0 \]

[[_homogeneous, ‘class G‘]]

16787

\[ {}y \left (y-2 y^{\prime } x \right )^{2} = 2 y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational]

16799

\[ {}x^{3}-3 x y^{2}+\left (y^{3}-3 x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

16800

\[ {}5 x y-4 y^{2}-6 x^{2}+\left (y^{2}-8 x y+\frac {5 x^{2}}{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

16806

\[ {}x y^{\prime } y-y^{2} = x^{4} \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

16807

\[ {}\frac {1}{x^{2}-x y+y^{2}} = \frac {y^{\prime }}{2 y^{2}-x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

16812

\[ {}x y^{2} y^{\prime }-y^{3} = \frac {x^{4}}{3} \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

16813

\[ {}1+{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _dAlembert]

16814

\[ {}x^{2}+y^{2}-x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

16816

\[ {}y+x y^{2}-y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

16822

\[ {}2 x^{5}+4 x^{3} y-2 x y^{2}+\left (y^{2}+2 x^{2} y-x^{4}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

16825

\[ {}x -y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

16831

\[ {}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

17229

\[ {}y^{\prime } = \frac {x^{4}}{y} \]

[_separable]

17234

\[ {}y^{\prime } x = \sqrt {1-y^{2}} \]

[_separable]

17239

\[ {}y^{\prime } = 4 \sqrt {x y} \]

[[_homogeneous, ‘class G‘]]

17244

\[ {}r^{\prime } = \frac {r^{2}}{\theta } \]
i.c.

[_separable]

17307

\[ {}y^{\prime } = \frac {t -y}{2 t +5 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17314

\[ {}y^{\prime } = -\frac {t}{2}+\frac {\sqrt {t^{2}+4 y}}{2} \]
i.c.

[[_1st_order, _with_linear_symmetries], _Clairaut]

17315

\[ {}y^{\prime } = -\frac {4 t}{y} \]
i.c.

[_separable]

17325

\[ {}2 x +4 y+\left (2 x -2 y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17328

\[ {}y^{\prime } = -\frac {4 x +2 y}{2 x +3 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17329

\[ {}y^{\prime } = -\frac {4 x -2 y}{2 x -3 y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17335

\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}} = 0 \]

[_separable]

17336

\[ {}2 x -y+\left (-x +2 y\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17349

\[ {}3 x y+y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

17352

\[ {}\frac {\left (3 x^{3}-x y^{2}\right ) y^{\prime }}{y^{3}+3 x^{2} y} = 1 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

17354

\[ {}y+\sqrt {x^{2}-y^{2}} = y^{\prime } x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

17355

\[ {}x y^{\prime } y = \left (x +y\right )^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

17356

\[ {}y^{\prime } = \frac {4 y-7 x}{5 x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17357

\[ {}y^{\prime } x -4 \sqrt {y^{2}-x^{2}} = y \]

[[_homogeneous, ‘class A‘], _dAlembert]

17358

\[ {}y^{\prime } = \frac {y^{4}+2 x y^{3}-3 x^{2} y^{2}-2 x^{3} y}{2 x^{2} y^{2}-2 x^{3} y-2 x^{4}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

17359

\[ {}\left (y+x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = y \,{\mathrm e}^{\frac {x}{y}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

17360

\[ {}x y^{\prime } y = x^{2}+y^{2} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

17361

\[ {}y^{\prime } = \frac {x +y}{x -y} \]
i.c.

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17362

\[ {}y^{\prime } t +y = t^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17364

\[ {}y^{\prime }+\frac {3 y}{t} = t^{2} y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17365

\[ {}t^{2} y^{\prime }+2 t y-y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17367

\[ {}3 y^{\prime } t +9 y = 2 t y^{{5}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17372

\[ {}\left (3 x-y \right ) x^{\prime }+9 y -2 x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17382

\[ {}x^{\prime } = \frac {2 x y +x^{2}}{3 y^{2}+2 x y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

17383

\[ {}4 x y^{\prime } y = 8 x^{2}+5 y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

17826

\[ {}y^{\prime } = \frac {2 x y}{x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

17827

\[ {}y^{\prime } = \frac {y \left (1+\ln \left (y\right )-\ln \left (x \right )\right )}{x} \]

[[_homogeneous, ‘class A‘], _dAlembert]

17828

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

17829

\[ {}\left (x +y\right ) y^{\prime } = y-x \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

17830

\[ {}x -y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

17835

\[ {}y^{\prime } x -4 y = x^{2} \sqrt {y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17843

\[ {}x -y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

17844

\[ {}y^{\prime } = \frac {y^{2}}{3}+\frac {2}{3 x^{2}} \]

[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]]

17845

\[ {}y^{\prime }+y^{2}+\frac {y}{x}-\frac {4}{x^{2}} = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Riccati]

17850

\[ {}y^{\prime } = \frac {x -y^{2}}{2 y \left (x +y^{2}\right )} \]

[[_homogeneous, ‘class G‘], _rational]

17855

\[ {}\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

17858

\[ {}y^{3}+2 \left (x^{2}-x y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

17859

\[ {}\left (x^{2} y^{2}-1\right ) y^{\prime }+2 x y^{3} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

17876

\[ {}y = 2 y^{\prime } x +\frac {x^{2}}{2}+{y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘]]

17881

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

17889

\[ {}y^{\prime } = -x +\sqrt {x^{2}+2 y} \]

[[_1st_order, _with_linear_symmetries], _Clairaut]

17890

\[ {}y^{\prime } = -x -\sqrt {x^{2}+2 y} \]

[[_1st_order, _with_linear_symmetries], _Clairaut]

17897

\[ {}y = 2 y^{\prime } x +\frac {x^{2}}{2}+{y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘]]

17993

\[ {}y^{\prime } = \frac {x y}{x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

17994

\[ {}2 x y^{\prime } y = x^{2}+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

17995

\[ {}y^{\prime } x +y = x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

17996

\[ {}y^{\prime } = \frac {y^{2}}{x y-x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18007

\[ {}x y^{\prime } y = -1+y \]

[_separable]

18008

\[ {}x^{5} y^{\prime }+y^{5} = 0 \]

[_separable]

18016

\[ {}y \ln \left (y\right )-y^{\prime } x = 0 \]

[_separable]

18033

\[ {}x^{2}-2 y^{2}+x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18034

\[ {}x^{2} y^{\prime }-3 x y-2 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18035

\[ {}x^{2} y^{\prime } = 3 \left (x^{2}+y^{2}\right ) \arctan \left (\frac {y}{x}\right )+x y \]

[[_homogeneous, ‘class A‘], _dAlembert]

18036

\[ {}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )+x \]

[[_homogeneous, ‘class A‘], _dAlembert]

18037

\[ {}y^{\prime } x = y+2 \,{\mathrm e}^{-\frac {y}{x}} x \]

[[_homogeneous, ‘class A‘], _dAlembert]

18038

\[ {}x -y-\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18039

\[ {}y^{\prime } x = 2 x +3 y \]

[_linear]

18040

\[ {}y^{\prime } x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18041

\[ {}x^{2} y^{\prime } = y^{2}+2 x y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18042

\[ {}x^{3}+y^{3}-x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18050

\[ {}y^{\prime } = \frac {1-x y^{2}}{2 x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18051

\[ {}y^{\prime } = \frac {2+3 x y^{2}}{4 x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18052

\[ {}y^{\prime } = \frac {y-x y^{2}}{x +x^{2} y} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18053

\[ {}\left (x +\frac {2}{y}\right ) y^{\prime }+y = 0 \]

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18069

\[ {}\frac {x}{\left (x^{2}+y^{2}\right )^{{3}/{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{{3}/{2}}} = 0 \]

[_separable]

18072

\[ {}\frac {4 y^{2}-2 x^{2}}{4 x y^{2}-x^{3}}+\frac {\left (8 y^{2}-x^{2}\right ) y^{\prime }}{4 y^{3}-x^{2} y} = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

18073

\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18075

\[ {}y^{\prime } x +y+3 x^{3} y^{4} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18078

\[ {}y+\left (x -2 x^{2} y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18079

\[ {}x +3 y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18085

\[ {}y-y^{\prime } x = x y^{3} y^{\prime } \]

[_separable]

18087

\[ {}\left (x +y\right ) y^{\prime } = y-x \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18089

\[ {}y^{\prime } x -y+y^{2} = 0 \]

[_separable]

18091

\[ {}y^{\prime } x +y = y^{\prime } \sqrt {x y} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18093

\[ {}y^{\prime } x -y = x^{2} y^{4} \left (y^{\prime } x +y\right ) \]

[[_homogeneous, ‘class G‘], _rational]

18094

\[ {}y^{\prime } x +y+x^{2} y^{5} y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18095

\[ {}2 x y^{2}-y+y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

18097

\[ {}y^{\prime } = \frac {2 y}{x}+\frac {x^{3}}{y}+x \tan \left (\frac {y}{x^{2}}\right ) \]

[[_homogeneous, ‘class G‘]]

18098

\[ {}y^{\prime } x -3 y = x^{4} \]

[_linear]

18103

\[ {}2 y-x^{3} = y^{\prime } x \]

[_linear]

18108

\[ {}y^{\prime } x +y = x^{4} y^{3} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18110

\[ {}y^{\prime } x +y = x y^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18129

\[ {}\left (1-x y\right ) y^{\prime } = y^{2} \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18131

\[ {}y^{\prime } x = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18132

\[ {}y^{2} = \left (x^{3}-x y\right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18133

\[ {}x^{2} y^{3}+y = \left (x^{3} y^{2}-x \right ) y^{\prime } \]

[[_homogeneous, ‘class G‘], _rational]

18136

\[ {}x y^{\prime } y = x^{2} y^{\prime }+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18139

\[ {}y+x^{2} = y^{\prime } x \]

[_linear]

18147

\[ {}y^{2}-3 x y-2 x^{2} = \left (x^{2}-x y\right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18153

\[ {}x^{2} y^{4}+x^{6}-x^{3} y^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18155

\[ {}y^{\prime } = 1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

18156

\[ {}y^{\prime } = \frac {2 x y \,{\mathrm e}^{\frac {x^{2}}{y^{2}}}}{y^{2}+y^{2} {\mathrm e}^{\frac {x^{2}}{y^{2}}}+2 x^{2} {\mathrm e}^{\frac {x^{2}}{y^{2}}}} \]

[[_homogeneous, ‘class A‘], _dAlembert]

18158

\[ {}3 x^{2} \ln \left (y\right )+\frac {x^{3} y^{\prime }}{y} = 0 \]

[_separable]

18160

\[ {}\frac {y-x}{\left (x +y\right )^{3}}-\frac {2 x y^{\prime }}{\left (x +y\right )^{3}} = 0 \]

[_linear]

18161

\[ {}x y^{2}+y+y^{\prime } x = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18163

\[ {}3 x^{2} y-y^{3}-\left (3 x y^{2}-x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

18169

\[ {}3 x y+y^{2}+\left (3 x y+x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18170

\[ {}x^{2} y^{\prime } = x^{2}+x y+y^{2} \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

18175

\[ {}-y^{2}+x^{2} y^{\prime } = 2 x y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18431

\[ {}x^{\prime } = \cos \left (\frac {x}{t}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

18432

\[ {}\left (t^{2}-x^{2}\right ) x^{\prime } = t x \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18464

\[ {}y^{\prime } = \frac {\sqrt {1-y^{2}}\, \arcsin \left (y\right )}{x} \]

[_separable]

18468

\[ {}v^{\prime }+\frac {2 v}{u} = 3 \]

[_linear]

18474

\[ {}y^{2} = x \left (y-x \right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18475

\[ {}2 x^{2} y+y^{3}-x^{3} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18478

\[ {}y y^{\prime }+x = m y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18479

\[ {}\frac {2 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

18491

\[ {}\sqrt {t^{2}+T} = T^{\prime } \]

[[_homogeneous, ‘class G‘]]

18504

\[ {}y^{\prime } = 1+\frac {2 y}{x -y} \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18508

\[ {}4 y {y^{\prime }}^{3}-2 x^{2} {y^{\prime }}^{2}+4 x y^{\prime } y+x^{3} = 16 y^{2} \]

[[_1st_order, _with_linear_symmetries]]

18569

\[ {}x \left (x -2 y\right ) y^{\prime }+x^{2}+2 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18570

\[ {}5 x y^{\prime } y-x^{2}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18571

\[ {}\left (x^{2}+3 x y-y^{2}\right ) y^{\prime }-3 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18572

\[ {}\left (x^{2}+2 x y\right ) y^{\prime }-3 x^{2}+2 x y-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18573

\[ {}5 x y^{\prime } y-4 x^{2}-y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18575

\[ {}3 x^{2} y^{\prime }+2 x^{2}-3 y^{2} = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Riccati]

18660

\[ {}x^{2}+y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18661

\[ {}y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18662

\[ {}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18663

\[ {}\left (3 x +4 y\right ) y^{\prime }+y-2 x = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18666

\[ {}x^{2}-4 x y-2 y^{2}+\left (y^{2}-4 x y-2 x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

18672

\[ {}\left (x y+1\right ) y-x \left (1-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18673

\[ {}a \left (y^{\prime } x +2 y\right ) = x y^{\prime } y \]

[_separable]

18676

\[ {}x^{2} y-2 x y^{2}-\left (x^{3}-3 x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18677

\[ {}y \left (x y+2 x^{2} y^{2}\right )+x \left (x y-x^{2} y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18680

\[ {}3 x^{2} y^{4}+2 x y+\left (2 x^{3} y^{3}-x^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18682

\[ {}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18683

\[ {}2 x^{2} y-3 y^{4}+\left (3 x^{3}+2 x y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18684

\[ {}y^{2}+2 x^{2} y+\left (2 x^{3}-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18690

\[ {}y^{\prime }+\frac {y}{x} = y^{6} x^{2} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18692

\[ {}y^{\prime }+\frac {2 y}{x} = 3 x^{2} y^{{1}/{3}} \]

[[_homogeneous, ‘class G‘], _rational, _Bernoulli]

18696

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18704

\[ {}y^{\prime } x +\frac {y^{2}}{x} = y \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18707

\[ {}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18710

\[ {}y y^{\prime }+x = m \left (y^{\prime } x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18718

\[ {}y y^{\prime } = a x \]

[_separable]

18720

\[ {}\left (x +y\right ) y^{\prime }+x -y = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18722

\[ {}\left (y^{2}-x^{2}\right ) y^{\prime }+2 x y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

18725

\[ {}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18726

\[ {}2 x^{2} y^{2}+y-\left (x^{3} y-3 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18727

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

18738

\[ {}4 y = {y^{\prime }}^{2}+x^{2} \]

[[_homogeneous, ‘class G‘]]

18747

\[ {}x^{2} \left (y-y^{\prime } x \right ) = y {y^{\prime }}^{2} \]

[[_1st_order, _with_linear_symmetries]]

18755

\[ {}y = -y^{\prime } x +x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

18769

\[ {}{y^{\prime }}^{3}-4 x y^{\prime } y+8 y^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

18776

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

18779

\[ {}\sqrt {x}\, y^{\prime } = \sqrt {y} \]

[_separable]

18780

\[ {}x^{2} {y^{\prime }}^{2}-3 x y^{\prime } y+x^{3}+2 y^{2} = 0 \]

[[_homogeneous, ‘class G‘], _rational]

18787

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a^{3} = 0 \]

[[_homogeneous, ‘class G‘]]

18976

\[ {}y^{\prime } x +x +y = 0 \]

[_linear]

18977

\[ {}\left (x y+1\right ) y-y^{\prime } x = 0 \]

[[_homogeneous, ‘class D‘], _rational, _Bernoulli]

18979

\[ {}y-x +\left (x +y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

18981

\[ {}x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

18985

\[ {}y y^{\prime }+x = m \left (y^{\prime } x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

19005

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = x^{2}+x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19006

\[ {}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y-\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _dAlembert]

19007

\[ {}x^{2}-y^{2}+2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19008

\[ {}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right ) \]

[[_homogeneous, ‘class A‘], _dAlembert]

19011

\[ {}y^{2} = \left (x y-x^{2}\right ) y^{\prime } \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19012

\[ {}x \sin \left (\frac {y}{x}\right ) y^{\prime } = y \sin \left (\frac {y}{x}\right )-x \]

[[_homogeneous, ‘class A‘], _dAlembert]

19013

\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = x y \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19014

\[ {}x^{2} y^{\prime }+y \left (x +y\right ) = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19015

\[ {}2 y^{\prime } = \frac {y}{x}+\frac {y^{2}}{x^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19019

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19020

\[ {}\left (3 x^{2}+y^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right ) = 0 \]

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

19021

\[ {}x^{2}+3 y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19048

\[ {}x^{2} y-2 x y^{2}-\left (x^{3}-3 x^{2} y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19049

\[ {}\left (x^{4} y^{4}+x^{2} y^{2}+x y\right ) y+\left (x^{4} y^{4}-x^{2} y^{2}+x y\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

19050

\[ {}y \left (x y+2 x^{2} y^{2}\right )+x \left (x y-x^{2} y^{2}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19052

\[ {}x^{2}+y^{2}-2 x y^{\prime } y = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19054

\[ {}y^{2}+2 x^{2} y+\left (2 x^{3}-x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19055

\[ {}2 y+3 y^{\prime } x +2 x y \left (3 y+4 y^{\prime } x \right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19063

\[ {}y^{\prime } x -y = \sqrt {x^{2}+y^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19066

\[ {}y y^{\prime }+x = m \left (y^{\prime } x -y\right ) \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

19073

\[ {}2 y+3 y^{\prime } x +2 x y \left (3 y+4 y^{\prime } x \right ) = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19083

\[ {}x^{2} y^{\prime }+y^{2} = x y^{\prime } y \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19187

\[ {}y = y^{\prime } x +x \sqrt {1+{y^{\prime }}^{2}} \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

19191

\[ {}4 x {y^{\prime }}^{2}+4 y y^{\prime } = y^{4} \]

[[_homogeneous, ‘class G‘]]

19194

\[ {}y-y^{\prime } x = y y^{\prime }+x \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

19196

\[ {}x^{2} \left (y-y^{\prime } x \right ) = y {y^{\prime }}^{2} \]

[[_1st_order, _with_linear_symmetries]]

19208

\[ {}\left (x^{3} y^{3}+x^{2} y^{2}+x y+1\right ) y+\left (x^{3} y^{3}-x^{2} y^{2}-x y+1\right ) x y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational]

19209

\[ {}\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y = \left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime } \]

[[_homogeneous, ‘class A‘], _dAlembert]

19218

\[ {}3 y = 2 y^{\prime } x -\frac {2 {y^{\prime }}^{2}}{x} \]

[[_homogeneous, ‘class G‘]]

19225

\[ {}{y^{\prime }}^{2}+2 x^{3} y^{\prime }-4 x^{2} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

19226

\[ {}y^{2} \left (y-y^{\prime } x \right ) = x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘]]

19230

\[ {}y+x^{2} = {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘]]

19231

\[ {}{y^{\prime }}^{3} = y^{4} \left (y^{\prime } x +y\right ) \]

[[_1st_order, _with_linear_symmetries]]

19441

\[ {}\left (x +2 y^{3}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class G‘], _rational]

19443

\[ {}1+y^{2}-x y^{\prime } y = 0 \]

[_separable]

19444

\[ {}y^{2}+\left (x^{2}+x y\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

19449

\[ {}\left (x +2 y^{3}\right ) y^{\prime } = y \]

[[_homogeneous, ‘class G‘], _rational]

19453

\[ {}x^{2} y-\left (x^{3}+y^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19457

\[ {}y^{3}-2 x^{2} y+\left (2 x y^{2}-x^{3}\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

19485

\[ {}y = 2 y^{\prime } x +y^{2} {y^{\prime }}^{3} \]

[[_1st_order, _with_linear_symmetries]]

19486

\[ {}y = -y^{\prime } x +x^{4} {y^{\prime }}^{2} \]

[[_homogeneous, ‘class G‘], _rational]

19488

\[ {}x^{2} \left (y-y^{\prime } x \right ) = y {y^{\prime }}^{2} \]

[[_1st_order, _with_linear_symmetries]]

19500

\[ {}{y^{\prime }}^{3}-4 x y^{\prime } y+8 y^{2} = 0 \]

[[_1st_order, _with_linear_symmetries]]

19502

\[ {}x^{3} {y^{\prime }}^{2}+x^{2} y y^{\prime }+a^{3} = 0 \]

[[_homogeneous, ‘class G‘]]