Problems 5101 to 5200

Table 2.105: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

5101	$x y y^{'} + 1 + y^{2} = 0$	[_separable]	✓	3.295

5102	$x y y^{'} = x + y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	3.226

5103	$x y y^{'} + x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	76.325

5104	$x y y^{'} + x^{4} - y^{2} = 0$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	3.113

5105	$x y y^{'} = a x^{3} \cos (x) + y^{2}$	[[_homogeneous, ‘class D‘], _Bernoulli]	✓	3.966

5106	$x y y^{'} = x^{2} - x y + y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	6.229

5107	$x y y^{'} + 2 x^{2} - 2 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	78.369

5108	$x y y^{'} = a + b y^{2}$	[_separable]	✓	2.233

5109	$x y y^{'} = a x^{n} + b y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.739

5110	$x y y^{'} = (x^{2} + 1) (1 - y^{2})$	[_separable]	✓	2.924

5111	$x y y^{'} + x^{2} arccot (\frac{y}{x}) - y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	7.293

5112	$x y y^{'} + x^{2} e^{- \frac{2 y}{x}} - y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	3.470

5113	$(1 + x y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.639

5114	$x (y + 1) y^{'} - (1 - x) y = 0$	[_separable]	✓	1.438

5115	$x (1 - y) y^{'} + (x + 1) y = 0$	[_separable]	✓	1.514

5116	$x (1 - y) y^{'} + (1 - x) y = 0$	[_separable]	✓	1.523

5117	$x (y + 2) y^{'} + a x = 0$	[_quadrature]	✓	1.204

5118	$(2 + 3 x - x y) y^{'} + y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.256

5119	$x (4 + y) y^{'} = 2 x + 2 y + y^{2}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.281

5120	$x (a + y) y^{'} + b x + c y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.021

5121	$x (a + y) y^{'} = y (B x + A)$	[_separable]	✓	1.633

5122	$x (x + y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	127.566

5123	$x (x - y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	40.945

5124	$x (x + y) y^{'} = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	16.569

5125	$x (x - y) y^{'} + 2 x^{2} + 3 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	78.125

5126	$x (x + y) y^{'} - y (x + y) + x \sqrt{x^{2} - y^{2}} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	108.618

5127	$(a + (x + y) x) y^{'} = b (x + y) y$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.522

5128	$x (y + 2 x) y^{'} = x^{2} + x y - y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	120.337

5129	$x (4 x - y) y^{'} + 4 x^{2} - 6 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	179.054

5130	$x (x^{3} + y) y^{'} = (x^{3} - y) y$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.180

5131	$x (2 x^{3} + y) y^{'} = (2 x^{3} - y) y$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.736

5132	$x (2 x^{3} + y) y^{'} = 6 y^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.810

5133	$y (1 - x) y^{'} + x (1 - y) = 0$	[_separable]	✓	1.641

5134	$(x + a) (x + b) y^{'} = x y$	[_separable]	✓	1.772

5135	$2 x y y^{'} + 1 - 2 x^{3} - y^{2} = 0$	[_rational, _Bernoulli]	✓	1.895

5136	$2 x y y^{'} + a + y^{2} = 0$	[_separable]	✓	2.239

5137	$2 x y y^{'} = a x + y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.022

5138	$2 x y y^{'} + x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]	✓	60.602

5139	$2 x y y^{'} = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	7.379

5140	$2 x y y^{'} = 4 x^{2} (2 x + 1) + y^{2}$	[_rational, _Bernoulli]	✓	4.678

5141	$2 x y y^{'} + x^{2} (a x^{3} + 1) = 6 y^{2}$	[_rational, _Bernoulli]	✓	1.674

5142	$(3 - x + 2 x y) y^{'} + 3 x^{2} - y + y^{2} = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.827

5143	$x (x - 2 y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	6.478

5144	$x (x + 2 y) y^{'} + (2 x - y) y = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	221.470

5145	$x (x - 2 y) y^{'} + (2 x - y) y = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	99.681

5146	$x (1 + x - 2 y) y^{'} + (1 - 2 x + y) y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.496

5147	$x (1 - x - 2 y) y^{'} + (1 + 2 x + y) y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.532

5148	$2 x (2 x^{2} + y) y^{'} + (12 x^{2} + y) y = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.427

5149	$2 (x + 1) y y^{'} + 2 x - 3 x^{2} + y^{2} = 0$	[_exact, _rational, _Bernoulli]	✓	2.253

5150	$x (2 x + 3 y) y^{'} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	39.581

5151	$x (2 x + 3 y) y^{'} + 3 {(x + y)}^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	44.061

5152	$(3 + 6 x y + x^{2}) y^{'} + 2 x + 2 x y + 3 y^{2} = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.638

5153	$3 x (x + 2 y) y^{'} + x^{3} + 3 y (y + 2 x) = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.764

5154	$a x y y^{'} = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.104

5155	$a x y y^{'} + x^{2} - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.161

5156	$x (a + b y) y^{'} = c y$	[_separable]	✓	1.815

5157	$x (x - a y) y^{'} = y (y - a x)$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	142.997

5158	$x (x^{n} + a y) y^{'} + (b + c y) y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]]	✓	5.977

5159	$(1 - x^{2} y) y^{'} + 1 - x y^{2} = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.514

5160	$(1 - x^{2} y) y^{'} - 1 + x y^{2} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.845

5161	$x (1 - x y) y^{'} + (1 + x y) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.064

5162	$x (2 + x y) y^{'} = 3 + 2 x^{3} - 2 y - x y^{2}$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.833

5163	$x (2 - x y) y^{'} + 2 y - x y^{2} (1 + x y) = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]	✓	2.752

5164	$x (3 - x y) y^{'} = y (x y - 1)$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.492

5165	$x^{2} (1 - y) y^{'} + (1 - x) y = 0$	[_separable]	✓	1.744

5166	$x^{2} (1 - y) y^{'} + (x + 1) y^{2} = 0$	[_separable]	✓	1.726

5167	$(x^{2} + 1) y y^{'} + x (1 - y^{2}) = 0$	[_separable]	✓	4.401

5168	$(- x^{2} + 1) y y^{'} + 2 x^{2} + x y^{2} = 0$	[_rational, _Bernoulli]	✓	2.161

5169	$2 x^{2} y y^{'} = x^{2} (2 x + 1) - y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	2.613

5170	$x (1 - 2 x y) y^{'} + (2 x y + 1) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.116

5171	$x (2 x y + 1) y^{'} + (2 + 3 x y) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	11.414

5172	$x (2 x y + 1) y^{'} + (1 + 2 x y - y^{2} x^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]	✓	2.977

5173	$x^{2} (x - 2 y) y^{'} = 2 x^{3} - 4 x y^{2} + y^{3}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]	✓	39.067

5174	$2 (x + 1) x y y^{'} = 1 + y^{2}$	[_separable]	✓	3.461

5175	$3 x^{2} y y^{'} + 1 + 2 x y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.662

5176	$x^{2} (4 x - 3 y) y^{'} = (6 x^{2} - 3 x y + 2 y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]	✓	23.202

5177	$(1 - x^{3} y) y^{'} = y^{2} x^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.398

5178	$2 x^{3} y y^{'} + a + 3 y^{2} x^{2} = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]	✓	2.014

5179	$x (3 - 2 x^{2} y) y^{'} = 4 x - 3 y + 3 y^{2} x^{2}$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.820

5180	$x (3 + 2 x^{2} y) y^{'} + (4 + 3 x^{2} y) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.052

5181	$8 x^{3} y y^{'} + 3 x^{4} - 6 y^{2} x^{2} - y^{4} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.036

5182	$x y (b x^{2} + a) y^{'} = A + B y^{2}$	[_separable]	✓	4.823

5183	$3 x^{4} y y^{'} = 1 - 2 x^{3} y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	3.032

5184	$x^{7} y y^{'} = 2 x^{2} + 2 + 5 x^{3} y$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	0.888

5185	$y y^{'} \sqrt{x^{2} + 1} + x \sqrt{1 + y^{2}} = 0$	[_separable]	✓	5.227

5186	$(y + 1) y^{'} \sqrt{x^{2} + 1} = y^{3}$	[_separable]	✓	4.010

5187	$(g 0 (x) + y g 1 (x)) y^{'} = f 0 (x) + f 1 (x) y + f 2 (x) y^{2} + f 3 (x) y^{3}$	[[_Abel, ‘2nd type‘, ‘class C‘]]	✗	5.414

5188	$y^{2} y^{'} + x (2 - y) = 0$	[_separable]	✓	1.326

5189	$y^{2} y^{'} = x (1 + y^{2})$	[_separable]	✓	1.511

5190	$(x + y^{2}) y^{'} + y = b x + a$	[_exact, _rational]	✓	1.343

5191	$(x - y^{2}) y^{'} = x^{2} - y$	[_exact, _rational]	✓	1.187

5192	$(x^{2} + y^{2}) y^{'} + x y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	129.095

5193	$(x^{2} + y^{2}) y^{'} = x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	16.027

5194	$(x^{2} - y^{2}) y^{'} = 2 x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	5.877

5195	$(x^{2} - y^{2}) y^{'} + x (x + 2 y) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	166.025

5196	$(x^{2} + y^{2}) y^{'} + 2 x (y + 2 x) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	4.308

5197	$(1 - x^{2} + y^{2}) y^{'} = 1 + x^{2} - y^{2}$	[[_1st_order, _with_linear_symmetries], _rational]	✓	1.180

5198	$(a^{2} + x^{2} + y^{2}) y^{'} + 2 x y = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.093

5199	$(a^{2} + x^{2} + y^{2}) y^{'} + b^{2} + x^{2} + 2 x y = 0$	[_exact, _rational]	✓	1.309

5200	$(x + x^{2} + y^{2}) y^{'} = y$	[_rational]	✓	1.320

2.2.52 Problems 5101 to 5200