Problems 8401 to 8500

Table 2.171: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

8401	$[\begin{matrix} x^{'} = x + y - z \\ y^{'} = 2 y \\ z^{'} = y - z \end{matrix}]$	system_of_ODEs	✓	0.395

8402	$[\begin{matrix} x^{'} = 2 x - 7 y \\ y^{'} = 5 x + 10 y + 4 z \\ z^{'} = 5 y + 2 z \end{matrix}]$	system_of_ODEs	✓	0.479

8403	$[\begin{matrix} x^{'} = y - x \\ y^{'} = x + 2 y + z \\ z^{'} = 3 y - z \end{matrix}]$	system_of_ODEs	✓	0.468

8404	$[\begin{matrix} x^{'} = x + z \\ y^{'} = y \\ z^{'} = x + z \end{matrix}]$	system_of_ODEs	✓	0.345

8405	$[\begin{matrix} x^{'} = - x - y \\ y^{'} = \frac{3 x}{4} - \frac{3 y}{2} + 3 z \\ z^{'} = \frac{x}{8} + \frac{y}{4} - \frac{z}{2} \end{matrix}]$	system_of_ODEs	✓	0.488

8406	$[\begin{matrix} x^{'} = - x - y \\ y^{'} = \frac{3 x}{4} - \frac{3 y}{2} + 3 z \\ z^{'} = \frac{x}{8} + \frac{y}{4} - \frac{z}{2} \end{matrix}]$	system_of_ODEs	✓	0.489

8407	$[\begin{matrix} x^{'} = - x + 4 y + 2 z \\ y^{'} = 4 x - y - 2 z \\ z^{'} = 6 z \end{matrix}]$	system_of_ODEs	✓	0.444

8408	$[\begin{matrix} x^{'} = \frac{x}{2} \\ y^{'} = x - \frac{y}{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.535

8409	$[\begin{matrix} x^{'} = x + y + 4 z \\ y^{'} = 2 y \\ z^{'} = x + y + z \end{matrix}]$ i.c.	system_of_ODEs	✓	0.441

8410	$[\begin{matrix} x^{'} = \frac{9 x}{10} + \frac{21 y}{10} + \frac{16 z}{5} \\ y^{'} = \frac{7 x}{10} + \frac{13 y}{2} + \frac{21 z}{5} \\ z^{'} = \frac{11 x}{10} + \frac{17 y}{10} + \frac{17 z}{5} \end{matrix}]$	system_of_ODEs	✓	62.882

8411	$[\begin{matrix} x_{1}^{'} = x_{1} + 2 x_{3} - \frac{9 x_{4}}{5} \\ x_{2}^{'} = \frac{51 x_{2}}{10} - x_{4} + 3 x_{5} \\ x_{3}^{'} = x_{1} + 2 x_{2} - 3 x_{3} \\ x_{4}^{'} = x_{2} - \frac{31 x_{3}}{10} + 4 x_{4} \\ x_{5}^{'} = - \frac{14 x_{1}}{5} + \frac{3 x_{4}}{2} - x_{5} \end{matrix}]$	system_of_ODEs	✓	87.845

8412	$[\begin{matrix} x^{'} = 3 x - y \\ y^{'} = 9 x - 3 y \end{matrix}]$	system_of_ODEs	✓	0.333

8413	$[\begin{matrix} x^{'} = - 6 x + 5 y \\ y^{'} = - 5 x + 4 y \end{matrix}]$	system_of_ODEs	✓	0.429

8414	$[\begin{matrix} x^{'} = - x + 3 y \\ y^{'} = - 3 x + 5 y \end{matrix}]$	system_of_ODEs	✓	0.427

8415	$[\begin{matrix} x^{'} = 12 x - 9 y \\ y^{'} = 4 x \end{matrix}]$	system_of_ODEs	✓	0.440

8416	$[\begin{matrix} x^{'} = 3 x - y - z \\ y^{'} = x + y - z \\ z^{'} = x - y + z \end{matrix}]$	system_of_ODEs	✓	0.358

8417	$[\begin{matrix} x^{'} = 3 x + 2 y + 4 z \\ y^{'} = 2 x + 2 z \\ z^{'} = 4 x + 2 y + 3 z \end{matrix}]$	system_of_ODEs	✓	0.409

8418	$[\begin{matrix} x^{'} = 5 x - 4 y \\ y^{'} = x + 2 z \\ z^{'} = 2 y + 5 z \end{matrix}]$	system_of_ODEs	✓	0.413

8419	$[\begin{matrix} x^{'} = x \\ y^{'} = 3 y + z \\ z^{'} = z - y \end{matrix}]$	system_of_ODEs	✓	0.351

8420	$[\begin{matrix} x^{'} = x \\ y^{'} = 2 x + 2 y - z \\ z^{'} = y \end{matrix}]$	system_of_ODEs	✓	0.343

8421	$[\begin{matrix} x^{'} = 4 x + y \\ y^{'} = 4 y + z \\ z^{'} = 4 z \end{matrix}]$	system_of_ODEs	✓	0.319

8422	$[\begin{matrix} x^{'} = 2 x + 4 y \\ y^{'} = - x + 6 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.546

8423	$[\begin{matrix} x^{'} = z \\ y^{'} = y \\ z^{'} = x \end{matrix}]$ i.c.	system_of_ODEs	✓	0.339

8424	$[\begin{matrix} x^{'} = 6 x - y \\ y^{'} = 5 x + 2 y \end{matrix}]$	system_of_ODEs	✓	0.539

8425	$[\begin{matrix} x^{'} = x + y \\ y^{'} = - 2 x - y \end{matrix}]$	system_of_ODEs	✓	0.511

8426	$[\begin{matrix} x^{'} = 5 x + y \\ y^{'} = - 2 x + 3 y \end{matrix}]$	system_of_ODEs	✓	0.530

8427	$[\begin{matrix} x^{'} = 4 x + 5 y \\ y^{'} = - 2 x + 6 y \end{matrix}]$	system_of_ODEs	✓	0.554

8428	$[\begin{matrix} x^{'} = 4 x - 5 y \\ y^{'} = 5 x - 4 y \end{matrix}]$	system_of_ODEs	✓	0.523

8429	$[\begin{matrix} x^{'} = x - 8 y \\ y^{'} = x - 3 y \end{matrix}]$	system_of_ODEs	✓	0.538

8430	$[\begin{matrix} x^{'} = z \\ y^{'} = - z \\ z^{'} = y \end{matrix}]$	system_of_ODEs	✓	0.403

8431	$[\begin{matrix} x^{'} = 2 x + y + 2 z \\ y^{'} = 3 x + 6 z \\ z^{'} = - 4 x - 3 z \end{matrix}]$	system_of_ODEs	✓	0.822

8432	$[\begin{matrix} x^{'} = x - 12 y - 14 z \\ y^{'} = x + 2 y - 3 z \\ z^{'} = x + y - 2 z \end{matrix}]$ i.c.	system_of_ODEs	✓	0.684

8433	$[\begin{matrix} x^{'} = 2 x + 3 y - 7 \\ y^{'} = - x - 2 y + 5 \end{matrix}]$	system_of_ODEs	✓	0.562

8434	$[\begin{matrix} x^{'} = 5 x + 9 y + 2 \\ y^{'} = - x + 11 y + 6 \end{matrix}]$	system_of_ODEs	✓	0.575

8435	$x^{2} {y^{'}}^{2} - y^{2} = 0$	[_separable]	✓	2.565

8436	$x {y^{'}}^{2} - (2 x + 3 y) y^{'} + 6 y = 0$	[_quadrature]	✓	0.459

8437	$x^{2} {y^{'}}^{2} - 5 x y y^{'} + 6 y^{2} = 0$	[_separable]	✓	1.062

8438	$x^{2} {y^{'}}^{2} + x y^{'} - y^{2} - y = 0$	[_separable]	✓	0.778

8439	$x {y^{'}}^{2} + (1 - x^{2} y) y^{'} - x y = 0$	[_quadrature]	✓	0.625

8440	${y^{'}}^{2} - (x^{2} y + 3) y^{'} + 3 x^{2} y = 0$	[_quadrature]	✓	0.487

8441	$x {y^{'}}^{2} - (1 + x y) y^{'} + y = 0$	[_quadrature]	✓	0.448

8442	${y^{'}}^{2} - y^{2} x^{2} = 0$	[_separable]	✓	2.334

8443	${(x + y)}^{2} {y^{'}}^{2} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.628

8444	$y {y^{'}}^{2} + (x - y^{2}) y^{'} - x y = 0$	[_quadrature]	✓	0.504

8445	${y^{'}}^{2} - x y (x + y) y^{'} + x^{3} y^{3} = 0$	[_separable]	✓	0.895

8446	$(4 x - y) {y^{'}}^{2} + 6 (x - y) y^{'} + 2 x - 5 y = 0$	[_quadrature]	✓	0.724

8447	${(x - y)}^{2} {y^{'}}^{2} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.323

8448	$x y {y^{'}}^{2} + (x y^{2} - 1) y^{'} - y = 0$	[_quadrature]	✓	0.528

8449	${(x^{2} + y^{2})}^{2} {y^{'}}^{2} = 4 y^{2} x^{2}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	45.930

8450	${(x + y)}^{2} {y^{'}}^{2} + (2 y^{2} + x y - x^{2}) y^{'} + y (y - x) = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.572

8451	$x y (x^{2} + y^{2}) ({y^{'}}^{2} - 1) = y^{'} (x^{4} + y^{2} x^{2} + y^{4})$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	418.068

8452	$x {y^{'}}^{3} - (x^{2} + x + y) {y^{'}}^{2} + (x^{2} + x y + y) y^{'} - x y = 0$	[_quadrature]	✓	0.825

8453	$x y {y^{'}}^{2} + (x + y) y^{'} + 1 = 0$	[_quadrature]	✓	0.511

8454	$x {y^{'}}^{2} - 2 y y^{'} + 4 x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.344

8455	$3 x^{4} {y^{'}}^{2} - x y^{'} - y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.725

8456	${y^{'}}^{2} - x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.438

8457	${y^{'}}^{2} - x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.387

8458	${y^{'}}^{2} + 4 x^{5} y^{'} - 12 x^{4} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	1.737

8459	$4 y^{3} {y^{'}}^{2} - 4 x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.297

8460	$4 y^{3} {y^{'}}^{2} + 4 x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.010

8461	${y^{'}}^{3} + x {y^{'}}^{2} - y = 0$	[_dAlembert]	✓	2.907

8462	$y^{4} {y^{'}}^{3} - 6 x y^{'} + 2 y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	108.414

8463	${y^{'}}^{2} + x^{3} y^{'} - 2 x^{2} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	1.871

8464	${y^{'}}^{2} + 4 x^{5} y^{'} - 12 x^{4} y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	1.715

8465	$2 x {y^{'}}^{3} - 6 y {y^{'}}^{2} + x^{4} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	120.142

8466	${y^{'}}^{2} - x y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.394

8467	$y = x y^{'} + k {y^{'}}^{2}$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.480

8468	$x^{8} {y^{'}}^{2} + 3 x y^{'} + 9 y = 0$	[[_homogeneous, ‘class G‘]]	✓	1.869

8469	$x^{4} {y^{'}}^{2} + 2 x^{3} y y^{'} - 4 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.203

8470	$x {y^{'}}^{2} - 2 y y^{'} + 4 x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.412

8471	$3 x^{4} {y^{'}}^{2} - x y^{'} - y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.775

8472	$x {y^{'}}^{2} + (x - y) y^{'} + 1 - y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	0.514

8473	$y^{'} (x y^{'} - y + k) + a = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.474

8474	$x^{6} {y^{'}}^{3} - 3 x y^{'} - 3 y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	10.691

8475	$y = x^{6} {y^{'}}^{3} - x y^{'}$	[[_1st_order, _with_linear_symmetries]]	✓	13.661

8476	$x {y^{'}}^{4} - 2 y {y^{'}}^{3} + 12 x^{3} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	2.431

8477	$x {y^{'}}^{3} - y {y^{'}}^{2} + 1 = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.549

8478	${y^{'}}^{2} - x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.440

8479	$2 {y^{'}}^{3} + x y^{'} - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.425

8480	$2 {y^{'}}^{2} + x y^{'} - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.588

8481	${y^{'}}^{3} + 2 x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.444

8482	$4 x {y^{'}}^{2} - 3 y y^{'} + 3 = 0$	[[_homogeneous, ‘class G‘], _rational, _dAlembert]	✓	0.483

8483	${y^{'}}^{3} - x y^{'} + 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.432

8484	$5 {y^{'}}^{2} + 6 x y^{'} - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.465

8485	$2 x {y^{'}}^{2} + (2 x - y) y^{'} + 1 - y = 0$	[_rational, _dAlembert]	✓	1.245

8486	$5 {y^{'}}^{2} + 3 x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.463

8487	${y^{'}}^{2} + 3 x y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.467

8488	$y = x y^{'} + x^{3} {y^{'}}^{2}$	[[_homogeneous, ‘class G‘], _rational]	✓	1.163

8489	$y^{''} = x {y^{'}}^{3}$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.481

8490	$x^{2} y^{''} + {y^{'}}^{2} - 2 x y^{'} = 0$ i.c.	[[_2nd_order, _missing_y]]	✓	0.435

8491	$x^{2} y^{''} + {y^{'}}^{2} - 2 x y^{'} = 0$ i.c.	[[_2nd_order, _missing_y]]	✓	0.483

8492	$y y^{''} + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.079

8493	$y^{2} y^{''} + {y^{'}}^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.355

8494	$(y + 1) y^{''} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.325

8495	$2 a y^{''} + {y^{'}}^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.719

8496	$x y^{''} = y^{'} + x^{5}$ i.c.	[[_2nd_order, _missing_y]]	✓	1.375

8497	$x y^{''} + y^{'} + x = 0$ i.c.	[[_2nd_order, _missing_y]]	✓	1.440

8498	$y^{''} = 2 y {y^{'}}^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.217

8499	$y y^{''} + {y^{'}}^{3} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.463

8500	$y^{''} + β^{2} y = 0$	[[_2nd_order, _missing_x]]	✓	1.649

2.2.85 Problems 8401 to 8500