Problems 6401 to 6500

Table 2.131: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

6401	$x^{2} y^{'} + 2 x y = \sinh (x)$ i.c.	[_linear]	✓	1.731

6402	$y^{'} + \frac{y}{1 - x} + 2 x - x^{2} = 0$	[_linear]	✓	1.383

6403	$y^{'} + \frac{y}{1 - x} + x - x^{2} = 0$	[_linear]	✓	1.321

6404	$(x^{2} + 1) y^{'} = 1 + x y$	[_linear]	✓	2.589

6405	$y^{'} + x y = x y^{2}$	[_separable]	✓	2.364

6406	$3 x y^{'} + y + x^{2} y^{4} = 0$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	3.141

6407	$x {(x + 1)}^{2} y^{''} + (- x^{2} + 1) y^{'} + (- 1 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.828

6408	$x (1 - x) y^{''} + 2 (1 - 2 x) y^{'} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.999

6409	$x^{2} y^{''} + x y^{'} - 9 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.837

6410	$x y^{''} + \frac{y^{'}}{2} + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.345

6411	$x^{2} y^{''} - x y^{'} + y = 0$	[[_Emden, _Fowler]]	✓	0.990

6412	$2 x y^{''} - y^{'} + 2 y = 0$	[[_Emden, _Fowler]]	✓	1.102

6413	$x y^{''} + x y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.017

6414	$x {(- 1 + x)}^{2} y^{''} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.793

6415	$y^{'} - \frac{2 y}{x} - x^{2} = 0$	[_linear]	✓	1.868

6416	$y^{'} + \frac{2 y}{x} - x^{3} = 0$	[_linear]	✓	1.648

6417	$x y^{''} + (1 - x) y^{'} + m y = 0$	[_Laguerre]	✗	0.702

6418	$x y^{'} = x^{2} + 2 x - 3$	[_quadrature]	✓	0.574

6419	${(x + 1)}^{2} y^{'} = 1 + y^{2}$	[_separable]	✓	2.413

6420	$y^{'} + 2 y = e^{3 x}$	[[_linear, ‘class A‘]]	✓	1.398

6421	$- y + x y^{'} = x^{2}$	[_linear]	✓	1.609

6422	$x^{2} y^{'} = x^{3} \sin (3 x) + 4$	[_quadrature]	✓	0.718

6423	$x \cos (y) y^{'} - \sin (y) = 0$	[_separable]	✓	4.815

6424	$(x^{3} + x y^{2}) y^{'} = 2 y^{3}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	81.730

6425	$(x^{2} - 1) y^{'} + 2 x y = x$	[_separable]	✓	1.745

6426	$y^{'} + y \tanh (x) = 2 \sinh (x)$	[_linear]	✓	2.021

6427	$x y^{'} - 2 y = x^{3} \cos (x)$	[_linear]	✓	1.865

6428	$y^{'} + \frac{y}{x} = y^{3}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	3.724

6429	$x y^{'} + 3 y = y^{2} x^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	1.927

6430	$x (y - 3) y^{'} = 4 y$	[_separable]	✓	2.255

6431	$(x^{3} + 1) y^{'} = x^{2} y$ i.c.	[_separable]	✓	2.500

6432	$x^{3} + {(y + 1)}^{2} y^{'} = 0$	[_separable]	✓	1.896

6433	$\cos (y) + (1 + e^{- x}) \sin (y) y^{'} = 0$ i.c.	[_separable]	✓	3.280

6434	$x^{2} (y + 1) + y^{2} (- 1 + x) y^{'} = 0$	[_separable]	✓	1.714

6435	$(2 y - x) y^{'} = y + 2 x$	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	4.258

6436	$x y + y^{2} + (x^{2} - x y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	90.108

6437	$x^{3} + y^{3} = 3 x y^{2} y^{'}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	11.470

6438	$y - 3 x + (4 y + 3 x) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	34.647

6439	$(x^{3} + 3 x y^{2}) y^{'} = y^{3} + 3 x^{2} y$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	48.929

6440	$- y + x y^{'} = x^{3} + 3 x^{2} - 2 x$	[_linear]	✓	0.240

6441	$y^{'} + y \tan (x) = \sin (x)$	[_linear]	✓	0.307

6442	$- y + x y^{'} = x^{3} \cos (x)$ i.c.	[_linear]	✓	0.507

6443	$(x^{2} + 1) y^{'} + 3 x y = 5 x$ i.c.	[_separable]	✓	0.763

6444	$y^{'} + y \cot (x) = 5 e^{\cos (x)}$ i.c.	[_linear]	✓	0.526

6445	$(3 x + 3 y - 4) y^{'} = - x - y$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.842

6446	$x - x y^{2} = (x + x^{2} y) y^{'}$	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.569

6447	$x - y - 1 + (4 y + x - 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	37.043

6448	$3 y - 7 x + 7 + (3 - 3 x + 7 y) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.501

6449	$y (1 + x y) + x (1 + x y + y^{2} x^{2}) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.961

6450	$y^{'} + y = x y^{3}$	[_Bernoulli]	✓	0.625

6451	$y^{'} + y = y^{4} e^{x}$	[[_1st_order, _with_linear_symmetries], _Bernoulli]	✓	0.444

6452	$2 y^{'} + y = y^{3} (- 1 + x)$	[_Bernoulli]	✓	0.695

6453	$y^{'} - 2 y \tan (x) = y^{2} \tan {(x)}^{2}$	[_Bernoulli]	✓	0.527

6454	$y^{'} + y \tan (x) = y^{3} \sec {(x)}^{4}$	[_Bernoulli]	✓	0.583

6455	$(- x^{2} + 1) y^{'} = 1 + x y$	[_linear]	✓	1.323

6456	$x y y^{'} - (x + 1) \sqrt{y - 1} = 0$	[_separable]	✓	1.898

6457	$x^{2} - 2 x y + 5 y^{2} = (x^{2} + 2 x y + y^{2}) y^{'}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.169

6458	$y^{'} - y \cot (x) = y^{2} \sec {(x)}^{2}$ i.c.	[_Bernoulli]	✓	2.944

6459	$y + (x^{2} - 4 x) y^{'} = 0$	[_separable]	✓	1.683

6460	$y^{'} - y \tan (x) = \cos (x) - 2 x \sin (x)$ i.c.	[_linear]	✓	2.829

6461	$y^{'} = \frac{2 x y + y^{2}}{x^{2} + 2 x y}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	149.484

6462	$(x^{2} + 1) y^{'} = x (y + 1)$	[_separable]	✓	1.562

6463	$x y^{'} + 2 y = 3 x - 1$ i.c.	[_linear]	✓	2.378

6464	$x^{2} y^{'} = y^{2} - x y y^{'}$ i.c.	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	49.591

6465	$y^{'} = e^{- 2 y + 3 x}$ i.c.	[_separable]	✓	5.283

6466	$y^{'} + \frac{y}{x} = \sin (2 x)$ i.c.	[_linear]	✓	2.131

6467	$y^{2} + x^{2} y^{'} = x y y^{'}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	7.364

6468	$2 x y y^{'} = x^{2} - y^{2}$	[[_homogeneous, ‘class A‘], _exact, _rational, _Bernoulli]	✓	68.430

6469	$y^{'} = \frac{x - 2 y + 1}{2 x - 4 y}$ i.c.	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.345

6470	$(- x^{3} + 1) y^{'} + x^{2} y = x^{2} (- x^{3} + 1)$	[_linear]	✓	2.940

6471	$y^{'} + \frac{y}{x} = \sin (x)$ i.c.	[_linear]	✓	1.757

6472	$y^{'} + x + x y^{2} = 0$ i.c.	[_separable]	✓	2.692

6473	$y^{'} + (\frac{1}{x} - \frac{2 x}{- x^{2} + 1}) y = \frac{1}{- x^{2} + 1}$	[_linear]	✓	1.392

6474	$(x^{2} + 1) y^{'} + x y = {(x^{2} + 1)}^{3 / 2}$	[_linear]	✓	1.850

6475	$x (1 + y^{2}) - y (x^{2} + 1) y^{'} = 0$	[_separable]	✓	4.468

6476	$\frac{r \tan (θ) r^{'}}{a^{2} - r^{2}} = 1$ i.c.	[_separable]	✓	2.770

6477	$y^{'} + y \cot (x) = \cos (x)$ i.c.	[_linear]	✓	2.198

6478	$y^{'} + \frac{y}{x} = x y^{2}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.076

6479	$y^{''} - y^{'} - 2 y = 8$	[[_2nd_order, _missing_x]]	✓	0.965

6480	$y^{''} - 4 y = 10 e^{3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.242

6481	$y^{''} + 2 y^{'} + y = e^{- 2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.089

6482	$y^{''} + 25 y = 5 x^{2} + x$	[[_2nd_order, _with_linear_symmetries]]	✓	3.383

6483	$y^{''} - 2 y^{'} + y = 4 \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.395

6484	$y^{''} + 4 y^{'} + 5 y = 2 e^{- 2 x}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	2.527

6485	$3 y^{''} - 2 y^{'} - y = 2 x - 3$	[[_2nd_order, _with_linear_symmetries]]	✓	1.141

6486	$y^{''} - 6 y^{'} + 8 y = 8 e^{4 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.115

6487	$2 y^{''} - 7 y^{'} - 4 y = e^{3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.123

6488	$y^{''} - 6 y^{'} + 9 y = 54 x + 18$	[[_2nd_order, _with_linear_symmetries]]	✓	1.218

6489	$y^{''} - 5 y^{'} + 6 y = 100 \sin (4 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.438

6490	$y^{''} + 2 y^{'} + y = 4 \sinh (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.548

6491	$y^{''} + y^{'} - 2 y = 2 \cosh (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	4.200

6492	$y^{''} - y^{'} + 10 y = 20 - e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	22.611

6493	$y^{''} + 4 y^{'} + 4 y = 2 \cos {(x)}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.832

6494	$y^{''} - 4 y^{'} + 3 y = x + e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.212

6495	$y^{''} - 2 y^{'} + 3 y = x^{2} - 1$	[[_2nd_order, _with_linear_symmetries]]	✓	20.257

6496	$y^{''} - 9 y = e^{3 x} + \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.052

6497	$x^{''} + 4 x^{'} + 3 x = e^{- 3 t}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	1.204

6498	$y^{''} + 4 y^{'} + 5 y = 6 \sin (t)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	13.601

6499	$x^{''} - 3 x^{'} + 2 x = \sin (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.329

6500	$y^{''} + 3 y^{'} + 2 y = 3 \sin (x)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.800

2.2.65 Problems 6401 to 6500