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]	✓	2.410

5102	$x y^{'} y = x + y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	2.636

5103	$x y^{'} y + x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	3.852

5104	$x y^{'} y + x^{4} - y^{2} = 0$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	2.363

5105	$x y^{'} y = a x^{3} \cos (x) + y^{2}$	[[_homogeneous, ‘class D‘], _Bernoulli]	✓	3.436

5106	$x y^{'} y = x^{2} - x y + y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	8.587

5107	$x y^{'} y + 2 x^{2} - 2 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.471

5108	$x y^{'} y = a + b y^{2}$	[_separable]	✓	2.514

5109	$x y^{'} y = a x^{n} + b y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.824

5110	$x y^{'} y = (x^{2} + 1) (1 - y^{2})$	[_separable]	✓	2.699

5111	$x y^{'} y + x^{2} arccot (\frac{y}{x}) - y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	8.456

5112	$x y^{'} y + x^{2} e^{- \frac{2 y}{x}} - y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	2.934

5113	$(x y + 1) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.494

5114	$x (1 + y) y^{'} - (1 - x) y = 0$	[_separable]	✓	1.474

5115	$x (1 - y) y^{'} + (x + 1) y = 0$	[_separable]	✓	1.552

5116	$x (1 - y) y^{'} + (1 - x) y = 0$	[_separable]	✓	1.571

5117	$x (y + 2) y^{'} + a x = 0$	[_quadrature]	✓	1.036

5118	$(2 + 3 x - x y) y^{'} + y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.291

5119	$x (4 + y) y^{'} = 2 x + 2 y + y^{2}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	6.186

5120	$x (a + y) y^{'} + b x + c y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.051

5121	$x (a + y) y^{'} = y (B x + A)$	[_separable]	✓	1.803

5122	$x (x + y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	6.984

5123	$x (x - y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	36.496

5124	$x (x + y) y^{'} = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.896

5125	$x (x - y) y^{'} + 2 x^{2} + 3 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.504

5126	$x (x + y) y^{'} - y (x + y) + x \sqrt{x^{2} - y^{2}} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	4.203

5127	$(a + (x + y) x) y^{'} = b (x + y) y$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.518

5128	$x (2 x + y) y^{'} = x^{2} + x y - y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.117

5129	$x (4 x - y) y^{'} + 4 x^{2} - 6 x y - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	16.024

5130	$x (x^{3} + y) y^{'} = (x^{3} - y) y$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.914

5131	$x (2 x^{3} + y) y^{'} = (2 x^{3} - y) y$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	7.525

5132	$x (2 x^{3} + y) y^{'} = 6 y^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.330

5133	$y (1 - x) y^{'} + x (1 - y) = 0$	[_separable]	✓	1.451

5134	$(x + a) (x + b) y^{'} = x y$	[_separable]	✓	2.039

5135	$2 x y^{'} y + 1 - 2 x^{3} - y^{2} = 0$	[_rational, _Bernoulli]	✓	1.694

5136	$2 x y^{'} y + a + y^{2} = 0$	[_separable]	✓	2.390

5137	$2 x y^{'} y = a x + y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	1.840

5138	$x^{2} + y^{2} + 2 x y^{'} y = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]	✓	3.642

5139	$2 x y^{'} y = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	7.189

5140	$2 x y^{'} y = 4 x^{2} (2 x + 1) + y^{2}$	[_rational, _Bernoulli]	✓	4.190

5141	$2 x y^{'} y + x^{2} (a x^{3} + 1) = 6 y^{2}$	[_rational, _Bernoulli]	✓	1.722

5142	$(3 - x + 2 x y) y^{'} + 3 x^{2} - y + y^{2} = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.793

5143	$x (x - 2 y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	22.306

5144	$x (x + 2 y) y^{'} + (2 x - y) y = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	36.503

5145	$x (x - 2 y) y^{'} + (2 x - y) y = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.634

5146	$x (x - 2 y + 1) y^{'} + (1 - 2 x + y) y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.537

5147	$x (1 - x - 2 y) y^{'} + (2 x + y + 1) y = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.472

5148	$2 x (2 x^{2} + y) y^{'} + (12 x^{2} + y) y = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.373

5149	$2 (x + 1) y y^{'} + 2 x - 3 x^{2} + y^{2} = 0$	[_exact, _rational, _Bernoulli]	✓	2.161

5150	$x (2 x + 3 y) y^{'} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.515

5151	$x (2 x + 3 y) y^{'} + 3 {(x + y)}^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.326

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	$3 x (x + 2 y) y^{'} + x^{3} + 3 y (2 x + y) = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.759

5154	$a x y y^{'} = x^{2} + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.568

5155	$a x y y^{'} + x^{2} - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.710

5156	$x (a + b y) y^{'} = c y$	[_separable]	✓	2.012

5157	$x (x - a y) y^{'} = y (y - a x)$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	5.192

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‘]]	✓	5.080

5159	$(1 - x^{2} y) y^{'} + 1 - x y^{2} = 0$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.533

5160	$(1 - x^{2} y) y^{'} - 1 + x y^{2} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.903

5161	$x (1 - x y) y^{'} + (x y + 1) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.895

5162	$x (2 + x y) y^{'} = 3 + 2 x^{3} - 2 y - x y^{2}$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.853

5163	$x (2 - x y) y^{'} + 2 y - x y^{2} (x y + 1) = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]	✓	2.402

5164	$x (3 - x y) y^{'} = y (x y - 1)$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.690

5165	$x^{2} (1 - y) y^{'} + (1 - x) y = 0$	[_separable]	✓	1.816

5166	$x^{2} (1 - y) y^{'} + (x + 1) y^{2} = 0$	[_separable]	✓	1.851

5167	$(x^{2} + 1) y y^{'} + x (1 - y^{2}) = 0$	[_separable]	✓	3.858

5168	$(- x^{2} + 1) y y^{'} + 2 x^{2} + x y^{2} = 0$	[_rational, _Bernoulli]	✓	1.933

5169	$2 x^{2} y y^{'} = x^{2} (2 x + 1) - y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	2.506

5170	$x (1 - 2 x y) y^{'} + y (1 + 2 x y) = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.044

5171	$x (1 + 2 x y) y^{'} + (2 + 3 x y) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	10.342

5172	$x (1 + 2 x y) y^{'} + (1 + 2 x y - x^{2} y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]]	✓	2.632

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]	✓	43.818

5174	$2 (x + 1) x y y^{'} = 1 + y^{2}$	[_separable]	✓	3.060

5175	$3 x^{2} y y^{'} + 1 + 2 x y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.401

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]	✓	18.332

5177	$(1 - x^{3} y) y^{'} = x^{2} y^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.845

5178	$2 x^{3} y y^{'} + a + 3 x^{2} y^{2} = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]	✓	2.019

5179	$x (3 - 2 x^{2} y) y^{'} = 4 x - 3 y + 3 x^{2} y^{2}$	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.688

5180	$x (3 + 2 x^{2} y) y^{'} + (4 + 3 x^{2} y) y = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.947

5181	$8 x^{3} y y^{'} + 3 x^{4} - 6 x^{2} y^{2} - y^{4} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.411

5182	$x y (b x^{2} + a) y^{'} = A + B y^{2}$	[_separable]	✓	4.402

5183	$3 x^{4} y y^{'} = 1 - 2 x^{3} y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.358

5184	$x^{7} y y^{'} = 2 x^{2} + 2 + 5 x^{3} y$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.072

5185	$y y^{'} \sqrt{x^{2} + 1} + x \sqrt{1 + y^{2}} = 0$	[_separable]	✓	2.758

5186	$(1 + y) y^{'} \sqrt{x^{2} + 1} = y^{3}$	[_separable]	✓	2.885

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‘]]	✗	4.786

5188	$y^{2} y^{'} + x (2 - y) = 0$	[_separable]	✓	1.306

5189	$y^{2} y^{'} = x (1 + y^{2})$	[_separable]	✓	1.367

5190	$(x + y^{2}) y^{'} + y = b x + a$	[_exact, _rational]	✓	1.407

5191	$(x - y^{2}) y^{'} = x^{2} - y$	[_exact, _rational]	✓	1.223

5192	$x y + (x^{2} + y^{2}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	4.217

5193	$(x^{2} + y^{2}) y^{'} = x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.419

5194	$(x^{2} - y^{2}) y^{'} = 2 x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	4.709

5195	$(x^{2} - y^{2}) y^{'} + x (x + 2 y) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	135.969

5196	$(x^{2} + y^{2}) y^{'} + 2 x (2 x + y) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	4.158

5197	$(1 - x^{2} + y^{2}) y^{'} = 1 + x^{2} - y^{2}$	[[_1st_order, _with_linear_symmetries], _rational]	✓	1.171

5198	$(a^{2} + x^{2} + y^{2}) y^{'} + 2 x y = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.223

5199	$(a^{2} + x^{2} + y^{2}) y^{'} + b^{2} + x^{2} + 2 x y = 0$	[_exact, _rational]	✓	1.393

5200	$(x^{2} + y^{2} + x) y^{'} = y$	[_rational]	✓	1.229

2.2.52 Problems 5101 to 5200