Problems 101 to 200

2.2.2 Problems 101 to 200

Table 2.17: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy	time(sec)

101	\begin{align} y^{\prime }&=2 y x +1 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.211

102	\begin{align} 2 y^{\prime } x&=y+2 \cos \left (x \right ) x \\ y \left (1\right ) &= 0 \\ \end{align}	[_linear]	✓	✓	✓	✓	2.192

103	\begin{align} y^{\prime }+p \left (x \right ) y&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	1.868

104	\begin{align} y^{\prime }+p \left (x \right ) y&=q \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✓	1.523

105	\begin{align} \left (x +y\right ) y^{\prime }&=x -y \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.376

106	\begin{align} 2 y y^{\prime } x&=x^{2}+2 y^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	4.990

107	\begin{align} y^{\prime } x&=y+2 \sqrt {y x} \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓	5.995

108	\begin{align} \left (x -y\right ) y^{\prime }&=x +y \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.173

109	\begin{align} x \left (x +y\right ) y^{\prime }&=y \left (x -y\right ) \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓	5.556

110	\begin{align} \left (x +2 y\right ) y^{\prime }&=y \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.270

111	\begin{align} y^{\prime } y^{2} x&=x^{3}+y^{3} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	6.564

112	\begin{align} x^{2} y^{\prime }&=y x +x^{2} {\mathrm e}^{\frac {y}{x}} \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗	4.318

113	\begin{align} x^{2} y^{\prime }&=y x +y^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	2.789

114	\begin{align} y y^{\prime } x&=x^{2}+3 y^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	5.297

115	\begin{align} \left (x^{2}-y^{2}\right ) y^{\prime }&=2 y x \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	4.258

116	\begin{align} y y^{\prime } x&=y^{2}+x \sqrt {4 x^{2}+y^{2}} \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗	11.644

117	\begin{align} y^{\prime } x&=y+\sqrt {x^{2}+y^{2}} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	6.611

118	\begin{align} y y^{\prime }+x&=\sqrt {x^{2}+y^{2}} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	7.003

119	\begin{align} x \left (x +y\right ) y^{\prime }+y \left (3 x +y\right )&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓	5.583

120	\begin{align} y^{\prime }&=\sqrt {x +y+1} \\ \end{align}	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✓	✓	2.307

121	\begin{align} y^{\prime }&=\left (4 x +y\right )^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _Riccati]	✓	✓	✓	✓	1.344

122	\begin{align} \left (x +y\right ) y^{\prime }&=1 \\ \end{align}	[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]	✓	✓	✓	✓	1.704

123	\begin{align} x^{2} y^{\prime }+2 y x&=5 y^{3} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	5.815

124	\begin{align} y^{2} y^{\prime }+2 x y^{3}&=6 x \\ \end{align}	[_separable]	✓	✓	✓	✓	1.939

125	\begin{align} y^{\prime }&=y+y^{3} \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.182

126	\begin{align} x^{2} y^{\prime }+2 y x&=5 y^{4} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	3.711

127	\begin{align} y^{\prime } x +6 y&=3 x y^{{4}/{3}} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	12.403

128	\begin{align} 2 y^{\prime } x +y^{3} {\mathrm e}^{-2 x}&=2 y x \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	3.311

129	\begin{align} y^{2} \left (y^{\prime } x +y\right ) \sqrt {x^{4}+1}&=x \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	8.537

130	\begin{align} 3 y^{2} y^{\prime }+y^{3}&={\mathrm e}^{-x} \\ \end{align}	[[_1st_order, _with_linear_symmetries], _Bernoulli]	✓	✓	✓	✓	1.452

131	\begin{align} 3 y^{\prime } y^{2} x&=3 x^{4}+y^{3} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	2.755

132	\begin{align} x \,{\mathrm e}^{y} y^{\prime }&=2 \,{\mathrm e}^{y}+2 x^{3} {\mathrm e}^{2 x} \\ \end{align}	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]	✓	✓	✓	✗	1.907

133	\begin{align} 2 x \sin \left (y\right ) \cos \left (y\right ) y^{\prime }&=4 x^{2}+\sin \left (y\right )^{2} \\ \end{align}	[‘y=_G(x,y’)‘]	✓	✓	✓	✓	3.314

134	\begin{align} \left ({\mathrm e}^{y}+x \right ) y^{\prime }&=x \,{\mathrm e}^{-y}-1 \\ \end{align}	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✓	2.617

135	\begin{align} 2 x +3 y+\left (3 x +2 y\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.421

136	\begin{align} 4 x -y+\left (6 y-x \right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.254

137	\begin{align} 3 x^{2}+2 y^{2}+\left (4 y x +6 y^{2}\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	✓	✓	✗	8.711

138	\begin{align} 2 x y^{2}+3 x^{2}+\left (2 x^{2} y+4 y^{3}\right ) y^{\prime }&=0 \\ \end{align}	[_exact, _rational]	✓	✓	✓	✗	1.961

139	\begin{align} x^{3}+\frac {y}{x}+\left (y^{2}+\ln \left (x \right )\right ) y^{\prime }&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	2.043

140	\begin{align} 1+y \,{\mathrm e}^{y x}+\left (2 y+x \,{\mathrm e}^{y x}\right ) y^{\prime }&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	55.048

141	\begin{align} \cos \left (x \right )+\ln \left (y\right )+\left (\frac {x}{y}+{\mathrm e}^{y}\right ) y^{\prime }&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	4.005

142	\begin{align} x +\arctan \left (y\right )+\frac {\left (x +y\right ) y^{\prime }}{1+y^{2}}&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	2.163

143	\begin{align} 3 x^{2} y^{3}+y^{4}+\left (3 x^{3} y^{2}+y^{4}+4 x y^{3}\right ) y^{\prime }&=0 \\ \end{align}	[_exact, _rational]	✓	✓	✓	✗	0.247

144	\begin{align} {\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 \\ \end{align}	[_exact]	✓	✓	✓	✗	9.164

145	\begin{align} \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 \\ \end{align}	[_exact, _rational]	✓	✓	✗	✗	10.961

146	\begin{align} \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 \\ \end{align}	[[_1st_order, _with_linear_symmetries], _exact, _rational]	✓	✓	✓	✗	3.027

147	\begin{align} y^{\prime \prime } x&=y^{\prime } \\ \end{align}	[[_2nd_order, _missing_y]]	✓	✓	✓	✓	13.184

148	\begin{align} y y^{\prime \prime }+{y^{\prime }}^{2}&=0 \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗	1.299

149	\begin{align} y^{\prime \prime }+4 y&=0 \\ \end{align}	[[_2nd_order, _missing_x]]	✓	✓	✓	✓	3.053

150	\begin{align} y^{\prime \prime } x +y^{\prime }&=4 x \\ \end{align}	[[_2nd_order, _missing_y]]	✓	✓	✓	✓	2.045

151	\begin{align} y^{\prime \prime }&={y^{\prime }}^{2} \\ \end{align}	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓	0.941

152	\begin{align} x^{2} y^{\prime \prime }+3 y^{\prime } x&=2 \\ \end{align}	[[_2nd_order, _missing_y]]	✓	✓	✓	✓	2.136

153	\begin{align} y y^{\prime \prime }+{y^{\prime }}^{2}&=y y^{\prime } \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗	1.487

154	\begin{align} y^{\prime \prime }&=\left (x +y^{\prime }\right )^{2} \\ \end{align}	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓	631.272

155	\begin{align} y^{\prime \prime }&=2 y {y^{\prime }}^{3} \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✗	0.523

156	\begin{align} y^{3} y^{\prime \prime }&=1 \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✓	2.007

157	\begin{align} y^{\prime \prime }&=2 y y^{\prime } \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗	0.919

158	\begin{align} y y^{\prime \prime }&=3 {y^{\prime }}^{2} \\ \end{align}	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗	1.310

159	\begin{align} y^{\prime }&=f \left (a x +b y+c \right ) \\ \end{align}	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✓	✓	0.986

160	\begin{align} y^{\prime }+p \left (x \right ) y&=q \left (x \right ) y^{n} \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	3.135

161	\begin{align} y^{\prime }+p \left (x \right ) y&=q \left (x \right ) y \ln \left (y\right ) \\ \end{align}	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✗	2.454

162	\begin{align} y^{\prime } x -4 x^{2} y+2 y \ln \left (y\right )&=0 \\ \end{align}	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	✓	✓	✓	4.126

163	\begin{align} y^{\prime }&=\frac {x -y-1}{x +y+3} \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.174

164	\begin{align} y^{\prime }&=\frac {2 y-x +7}{4 x -3 y-18} \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	24.062

165	\begin{align} y^{\prime }&=\sin \left (x -y\right ) \\ \end{align}	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✓	✓	1.428

166	\begin{align} y^{\prime }&=-\frac {y \left (2 x^{3}-y^{3}\right )}{x \left (2 y^{3}-x^{3}\right )} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	5.890

167	\begin{align} y^{\prime }+y^{2}&=x^{2}+1 \\ \end{align}	[_Riccati]	✓	✓	✓	✗	1.866

168	\begin{align} y^{\prime }+2 y x&=1+x^{2}+y^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _Riccati]	✓	✓	✓	✓	1.791

169	\begin{align} y&=y^{\prime } x -\frac {{y^{\prime }}^{2}}{4} \\ \end{align}	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	✓	✓	✓	0.205

170	\begin{align} r y^{\prime \prime }&=\left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \\ \end{align}	[[_2nd_order, _missing_x]]	✓	✓	✓	✗	55.293

171	\begin{align} x^{\prime }&=x-x^{2} \\ x \left (0\right ) &= 2 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.253

172	\begin{align} x^{\prime }&=10 x-x^{2} \\ x \left (0\right ) &= 1 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.014

173	\begin{align} x^{\prime }&=1-x^{2} \\ x \left (0\right ) &= 3 \\ \end{align}	[_quadrature]	✓	✓	✓	✗	2.117

174	\begin{align} x^{\prime }&=9-4 x^{2} \\ x \left (0\right ) &= 0 \\ \end{align}	[_quadrature]	✓	✓	✓	✗	5.523

175	\begin{align} x^{\prime }&=3 x \left (5-x\right ) \\ x \left (0\right ) &= 8 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.082

176	\begin{align} x^{\prime }&=3 x \left (5-x\right ) \\ x \left (0\right ) &= 2 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	0.982

177	\begin{align} x^{\prime }&=4 x \left (7-x\right ) \\ x \left (0\right ) &= 11 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.011

178	\begin{align} x^{\prime }&=7 x \left (x-13\right ) \\ x \left (0\right ) &= 17 \\ \end{align}	[_quadrature]	✓	✓	✓	✓	1.262

179	\begin{align} x^{3}+3 y-y^{\prime } x&=0 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.852

180	\begin{align} x y^{2}+3 y^{2}-x^{2} y^{\prime }&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	2.513

181	\begin{align} y x +y^{2}-x^{2} y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	3.178

182	\begin{align} 2 x y^{3}+{\mathrm e}^{x}+\left (3 y^{2} x^{2}+\sin \left (y\right )\right ) y^{\prime }&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	2.960

183	\begin{align} 3 y+x^{4} y^{\prime }&=2 y x \\ \end{align}	[_separable]	✓	✓	✓	✓	2.615

184	\begin{align} 2 x y^{2}+x^{2} y^{\prime }&=y^{2} \\ \end{align}	[_separable]	✓	✓	✓	✓	2.549

185	\begin{align} 2 x^{2} y+x^{3} y^{\prime }&=1 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.896

186	\begin{align} x^{2} y^{\prime }+2 y x&=y^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	6.480

187	\begin{align} y^{\prime } x +2 y&=6 \sqrt {y}\, x^{2} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	6.243

188	\begin{align} y^{\prime }&=1+x^{2}+y^{2}+y^{2} x^{2} \\ \end{align}	[_separable]	✓	✓	✓	✓	2.970

189	\begin{align} x^{2} y^{\prime }&=y x +3 y^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	3.385

190	\begin{align} 6 x y^{3}+2 y^{4}+\left (9 y^{2} x^{2}+8 x y^{3}\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✗	0.253

191	\begin{align} 4 x y^{2}+y^{\prime }&=5 y^{2} x^{4} \\ \end{align}	[_separable]	✓	✓	✓	✓	2.559

192	\begin{align} x^{3} y^{\prime }&=x^{2} y-y^{3} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	129.730

193	\begin{align} y^{\prime }+3 y&=3 x^{2} {\mathrm e}^{-3 x} \\ \end{align}	[[_linear, ‘class A‘]]	✓	✓	✓	✓	2.642

194	\begin{align} y^{\prime }&=x^{2}-2 y x +y^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _Riccati]	✓	✓	✓	✓	1.595

195	\begin{align} {\mathrm e}^{x}+y \,{\mathrm e}^{y x}+\left ({\mathrm e}^{y}+x \,{\mathrm e}^{y x}\right ) y^{\prime }&=0 \\ \end{align}	[_exact]	✓	✓	✓	✗	3.787

196	\begin{align} 2 x^{2} y-x^{3} y^{\prime }&=y^{3} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	134.437

197	\begin{align} 3 x^{5} y^{2}+x^{3} y^{\prime }&=2 y^{2} \\ \end{align}	[_separable]	✓	✓	✓	✓	2.440

198	\begin{align} y^{\prime } x +3 y&=\frac {3}{x^{{3}/{2}}} \\ \end{align}	[_linear]	✓	✓	✓	✓	3.286

199	\begin{align} \left (x^{2}-1\right ) y^{\prime }+\left (x -1\right ) y&=1 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.560

200	\begin{align} y^{\prime } x&=6 y+12 x^{4} y^{{2}/{3}} \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	✓	✓	✓	7.717

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