Problems 5401 to 5500

Table 2.111: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

5401	${y^{'}}^{2} - a y y^{'} - a x = 0$	[_dAlembert]	✓	85.914

5402	${y^{'}}^{2} + (a x + b y) y^{'} + a b x y = 0$	[_quadrature]	✓	0.503

5403	${y^{'}}^{2} - x y^{'} y + y^{2} \ln (a y) = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	5.178

5404	${y^{'}}^{2} - (1 + 2 x y) y^{'} + 2 x y = 0$	[_quadrature]	✓	0.403

5405	${y^{'}}^{2} - (4 + y^{2}) y^{'} + 4 + y^{2} = 0$	[_quadrature]	✓	1.751

5406	${y^{'}}^{2} - (x - y) y y^{'} - x y^{3} = 0$	[_separable]	✓	0.474

5407	${y^{'}}^{2} + x y^{2} y^{'} + y^{3} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.869

5408	${y^{'}}^{2} - 2 x^{3} y^{2} y^{'} - 4 x^{2} y^{3} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	2.188

5409	${y^{'}}^{2} - x y (x^{2} + y^{2}) y^{'} + x^{4} y^{4} = 0$	[_separable]	✓	0.670

5410	${y^{'}}^{2} + 2 x y^{3} y^{'} + y^{4} = 0$	[[_homogeneous, ‘class G‘]]	✓	1.753

5411	${y^{'}}^{2} + 2 y y^{'} \cot (x) - y^{2} = 0$	[_separable]	✓	1.305

5412	${y^{'}}^{2} - 3 x y^{2 / 3} y^{'} + 9 y^{5 / 3} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	4.230

5413	${y^{'}}^{2} = e^{4 x - 2 y} (y^{'} - 1)$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	1.224

5414	$2 {y^{'}}^{2} + y^{'} x - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.665

5415	$2 {y^{'}}^{2} - (1 - x) y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.511

5416	$2 {y^{'}}^{2} - 2 x^{2} y^{'} + 3 x y = 0$	[[_homogeneous, ‘class G‘]]	✓	2.150

5417	$2 {y^{'}}^{2} + 2 (6 y - 1) y^{'} + 3 y (6 y - 1) = 0$	[_quadrature]	✓	10.929

5418	$3 {y^{'}}^{2} - 2 y^{'} x + y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.538

5419	$3 {y^{'}}^{2} + 4 y^{'} x + x^{2} - y = 0$	[[_homogeneous, ‘class G‘]]	✓	1.784

5420	$4 {y^{'}}^{2} = 9 x$	[_quadrature]	✓	0.419

5421	$4 {y^{'}}^{2} + 2 x e^{- 2 y} y^{'} - e^{- 2 y} = 0$	[[_1st_order, _with_linear_symmetries]]	✗	19.778

5422	$4 {y^{'}}^{2} + 2 e^{2 x - 2 y} y^{'} - e^{2 x - 2 y} = 0$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	1.267

5423	$5 {y^{'}}^{2} + 3 y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.573

5424	$5 {y^{'}}^{2} + 6 y^{'} x - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.561

5425	$9 {y^{'}}^{2} + 3 x y^{4} y^{'} + y^{5} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	71.931

5426	$x {y^{'}}^{2} = a$	[_quadrature]	✓	0.362

5427	$x {y^{'}}^{2} = - x^{2} + a$	[_quadrature]	✓	0.763

5428	$x {y^{'}}^{2} = y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	5.093

5429	$x {y^{'}}^{2} + x - 2 y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.036

5430	$x {y^{'}}^{2} + y^{'} = y$	[_rational, _dAlembert]	✓	1.219

5431	$x {y^{'}}^{2} + 2 y^{'} - y = 0$	[_rational, _dAlembert]	✓	1.191

5432	$x {y^{'}}^{2} - 2 y^{'} - y = 0$	[_rational, _dAlembert]	✓	1.201

5433	$x {y^{'}}^{2} + 4 y^{'} - 2 y = 0$	[_rational, _dAlembert]	✓	1.319

5434	$x {y^{'}}^{2} + y^{'} x - y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.032

5435	$x {y^{'}}^{2} - (x^{2} + 1) y^{'} + x = 0$	[_quadrature]	✓	0.297

5436	$x {y^{'}}^{2} + y y^{'} + a = 0$	[[_homogeneous, ‘class G‘], _dAlembert]	✓	0.627

5437	$x {y^{'}}^{2} - y y^{'} + a = 0$	[[_homogeneous, ‘class G‘], _rational, _Clairaut]	✓	0.544

5438	$x {y^{'}}^{2} - y y^{'} + a x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.106

5439	$x {y^{'}}^{2} + y y^{'} + x^{3} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.366

5440	$x {y^{'}}^{2} - y y^{'} + a y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.830

5441	$x {y^{'}}^{2} + y y^{'} - y^{4} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.249

5442	$x {y^{'}}^{2} + (- y + a) y^{'} + b = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.594

5443	$x {y^{'}}^{2} + (x - y) y^{'} + 1 - y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	0.633

5444	$x {y^{'}}^{2} + (a + x - y) y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	0.687

5445	$x {y^{'}}^{2} - (3 x - y) y^{'} + y = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	3.938

5446	$x {y^{'}}^{2} + a + b x - y - b y = 0$	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	1.239

5447	$x {y^{'}}^{2} - 2 y y^{'} + a = 0$	[[_homogeneous, ‘class G‘], _rational, _dAlembert]	✓	0.682

5448	$x {y^{'}}^{2} - 2 y y^{'} + a x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.592

5449	$x {y^{'}}^{2} - 2 y y^{'} + x + 2 y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.881

5450	$x {y^{'}}^{2} - 3 y y^{'} + 9 x^{2} = 0$	[[_homogeneous, ‘class G‘]]	✓	4.553

5451	$x {y^{'}}^{2} - (2 x + 3 y) y^{'} + 6 y = 0$	[_quadrature]	✓	0.428

5452	$x {y^{'}}^{2} - a y y^{'} + b = 0$	[[_homogeneous, ‘class G‘], _rational, _dAlembert]	✓	0.888

5453	$x {y^{'}}^{2} + a y y^{'} + b x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.506

5454	$x {y^{'}}^{2} - (x y + 1) y^{'} + y = 0$	[_quadrature]	✓	0.310

5455	$x {y^{'}}^{2} + y (1 - x) y^{'} - y^{2} = 0$	[_quadrature]	✓	0.366

5456	$x {y^{'}}^{2} + (1 - x^{2} y) y^{'} - x y = 0$	[_quadrature]	✓	0.450

5457	$(x + 1) {y^{'}}^{2} = y$	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	1.003

5458	$(x + 1) {y^{'}}^{2} - (x + y) y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	0.654

5459	$(a - x) {y^{'}}^{2} + y y^{'} - b = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.602

5460	$2 x {y^{'}}^{2} + (2 x - y) y^{'} + 1 - y = 0$	[_rational, _dAlembert]	✓	1.454

5461	$3 x {y^{'}}^{2} - 6 y y^{'} + x + 2 y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.069

5462	$(3 x + 1) {y^{'}}^{2} - 3 (y + 2) y^{'} + 9 = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.652

5463	$(3 x + 5) {y^{'}}^{2} - (3 + 3 y) y^{'} + y = 0$	[_rational, _dAlembert]	✓	2.256

5464	$4 x {y^{'}}^{2} = {(a - 3 x)}^{2}$	[_quadrature]	✓	0.473

5465	$4 x {y^{'}}^{2} + 2 y^{'} x - y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	35.092

5466	$4 x {y^{'}}^{2} - 3 y y^{'} + 3 = 0$	[[_homogeneous, ‘class G‘], _rational, _dAlembert]	✓	0.581

5467	$4 x {y^{'}}^{2} + 4 y y^{'} = 1$	[[_homogeneous, ‘class G‘], _rational, _dAlembert]	✓	0.640

5468	$4 x {y^{'}}^{2} + 4 y y^{'} - y^{4} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.343

5469	$4 (2 - x) {y^{'}}^{2} + 1 = 0$	[_quadrature]	✓	0.326

5470	$16 x {y^{'}}^{2} + 8 y y^{'} + y^{6} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.652

5471	$x^{2} {y^{'}}^{2} = a^{2}$	[_quadrature]	✓	0.424

5472	$x^{2} {y^{'}}^{2} = y^{2}$	[_separable]	✓	1.368

5473	$x^{2} {y^{'}}^{2} + x^{2} - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	8.000

5474	$x^{2} {y^{'}}^{2} = {(x - y)}^{2}$	[_linear]	✓	1.037

5475	$x^{2} {y^{'}}^{2} + y^{2} - y^{4} = 0$	[_separable]	✓	2.762

5476	$x^{2} {y^{'}}^{2} - y^{'} x + y (1 - y) = 0$	[_separable]	✓	0.585

5477	$x^{2} {y^{'}}^{2} + 2 a x y^{'} + a^{2} + x^{2} - 2 a y = 0$	[_rational]	✗	70.604

5478	$x^{2} {y^{'}}^{2} - 2 x y^{'} y - x + y (1 + y) = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.488

5479	$x^{2} {y^{'}}^{2} - 2 x y^{'} y - x^{4} + (- x^{2} + 1) y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✗	9.357

5480	$x^{2} {y^{'}}^{2} - (1 + 2 x y) y^{'} + 1 + y^{2} = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.730

5481	$x^{2} {y^{'}}^{2} - (a + 2 x y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational, _Clairaut]	✓	0.820

5482	$x^{2} {y^{'}}^{2} - x (x - 2 y) y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.960

5483	$x^{2} {y^{'}}^{2} + 2 x (2 x + y) y^{'} - 4 a + y^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	4.612

5484	$x^{2} {y^{'}}^{2} + x (x^{3} - 2 y) y^{'} - (2 x^{3} - y) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.089

5485	$x^{2} {y^{'}}^{2} + 3 x y^{'} y + 2 y^{2} = 0$	[_separable]	✓	0.714

5486	$x^{2} {y^{'}}^{2} - 3 x y^{'} y + x^{3} + 2 y^{2} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	73.167

5487	$x^{2} {y^{'}}^{2} + 4 x y^{'} y - 5 y^{2} = 0$	[_separable]	✓	0.758

5488	$x^{2} {y^{'}}^{2} - 4 x (y + 2) y^{'} + 4 (y + 2) y = 0$	[_separable]	✓	0.614

5489	$x^{2} {y^{'}}^{2} - 5 x y^{'} y + 6 y^{2} = 0$	[_separable]	✓	0.682

5490	$x^{2} {y^{'}}^{2} + x (x^{2} + x y - 2 y) y^{'} + (1 - x) (x^{2} - y) y = 0$	[_rational]	✗	85.603

5491	$x^{2} {y^{'}}^{2} + (2 x + y) y y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	125.585

5492	$x^{2} {y^{'}}^{2} + (2 x - y) y y^{'} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	74.387

5493	$x^{2} {y^{'}}^{2} + (a + b x^{2} y^{3}) y^{'} + a b y^{3} = 0$	[_quadrature]	✓	4.462

5494	$(- x^{2} + 1) {y^{'}}^{2} = 1 - y^{2}$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	1.356

5495	$(- x^{2} + 1) {y^{'}}^{2} + 2 x y^{'} y + 4 x^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	110.494

5496	$(a^{2} + x^{2}) {y^{'}}^{2} = b^{2}$	[_quadrature]	✓	0.767

5497	$(a^{2} - x^{2}) {y^{'}}^{2} + b^{2} = 0$	[_quadrature]	✓	0.503

5498	$(a^{2} - x^{2}) {y^{'}}^{2} = b^{2}$	[_quadrature]	✓	0.382

5499	$(a^{2} - x^{2}) {y^{'}}^{2} = x^{2}$	[_quadrature]	✓	0.453

5500	$(a^{2} - x^{2}) {y^{'}}^{2} + 2 x y^{'} y + x^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✗	30.337

2.2.55 Problems 5401 to 5500