2.2.2 Problems 101 to 200

Table 2.5: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

101

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

[_linear]

1.229

102

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

[_linear]

2.192

103

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

[_separable]

0.815

104

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

[_linear]

1.088

105

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

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

4.463

106

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

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

5.319

107

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

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

5.209

108

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

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

39.531

109

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

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

97.852

110

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

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

4.605

111

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

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

12.138

112

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

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

215.689

113

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

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

2.552

114

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

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

76.911

115

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

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

6.072

116

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

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

114.656

117

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

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

8.904

118

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

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

8.281

119

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

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

65.291

120

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

[[_homogeneous, ‘class C‘], _dAlembert]

2.711

121

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

[[_homogeneous, ‘class C‘], _Riccati]

1.651

122

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

[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

1.926

123

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

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

3.323

124

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

[_separable]

2.600

125

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

[_quadrature]

5.211

126

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

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

3.186

127

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

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

88.531

128

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

[_Bernoulli]

2.780

129

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

[_Bernoulli]

6.399

130

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

[[_1st_order, _with_linear_symmetries], _Bernoulli]

2.174

131

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

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

2.730

132

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

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

1.706

133

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

[‘y=_G(x,y’)‘]

3.538

134

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

[[_1st_order, _with_linear_symmetries]]

2.144

135

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

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

4.398

136

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

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

4.052

137

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

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

5.167

138

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

[_exact, _rational]

2.134

139

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

[_exact]

1.678

140

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

[_exact]

2.165

141

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

[_exact]

3.691

142

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

[_exact]

1.965

143

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

[_exact, _rational]

1.671

144

\[ {}{\mathrm e}^{x} \sin \left (y\right )+\tan \left (y\right )+\left ({\mathrm e}^{x} \cos \left (y\right )+x \sec \left (y\right )^{2}\right ) y^{\prime } = 0 \]

[_exact]

17.579

145

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

[_exact, _rational]

17.730

146

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

[[_1st_order, _with_linear_symmetries], _exact, _rational]

2.426

147

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

[[_2nd_order, _missing_y]]

0.866

148

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

1.221

149

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

[[_2nd_order, _missing_x]]

2.293

150

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

[[_2nd_order, _missing_y]]

0.963

151

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

0.395

152

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

[[_2nd_order, _missing_y]]

0.837

153

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

1.781

154

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

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]

0.471

155

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

0.310

156

\[ {}y^{3} y^{\prime \prime } = 1 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

2.365

157

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

1.294

158

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.562

159

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

[[_homogeneous, ‘class C‘], _dAlembert]

1.244

160

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

[_Bernoulli]

1.862

161

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

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

0.212

162

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

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

2.635

163

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

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

3.346

164

\[ {}y^{\prime } = \frac {2 y-x +7}{4 x -3 y-18} \]

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

3.322

165

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

[[_homogeneous, ‘class C‘], _dAlembert]

2.378

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]

99.884

167

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

[_Riccati]

1.652

168

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

[[_homogeneous, ‘class C‘], _Riccati]

1.898

169

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.431

170

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

[[_2nd_order, _missing_x]]

19.418

171

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

[_quadrature]

3.316

172

\[ {}x^{\prime } = 10 x-x^{2} \]
i.c.

[_quadrature]

2.933

173

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

[_quadrature]

13.768

174

\[ {}x^{\prime } = 9-4 x^{2} \]
i.c.

[_quadrature]

5.368

175

\[ {}x^{\prime } = 3 x \left (5-x\right ) \]
i.c.

[_quadrature]

3.426

176

\[ {}x^{\prime } = 3 x \left (5-x\right ) \]
i.c.

[_quadrature]

3.034

177

\[ {}x^{\prime } = 4 x \left (7-x\right ) \]
i.c.

[_quadrature]

3.354

178

\[ {}x^{\prime } = 7 x \left (x-13\right ) \]
i.c.

[_quadrature]

3.877

179

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

[_linear]

1.362

180

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

[_separable]

1.949

181

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

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

2.603

182

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

[_exact]

2.480

183

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

[_separable]

2.135

184

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

[_separable]

1.995

185

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

[_linear]

1.381

186

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

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

3.428

187

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

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

5.765

188

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

[_separable]

2.408

189

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

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

2.607

190

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

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

477.563

191

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

[_separable]

2.056

192

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

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

74.504

193

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

[[_linear, ‘class A‘]]

2.011

194

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

[[_homogeneous, ‘class C‘], _Riccati]

1.959

195

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

[_exact]

2.661

196

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

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

77.921

197

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

[_separable]

2.065

198

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

[_linear]

2.102

199

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

[_linear]

1.483

200

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

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

5.967