Problems 5201 to 5300

Table 2.107: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

5201	$(3 x^{2} - y^{2}) y^{'} = 2 x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.666

5202	$(x^{4} + y^{2}) y^{'} = 4 x^{3} y$	[[_homogeneous, ‘class G‘], _rational]	✓	3.072

5203	$y (1 + y) y^{'} = x (x + 1)$	[_separable]	✓	1.506

5204	$(x + 2 y + y^{2}) y^{'} + y (1 + y) + {(x + y)}^{2} y^{2} = 0$	[_rational]	✗	3.893

5205	$(x^{2} + 2 y + y^{2}) y^{'} + 2 x = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.385

5206	$(x^{3} + 2 y - y^{2}) y^{'} + 3 x^{2} y = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.222

5207	$(1 + y + x y + y^{2}) y^{'} + 1 + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	1.484

5208	${(x + y)}^{2} y^{'} = a^{2}$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	12.796

5209	${(x - y)}^{2} y^{'} = a^{2}$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	4.316

5210	$(x^{2} + 2 x y - y^{2}) y^{'} + x^{2} - 2 x y + y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.153

5211	${(x + y)}^{2} y^{'} = x^{2} - 2 x y + 5 y^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	11.302

5212	${(a + b + x + y)}^{2} y^{'} = 2 {(a + y)}^{2}$	[[_homogeneous, ‘class C‘], _rational]	✓	2.350

5213	$(2 x^{2} + 4 x y - y^{2}) y^{'} = x^{2} - 4 x y - 2 y^{2}$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	4.390

5214	${(3 x + y)}^{2} y^{'} = 4 (3 x + 2 y) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	35.362

5215	${(1 - 3 x - y)}^{2} y^{'} = (1 - 2 y) (3 - 6 x - 4 y)$	[[_homogeneous, ‘class C‘], _rational]	✓	2.882

5216	$(\cot (x) - 2 y^{2}) y^{'} = y^{3} \csc (x) \sec (x)$	[‘y=_G(x,y’)‘]	✗	57.801

5217	$3 y^{2} y^{'} = 1 + x + a y^{3}$	[_rational, _Bernoulli]	✓	2.004

5218	$(x^{2} - 3 y^{2}) y^{'} + 1 + 2 x y = 0$	[_exact, _rational]	✓	1.218

5219	$(2 x^{2} + 3 y^{2}) y^{'} + x (3 x + y) = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.166

5220	$3 (x^{2} - y^{2}) y^{'} + 3 e^{x} + 6 x y (x + 1) - 2 y^{3} = 0$	[‘y=_G(x,y’)‘]	✓	2.027

5221	$(3 x^{2} + 2 x y + 4 y^{2}) y^{'} + 2 x^{2} + 6 x y + y^{2} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	6.907

5222	${(1 - 3 x + 2 y)}^{2} y^{'} = {(4 + 2 x - 3 y)}^{2}$	[[_homogeneous, ‘class C‘], _rational]	✓	36.310

5223	$(1 - 3 x^{2} y + 6 y^{2}) y^{'} + x^{2} - 3 x y^{2} = 0$	[_exact, _rational]	✓	1.607

5224	${(x - 6 y)}^{2} y^{'} + a + 2 x y - 6 y^{2} = 0$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]	✓	1.581

5225	$(x^{2} + a y^{2}) y^{'} = x y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.519

5226	$(x^{2} + x y + a y^{2}) y^{'} = a x^{2} + x y + y^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	25.095

5227	$(a x^{2} + 2 x y - a y^{2}) y^{'} + x^{2} - 2 y a x - y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	25.224

5228	$(a x^{2} + 2 b x y + c y^{2}) y^{'} + k x^{2} + 2 y a x + b y^{2} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	547.793

5229	$x (1 - y^{2}) y^{'} = (x^{2} + 1) y$	[_separable]	✓	2.032

5230	$x (3 x - y^{2}) y^{'} + (5 x - 2 y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.439

5231	$x (x^{2} + y^{2}) y^{'} = (x^{2} + x^{4} + y^{2}) y$	[[_homogeneous, ‘class D‘], _rational]	✓	1.802

5232	$x (1 - x^{2} + y^{2}) y^{'} + (1 + x^{2} - y^{2}) y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	5.343

5233	$x (a - x^{2} - y^{2}) y^{'} + (a + x^{2} + y^{2}) y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	6.561

5234	$x (2 x^{2} + y^{2}) y^{'} = (2 x^{2} + 3 y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	28.595

5235	$(x (a - x^{2} - y^{2}) + y) y^{'} + x - (a - x^{2} - y^{2}) y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	1.959

5236	$x {(a + y)}^{2} y^{'} = b y^{2}$	[_separable]	✓	1.920

5237	$x (x^{2} - x y + y^{2}) y^{'} + (x^{2} + x y + y^{2}) y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	174.604

5238	$x (x^{2} - x y - y^{2}) y^{'} = (x^{2} + x y - y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	170.251

5239	$x (x^{2} + y a x + y^{2}) y^{'} = (x^{2} + b x y + y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	396.954

5240	$x (x^{2} - 2 y^{2}) y^{'} = (2 x^{2} - y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	32.273

5241	$x (x^{2} + 2 y^{2}) y^{'} = (2 x^{2} + 3 y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	33.554

5242	$2 x (5 x^{2} + y^{2}) y^{'} = x^{2} y - y^{3}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	43.500

5243	$x (x^{2} + y a x + 2 y^{2}) y^{'} = (a x + 2 y) y^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	582.150

5244	$3 x y^{2} y^{'} = 2 x - y^{3}$	[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]	✓	2.319

5245	$(1 - 4 x + 3 x y^{2}) y^{'} = (2 - y^{2}) y$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.748

5246	$x (x - 3 y^{2}) y^{'} + (2 x - y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _exact, _rational]	✓	1.902

5247	$3 x (x + y^{2}) y^{'} + x^{3} - 3 x y - 2 y^{3} = 0$	[_rational]	✓	1.560

5248	$x (x^{3} - 3 x^{3} y + 4 y^{2}) y^{'} = 6 y^{3}$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.817

5249	$6 x y^{2} y^{'} + x + 2 y^{3} = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli]	✓	2.511

5250	$x (6 y^{2} + x) y^{'} + x y - 3 y^{3} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.658

5251	$x (x^{2} - 6 y^{2}) y^{'} = 4 (x^{2} + 3 y^{2}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	34.993

5252	$x (3 x - 7 y^{2}) y^{'} + (5 x - 3 y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.292

5253	$x^{2} y^{2} y^{'} + 1 - x + x^{3} = 0$	[_separable]	✓	3.121

5254	$(1 - x^{2} y^{2}) y^{'} = x y^{3}$	[[_homogeneous, ‘class G‘], _rational]	✓	2.205

5255	$(1 - x^{2} y^{2}) y^{'} = (x y + 1) y^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.732

5256	$x (x y^{2} + 1) y^{'} + y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	7.604

5257	$x (x y^{2} + 1) y^{'} = (2 - 3 x y^{2}) y$	[[_homogeneous, ‘class G‘], _rational]	✓	5.744

5258	$x^{2} {(a + y)}^{2} y^{'} = (x^{2} + 1) (y^{2} + a^{2})$	[_separable]	✓	1.562

5259	$(x^{2} + 1) (1 + y^{2}) y^{'} + 2 x y (1 - y^{2}) = 0$	[_separable]	✓	53.398

5260	$(x^{2} + 1) (1 + y^{2}) y^{'} + 2 x y {(1 - y)}^{2} = 0$	[_separable]	✓	2.244

5261	$(1 - x^{3} + 6 x^{2} y^{2}) y^{'} = (6 + 3 x y - 4 y^{3}) x$	[_exact, _rational]	✓	1.780

5262	$x (3 + 5 x - 12 x y^{2} + 4 x^{2} y) y^{'} + (3 + 10 x - 8 x y^{2} + 6 x^{2} y) y = 0$	[_exact, _rational]	✓	2.164

5263	$x^{3} (1 + y^{2}) y^{'} + 3 x^{2} y = 0$	[_separable]	✓	3.134

5264	$x {(1 - x y)}^{2} y^{'} + (1 + x^{2} y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.835

5265	$(1 - x^{4} y^{2}) y^{'} = x^{3} y^{3}$	[[_homogeneous, ‘class G‘], _rational]	✓	4.427

5266	$(3 x - y^{3}) y^{'} = x^{2} - 3 y$	[_exact, _rational]	✓	1.304

5267	$(x^{3} - y^{3}) y^{'} + x^{2} y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	141.925

5268	$(x^{3} + y^{3}) y^{'} + x^{2} (a x + 3 y) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	181.849

5269	$(x - x^{2} y - y^{3}) y^{'} = x^{3} - y + x y^{2}$	[_exact, _rational]	✓	1.771

5270	$(a^{2} x + y (x^{2} - y^{2})) y^{'} + x (x^{2} - y^{2}) = a^{2} y$	[_rational]	✓	1.852

5271	$(a + x^{2} + y^{2}) y y^{'} = x (a - x^{2} - y^{2})$	[_exact, _rational]	✓	2.143

5272	$(3 x^{2} + y^{2}) y y^{'} + x (x^{2} + 3 y^{2}) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	10.589

5273	$(a - 3 x^{2} - y^{2}) y y^{'} + x (a - x^{2} + y^{2}) = 0$	[_rational]	✗	2.712

5274	$2 y^{3} y^{'} = x^{3} - x y^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	23.018

5275	$y (1 + 2 y^{2}) y^{'} = x (2 x^{2} + 1)$	[_separable]	✓	2.725

5276	$(3 x^{2} + 2 y^{2}) y y^{'} + x^{3} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	176.731

5277	$(5 x^{2} + 2 y^{2}) y y^{'} + x (x^{2} + 5 y^{2}) = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	190.300

5278	$(x^{2} - x^{3} + 3 x y^{2} + 2 y^{3}) y^{'} + 2 x^{3} + 3 x^{2} y + y^{2} - y^{3} = 0$	[_rational]	✗	2.765

5279	$(3 x^{3} + 6 x^{2} y - 3 x y^{2} + 20 y^{3}) y^{'} + 4 x^{3} + 9 x^{2} y + 6 x y^{2} - y^{3} = 0$	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	251.397

5280	$(x^{3} + a y^{3}) y^{'} = x^{2} y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	21.618

5281	$x y^{3} y^{'} = (- x^{2} + 1) (1 + y^{2})$	[_separable]	✓	2.360

5282	$x (x - y^{3}) y^{'} = (3 x + y^{3}) y$	[[_homogeneous, ‘class G‘], _rational]	✓	2.467

5283	$x (2 x^{3} + y^{3}) y^{'} = (2 x^{3} - x^{2} y + y^{3}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	4.424

5284	$x (2 x^{3} - y^{3}) y^{'} = (x^{3} - 2 y^{3}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	21.054

5285	$x (x^{3} + 3 x^{2} y + y^{3}) y^{'} = (3 x^{2} + y^{2}) y^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	4.860

5286	$x (x^{3} - 2 y^{3}) y^{'} = (2 x^{3} - y^{3}) y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	8.284

5287	$x (x^{4} - 2 y^{3}) y^{'} + (2 x^{4} + y^{3}) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.261

5288	$x (2 y^{3} + y + x) y^{'} = y (x - y)$	[_rational]	✓	1.679

5289	$(5 x - y - 7 x y^{3}) y^{'} + 5 y - y^{4} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.762

5290	$x (1 - 2 x y^{3}) y^{'} + (1 - 2 x^{3} y) y = 0$	[_rational]	✓	1.761

5291	$x (2 - x y^{2} - 2 x y^{3}) y^{'} + 1 + 2 y = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	2.830

5292	$(2 - 10 x^{2} y^{3} + 3 y^{2}) y^{'} = x (1 + 5 y^{4})$	[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	1.663

5293	$x (a + b x y^{3}) y^{'} + (a + c x^{3} y) y = 0$	[_rational]	✓	1.894

5294	$x (1 - 2 x^{2} y^{3}) y^{'} + (1 - 2 x^{3} y^{2}) y = 0$	[_rational]	✓	1.508

5295	$x (1 - x y) (1 - x^{2} y^{2}) y^{'} + (x y + 1) (1 + x^{2} y^{2}) y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.880

5296	$(x^{2} - y^{4}) y^{'} = x y$	[[_homogeneous, ‘class G‘], _rational]	✓	4.998

5297	$(x^{3} - y^{4}) y^{'} = 3 x^{2} y$	[[_homogeneous, ‘class G‘], _rational]	✓	2.392

5298	$(a^{2} x^{2} + {(x^{2} + y^{2})}^{2}) y^{'} = a^{2} x y$	[_rational]	✓	3.617

5299	$2 (x - y^{4}) y^{'} = y$	[[_homogeneous, ‘class G‘], _rational]	✓	4.869

5300	$(4 x - x y^{3} - 2 y^{4}) y^{'} = (2 + y^{3}) y$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	2.321

2.2.53 Problems 5201 to 5300