Problems 101 to 200

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


#	ODE	CAS classiﬁcation	Solved?	time (sec)

101	$y^{'} = 1 + 2 x y$	[_linear]	✓	1.088

102	$2 y^{'} x = y + 2 x \cos (x)$ i.c.	[_linear]	✓	1.997

103	$y^{'} + p (x) y = 0$	[_separable]	✓	1.040

104	$y^{'} + p (x) y = q (x)$	[_linear]	✓	1.192

105	$(x + y) y^{'} = x - y$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.732

106	$2 x y^{'} y = x^{2} + 2 y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.922

107	$y^{'} x = y + 2 \sqrt{x y}$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	6.358

108	$(x - y) y^{'} = x + y$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.507

109	$x (x + y) y^{'} = y (x - y)$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.305

110	$(x + 2 y) y^{'} = y$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	4.571

111	$x y^{2} y^{'} = x^{3} + y^{3}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	11.498

112	$x^{2} y^{'} = x y + x^{2} e^{\frac{y}{x}}$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	5.061

113	$x^{2} y^{'} = x y + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.754

114	$x y^{'} y = x^{2} + 3 y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	4.234

115	$(x^{2} - y^{2}) y^{'} = 2 x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	4.940

116	$x y^{'} y = y^{2} + x \sqrt{4 x^{2} + y^{2}}$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	19.067

117	$y^{'} x = y + \sqrt{x^{2} + y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	5.968

118	$y y^{'} + x = \sqrt{x^{2} + y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.659

119	$x (x + y) y^{'} + y (3 x + y) = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.471

120	$y^{'} = \sqrt{x + y + 1}$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	3.161

121	$y^{'} = {(4 x + y)}^{2}$	[[_homogeneous, ‘class C‘], _Riccati]	✓	1.652

122	$(x + y) y^{'} = 1$	[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]	✓	1.664

123	$x^{2} y^{'} + 2 x y = 5 y^{3}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	3.915

124	$y^{2} y^{'} + 2 x y^{3} = 6 x$	[_separable]	✓	2.226

125	$y^{'} = y + y^{3}$	[_quadrature]	✓	4.998

126	$x^{2} y^{'} + 2 x y = 5 y^{4}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	3.467

127	$y^{'} x + 6 y = 3 x y^{4 / 3}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	56.393

128	$2 y^{'} x + y^{3} e^{- 2 x} = 2 x y$	[_Bernoulli]	✓	2.345

129	$y^{2} (y^{'} x + y) \sqrt{x^{4} + 1} = x$	[_Bernoulli]	✓	8.202

130	$3 y^{2} y^{'} + y^{3} = e^{- x}$	[[_1st_order, _with_linear_symmetries], _Bernoulli]	✓	1.882

131	$3 x y^{2} y^{'} = 3 x^{4} + y^{3}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.833

132	$x e^{y} y^{'} = 2 e^{y} + 2 x^{3} e^{2 x}$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]	✓	1.625

133	$2 x \sin (y) \cos (y) y^{'} = 4 x^{2} + \sin {(y)}^{2}$	[‘y=_G(x,y’)‘]	✓	3.128

134	$(e^{y} + x) y^{'} = x e^{- y} - 1$	[[_1st_order, _with_linear_symmetries]]	✓	2.001

135	$2 x + 3 y + (3 x + 2 y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.887

136	$4 x - y + (6 y - x) y^{'} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.459

137	$3 x^{2} + 2 y^{2} + (4 x y + 6 y^{2}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	8.009

138	$2 x y^{2} + 3 x^{2} + (2 x^{2} y + 4 y^{3}) y^{'} = 0$	[_exact, _rational]	✓	2.116

139	$x^{3} + \frac{y}{x} + (y^{2} + \ln (x)) y^{'} = 0$	[_exact]	✓	1.631

140	$1 + y e^{x y} + (2 y + x e^{x y}) y^{'} = 0$	[_exact]	✓	39.164

141	$\cos (x) + \ln (y) + (\frac{x}{y} + e^{y}) y^{'} = 0$	[_exact]	✓	3.296

142	$x + \arctan (y) + \frac{(x + y) y^{'}}{1 + y^{2}} = 0$	[_exact]	✓	1.757

143	$3 x^{2} y^{3} + y^{4} + (3 x^{3} y^{2} + y^{4} + 4 x y^{3}) y^{'} = 0$	[_exact, _rational]	✓	1.439

144	$e^{x} \sin (y) + \tan (y) + (e^{x} \cos (y) + x \sec {(y)}^{2}) y^{'} = 0$	[_exact]	✓	17.383

145	$\frac{2 x}{y} - \frac{3 y^{2}}{x^{4}} + (\frac{2 y}{x^{3}} - \frac{x^{2}}{y^{2}} + \frac{1}{\sqrt{y}}) y^{'} = 0$	[_exact, _rational]	✓	14.493

146	$\frac{2 x^{5 / 2} - 3 y^{5 / 3}}{2 x^{5 / 2} y^{2 / 3}} + \frac{(3 y^{5 / 3} - 2 x^{5 / 2}) y^{'}}{3 x^{3 / 2} y^{5 / 3}} = 0$	[[_1st_order, _with_linear_symmetries], _exact, _rational]	✓	2.099

147	$x y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓	1.001

148	$y y^{''} + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.260

149	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓	1.495

150	$x y^{''} + y^{'} = 4 x$	[[_2nd_order, _missing_y]]	✓	0.909

151	$y^{''} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	0.523

152	$x^{2} y^{''} + 3 y^{'} x = 2$	[[_2nd_order, _missing_y]]	✓	0.659

153	$y y^{''} + {y^{'}}^{2} = y y^{'}$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.710

154	$y^{''} = {(x + y^{'})}^{2}$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]	✓	0.588

155	$y^{''} = 2 y {y^{'}}^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.435

156	$y^{3} y^{''} = 1$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.938

157	$y^{''} = 2 y y^{'}$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.039

158	$y y^{''} = 3 {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.629

159	$y^{'} = f (a x + b y + c)$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	1.288

160	$y^{'} + p (x) y = q (x) y^{n}$	[_Bernoulli]	✓	2.021

161	$y^{'} + p (x) y = q (x) y \ln (y)$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	1.962

162	$y^{'} x - 4 x^{2} y + 2 y \ln (y) = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	2.583

163	$y^{'} = \frac{x - y - 1}{x + y + 3}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.845

164	$y^{'} = \frac{2 y - x + 7}{4 x - 3 y - 18}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.249

165	$y^{'} = \sin (x - y)$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	2.006

166	$y^{'} = - \frac{y (2 x^{3} - y^{3})}{x (2 y^{3} - x^{3})}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	8.789

167	$y^{'} + y^{2} = x^{2} + 1$	[_Riccati]	✓	1.470

168	$y^{'} + 2 x y = x^{2} + y^{2} + 1$	[[_homogeneous, ‘class C‘], _Riccati]	✓	1.914

169	$y = y^{'} x - \frac{{y^{'}}^{2}}{4}$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.523

170	$r y^{''} = {(1 + {y^{'}}^{2})}^{3 / 2}$	[[_2nd_order, _missing_x]]	✓	2.643

171	$x^{'} = x - x^{2}$ i.c.	[_quadrature]	✓	1.491

172	$x^{'} = 10 x - x^{2}$ i.c.	[_quadrature]	✓	1.393

173	$x^{'} = 1 - x^{2}$ i.c.	[_quadrature]	✓	2.397

174	$x^{'} = 9 - 4 x^{2}$ i.c.	[_quadrature]	✓	3.981

175	$x^{'} = 3 x (5 - x)$ i.c.	[_quadrature]	✓	1.470

176	$x^{'} = 3 x (5 - x)$ i.c.	[_quadrature]	✓	1.377

177	$x^{'} = 4 x (7 - x)$ i.c.	[_quadrature]	✓	1.635

178	$x^{'} = 7 x (x - 13)$ i.c.	[_quadrature]	✓	2.607

179	$x^{3} + 3 y - y^{'} x = 0$	[_linear]	✓	1.022

180	$x y^{2} + 3 y^{2} - x^{2} y^{'} = 0$	[_separable]	✓	1.515

181	$x y + y^{2} - x^{2} y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.715

182	$2 x y^{3} + e^{x} + (3 x^{2} y^{2} + \sin (y)) y^{'} = 0$	[_exact]	✓	2.284

183	$3 y + x^{4} y^{'} = 2 x y$	[_separable]	✓	1.648

184	$2 x y^{2} + x^{2} y^{'} = y^{2}$	[_separable]	✓	1.490

185	$2 x^{2} y + x^{3} y^{'} = 1$	[_linear]	✓	1.009

186	$x^{2} y^{'} + 2 x y = y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	2.561

187	$y^{'} x + 2 y = 6 x^{2} \sqrt{y}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	6.008

188	$y^{'} = 1 + x^{2} + y^{2} + x^{2} y^{2}$	[_separable]	✓	2.499

189	$x^{2} y^{'} = x y + 3 y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.739

190	$6 x y^{3} + 2 y^{4} + (9 x^{2} y^{2} + 8 x y^{3}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.918

191	$4 x y^{2} + y^{'} = 5 x^{4} y^{2}$	[_separable]	✓	1.529

192	$x^{3} y^{'} = x^{2} y - y^{3}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	145.428

193	$y^{'} + 3 y = 3 x^{2} e^{- 3 x}$	[[_linear, ‘class A‘]]	✓	1.589

194	$y^{'} = x^{2} - 2 x y + y^{2}$	[[_homogeneous, ‘class C‘], _Riccati]	✓	1.770

195	$e^{x} + y e^{x y} + (e^{y} + x e^{x y}) y^{'} = 0$	[_exact]	✓	2.519

196	$2 x^{2} y - x^{3} y^{'} = y^{3}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	152.893

197	$3 x^{5} y^{2} + x^{3} y^{'} = 2 y^{2}$	[_separable]	✓	1.450

198	$y^{'} x + 3 y = \frac{3}{x^{3 / 2}}$	[_linear]	✓	1.594

199	$(x^{2} - 1) y^{'} + (x - 1) y = 1$	[_linear]	✓	1.256

200	$y^{'} x = 6 y + 12 x^{4} y^{2 / 3}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	5.198

2.2.2 Problems 101 to 200