Problems 5501 to 5600

Table 2.113: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

5501	$(a^{2} - x^{2}) {y^{'}}^{2} - 2 x y y^{'} - y^{2} = 0$	[_separable]	✓	0.917

5502	$(a^{2} + x^{2}) {y^{'}}^{2} - 2 x y y^{'} + b + y^{2} = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.855

5503	$4 x^{2} {y^{'}}^{2} - 4 x y y^{'} = 8 x^{3} - y^{2}$	[_linear]	✓	0.531

5504	$a x^{2} {y^{'}}^{2} - 2 a x y y^{'} + a (1 - a) x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.658

5505	$(- a^{2} + 1) x^{2} {y^{'}}^{2} - 2 x y y^{'} - a^{2} x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	71.651

5506	$x^{3} {y^{'}}^{2} = a$	[_quadrature]	✓	0.335

5507	$x^{3} {y^{'}}^{2} + x y^{'} - y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.229

5508	$x^{3} {y^{'}}^{2} + x^{2} y y^{'} + a = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.262

5509	$x (- x^{2} + 1) {y^{'}}^{2} - 2 (- x^{2} + 1) y y^{'} + x (1 - y^{2}) = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	12.957

5510	$4 x (- x + a) (b - x) {y^{'}}^{2} = {(a b - 2 x (a + b) + 2 x^{2})}^{2}$	[_quadrature]	✓	1.055

5511	$x^{4} {y^{'}}^{2} - x y^{'} - y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.912

5512	$x^{4} {y^{'}}^{2} + 2 x^{3} y y^{'} - 4 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.254

5513	$x^{4} {y^{'}}^{2} + x y^{2} y^{'} - y^{3} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.961

5514	$x^{2} (a^{2} - x^{2}) {y^{'}}^{2} + 1 = 0$	[_quadrature]	✓	0.681

5515	$3 x^{4} {y^{'}}^{2} - x y - y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	1.028

5516	$4 x^{5} {y^{'}}^{2} + 12 x^{4} y y^{'} + 9 = 0$	[[_homogeneous, ‘class G‘]]	✓	4.053

5517	$x^{6} {y^{'}}^{2} - 2 x y^{'} - 4 y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.846

5518	$x^{8} {y^{'}}^{2} + 3 x y^{'} + 9 y = 0$	[[_homogeneous, ‘class G‘]]	✓	1.959

5519	$y {y^{'}}^{2} = a$	[_quadrature]	✓	0.389

5520	$y {y^{'}}^{2} = a^{2} x$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	1.536

5521	$y {y^{'}}^{2} = e^{2 x}$	[[_1st_order, _with_linear_symmetries]]	✓	1.169

5522	$y {y^{'}}^{2} + 2 a x y^{'} - a y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.865

5523	$y {y^{'}}^{2} - 4 a^{2} x y^{'} + a^{2} y = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	1.943

5524	$y {y^{'}}^{2} + a x y^{'} + b y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.574

5525	$y {y^{'}}^{2} - (- 2 b x + a) y^{'} - b y = 0$	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	1.461

5526	$y {y^{'}}^{2} + x^{3} y^{'} - x^{2} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	2.905

5527	$y {y^{'}}^{2} + (x - y) y^{'} - x = 0$	[_quadrature]	✓	0.549

5528	$y {y^{'}}^{2} - (x + y) y^{'} + y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.900

5529	$y {y^{'}}^{2} - (1 + x y) y^{'} + x = 0$	[_quadrature]	✓	0.436

5530	$y {y^{'}}^{2} + (x - y^{2}) y^{'} - x y = 0$	[_quadrature]	✓	0.630

5531	$y {y^{'}}^{2} + y = a$	[_quadrature]	✓	0.689

5532	$(x + y) {y^{'}}^{2} + 2 x y^{'} - y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.790

5533	$(2 x - y) {y^{'}}^{2} - 2 (1 - x) y^{'} + 2 - y = 0$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	0.804

5534	$2 y {y^{'}}^{2} + (5 - 4 x) y^{'} + 2 y = 0$	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	0.764

5535	$9 y {y^{'}}^{2} + 4 x^{3} y^{'} - 4 x^{2} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	3.119

5536	$(1 - a y) {y^{'}}^{2} = a y$	[_quadrature]	✓	0.586

5537	$(x^{2} - a y) {y^{'}}^{2} - 2 x y y^{'} = 0$	[_quadrature]	✓	0.834

5538	$x y {y^{'}}^{2} + (x + y) y^{'} + 1 = 0$	[_quadrature]	✓	0.494

5539	$x y {y^{'}}^{2} + (x^{2} + y^{2}) y^{'} + x y = 0$	[_separable]	✓	1.467

5540	$x y {y^{'}}^{2} + (x^{2} - y^{2}) y^{'} - x y = 0$	[_separable]	✓	1.208

5541	$x y {y^{'}}^{2} - (x^{2} - y^{2}) y^{'} - x y = 0$	[_separable]	✓	1.342

5542	$x y {y^{'}}^{2} + (a + x^{2} - y^{2}) y^{'} - x y = 0$	[_rational]	✓	141.127

5543	$x y {y^{'}}^{2} - (a - b x^{2} + y^{2}) y^{'} - b x y = 0$	[_rational]	✓	101.730

5544	$x y {y^{'}}^{2} + (3 x^{2} - 2 y^{2}) y^{'} - 6 x y = 0$	[_separable]	✓	1.424

5545	$x (x - 2 y) {y^{'}}^{2} - 2 x y y^{'} - 2 x y + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	150.813

5546	$x (x - 2 y) {y^{'}}^{2} + 6 x y y^{'} - 2 x y + y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	72.950

5547	$y^{2} {y^{'}}^{2} = a^{2}$	[_quadrature]	✓	0.651

5548	$y^{2} {y^{'}}^{2} - a^{2} + y^{2} = 0$	[_quadrature]	✓	0.546

5549	$y^{2} {y^{'}}^{2} - 3 x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.476

5550	$y^{2} {y^{'}}^{2} - 6 x^{3} y^{'} + 4 x^{2} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	2.993

5551	$y^{2} {y^{'}}^{2} - 4 a y y^{'} + 4 a^{2} - 4 a x + y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	72.300

5552	$y^{2} {y^{'}}^{2} - (x + 1) y y^{'} + x = 0$	[_quadrature]	✓	0.616

5553	$y^{2} {y^{'}}^{2} + 2 x y y^{'} + x^{2} = 0$	[_separable]	✓	0.720

5554	$y^{2} {y^{'}}^{2} + 2 x y y^{'} + a - y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	31.248

5555	$y^{2} {y^{'}}^{2} - 2 x y y^{'} - x^{2} + 2 y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	13.391

5556	$y^{2} {y^{'}}^{2} - 2 x y y^{'} + a - x^{2} + 2 y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	80.392

5557	$y^{2} {y^{'}}^{2} + 2 a x y y^{'} + (a - 1) b + a x^{2} + (1 - a) y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	10.843

5558	$(1 - y^{2}) {y^{'}}^{2} = 1$	[_quadrature]	✓	30.265

5559	$(a^{2} - y^{2}) {y^{'}}^{2} = y^{2}$	[_quadrature]	✓	0.681

5560	$(a^{2} - 2 y a x + y^{2}) {y^{'}}^{2} + 2 a y y^{'} + y^{2} = 0$	[‘y=_G(x,y’)‘]	✓	84.885

5561	$((1 - a) x^{2} + y^{2}) {y^{'}}^{2} + 2 a x y y^{'} + x^{2} + (1 - a) y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	305.651

5562	$((- 4 a^{2} + 1) x^{2} + y^{2}) {y^{'}}^{2} - 8 a^{2} x y y^{'} + x^{2} + (- 4 a^{2} + 1) y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	340.019

5563	$((- a^{2} + 1) x^{2} + y^{2}) {y^{'}}^{2} + 2 a^{2} x y y^{'} + x^{2} + (- a^{2} + 1) y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	191.463

5564	${(x + y)}^{2} {y^{'}}^{2} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.783

5565	${(x + y)}^{2} {y^{'}}^{2} - (x^{2} - x y - 2 y^{2}) y^{'} - (x - y) y = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.844

5566	$(a^{2} - {(x - y)}^{2}) {y^{'}}^{2} + 2 a^{2} y^{'} + a^{2} - {(x - y)}^{2} = 0$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	7.540

5567	$2 y^{2} {y^{'}}^{2} + 2 x y y^{'} - 1 + x^{2} + y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	143.572

5568	$3 y^{2} {y^{'}}^{2} - 2 x y y^{'} - x^{2} + 4 y^{2} = 0$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	3.462

5569	$4 y^{2} {y^{'}}^{2} + 2 (1 + 3 x) x y y^{'} + 3 x^{3} = 0$	[_separable]	✓	0.887

5570	$(x^{2} - 4 y^{2}) {y^{'}}^{2} + 6 x y y^{'} - 4 x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	10.924

5571	$9 y^{2} {y^{'}}^{2} - 3 x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.488

5572	${(2 - 3 y)}^{2} {y^{'}}^{2} = 4 - 4 y$	[_quadrature]	✓	0.398

5573	$(- a^{2} + 1) y^{2} {y^{'}}^{2} - 3 a^{2} x y y^{'} - a^{2} x^{2} + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	327.426

5574	$(a - b) y^{2} {y^{'}}^{2} - 2 b x y y^{'} - a b - b x^{2} + a y^{2} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	11.929

5575	$a^{2} (b^{2} - {(c x - a y)}^{2}) {y^{'}}^{2} + 2 a b^{2} c y^{'} + c^{2} (b^{2} - {(c x - a y)}^{2}) = 0$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	8.647

5576	$x y^{2} {y^{'}}^{2} - y^{3} y^{'} + a^{2} x = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.966

5577	$x y^{2} {y^{'}}^{2} + (a - x^{3} - y^{3}) y^{'} + x^{2} y = 0$	[_rational]	✓	18.773

5578	$2 x y^{2} {y^{'}}^{2} - y^{3} y^{'} - a = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.247

5579	$4 x^{2} y^{2} {y^{'}}^{2} = {(x^{2} + y^{2})}^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	150.332

5580	$4 y^{3} {y^{'}}^{2} - 4 x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.559

5581	$3 x y^{4} {y^{'}}^{2} - y^{5} y^{'} + 1 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.509

5582	$9 x y^{4} {y^{'}}^{2} - 3 y^{5} y^{'} - a = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.444

5583	$9 (- x^{2} + 1) y^{4} {y^{'}}^{2} + 6 x y^{5} y^{'} + 4 x^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	16.871

5584	${y^{'}}^{3} = b x + a$	[_quadrature]	✓	0.406

5585	${y^{'}}^{3} = a x^{n}$	[_quadrature]	✓	0.529

5586	${y^{'}}^{3} + x - y = 0$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	0.473

5587	${y^{'}}^{3} = (a + b y + c y^{2}) f (x)$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	1.311

5588	${y^{'}}^{3} = {(y - a)}^{2} {(y - b)}^{2}$	[_quadrature]	✓	6.161

5589	${y^{'}}^{3} + f (x) {(y - a)}^{2} {(y - b)}^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	1.569

5590	${y^{'}}^{3} + f (x) {(y - a)}^{2} {(y - b)}^{2} {(y - c)}^{2} = 0$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	3.332

5591	${y^{'}}^{3} + y^{'} + a - b x = 0$	[_quadrature]	✓	0.809

5592	${y^{'}}^{3} + y^{'} - y = 0$	[_quadrature]	✓	0.453

5593	${y^{'}}^{3} + y^{'} = e^{y}$	[_quadrature]	✓	0.687

5594	${y^{'}}^{3} - 7 y^{'} + 6 = 0$	[_quadrature]	✓	0.705

5595	${y^{'}}^{3} - x y^{'} + a y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.685

5596	${y^{'}}^{3} + 2 x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.469

5597	${y^{'}}^{3} - 2 x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.441

5598	${y^{'}}^{3} - a x y^{'} + x^{3} = 0$	[_quadrature]	✓	0.779

5599	${y^{'}}^{3} + a x y^{'} - a y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.503

5600	${y^{'}}^{3} - (b x + a) y^{'} + b y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.511

2.2.56 Problems 5501 to 5600