Problems 8501 to 8600

Table 2.173: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

8501	$y y^{''} + {y^{'}}^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.546

8502	$\cos (x) y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓	1.171

8503	$y^{''} = x {y^{'}}^{2}$ i.c.	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.556

8504	$y^{''} = x {y^{'}}^{2}$ i.c.	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.613

8505	$y^{''} = - e^{- 2 y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.558

8506	$y^{''} = - e^{- 2 y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	6.593

8507	$2 y^{''} = \sin (2 y)$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✗	5.241

8508	$2 y^{''} = \sin (2 y)$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✗	5.314

8509	$x^{3} y^{''} - x^{2} y^{'} = - x^{2} + 3$	[[_2nd_order, _missing_y]]	✓	0.781

8510	$y^{''} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	0.527

8511	$y^{''} = e^{x} {y^{'}}^{2}$	[[_2nd_order, _missing_y]]	✓	0.420

8512	$2 y^{''} = {y^{'}}^{3} \sin (2 x)$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	1.444

8513	$x^{2} y^{''} + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.434

8514	$y^{''} = 1 + {y^{'}}^{2}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	1.584

8515	$y^{''} = {(1 + {y^{'}}^{2})}^{3 / 2}$	[[_2nd_order, _missing_x]]	✓	2.410

8516	$y y^{''} = {y^{'}}^{2} (1 - y^{'} \sin (y) - y y^{'} \cos (y))$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]	✓	1.440

8517	$(1 + y^{2}) y^{''} + {y^{'}}^{3} + y^{'} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	2.428

8518	${(y y^{''} + 1 + {y^{'}}^{2})}^{2} = {(1 + {y^{'}}^{2})}^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	5.965

8519	$x^{2} y^{''} = y^{'} (2 x - y^{'})$ i.c.	[[_2nd_order, _missing_y]]	✓	0.828

8520	$x^{2} y^{''} = y^{'} (3 x - 2 y^{'})$	[[_2nd_order, _missing_y]]	✓	0.646

8521	$x y^{''} = y^{'} (2 - 3 y^{'} x)$	[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	0.793

8522	$x^{4} y^{''} = y^{'} (y^{'} + x^{3})$ i.c.	[[_2nd_order, _missing_y]]	✓	0.693

8523	$y^{''} = 2 x + {(x^{2} - y^{'})}^{2}$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]	✓	0.988

8524	${y^{''}}^{2} - 2 y^{''} + {y^{'}}^{2} - 2 y^{'} x + x^{2} = 0$ i.c.	[[_2nd_order, _missing_y]]	✓	2.048

8525	${y^{''}}^{2} - x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓	0.608

8526	${y^{''}}^{3} = 12 y^{'} (x y^{''} - 2 y^{'})$	[[_2nd_order, _missing_y]]	✓	8.799

8527	$3 y y^{'} y^{''} = {y^{'}}^{3} - 1$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.739

8528	$4 y {y^{'}}^{2} y^{''} = {y^{'}}^{4} + 3$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.654

8529	$y^{''} + y = - \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.663

8530	$y^{''} - 6 y^{'} + 9 y = e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.533

8531	$y^{''} + 3 y^{'} + 2 y = 12 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.515

8532	$y^{''} + 3 y^{'} + 2 y = x^{2} + 2 x + 1$	[[_2nd_order, _with_linear_symmetries]]	✓	0.433

8533	$x^{3} {y^{'}}^{2} + x^{2} y y^{'} + 4 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.679

8534	$6 x {y^{'}}^{2} - (3 x + 2 y) y^{'} + y = 0$	[_quadrature]	✓	0.418

8535	$9 {y^{'}}^{2} + 3 x y^{4} y^{'} + y^{5} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	72.141

8536	$4 y^{3} {y^{'}}^{2} - 4 y^{'} x + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	3.473

8537	$x^{6} {y^{'}}^{2} - 2 y^{'} x - 4 y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	1.864

8538	$5 {y^{'}}^{2} + 6 y^{'} x - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.540

8539	$y^{2} {y^{'}}^{2} - (x + 1) y y^{'} + x = 0$	[_quadrature]	✓	0.399

8540	$4 x^{5} {y^{'}}^{2} + 12 x^{4} y y^{'} + 9 = 0$	[[_homogeneous, ‘class G‘]]	✓	3.792

8541	$4 y^{2} {y^{'}}^{3} - 2 y^{'} x + y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	115.137

8542	${y^{'}}^{4} + y^{'} x - 3 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	1.752

8543	$x^{2} {y^{'}}^{3} - 2 x y {y^{'}}^{2} + y^{2} y^{'} + 1 = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	1.918

8544	$16 x {y^{'}}^{2} + 8 y y^{'} + y^{6} = 0$	[[_homogeneous, ‘class G‘]]	✓	2.714

8545	$x {y^{'}}^{2} - (x^{2} + 1) y^{'} + x = 0$	[_quadrature]	✓	0.392

8546	${y^{'}}^{3} - 2 y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.246

8547	$9 x y^{4} {y^{'}}^{2} - 3 y^{5} y^{'} - 1 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	3.322

8548	$x^{2} {y^{'}}^{2} - (1 + 2 x y) y^{'} + 1 + y^{2} = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.732

8549	$x^{6} {y^{'}}^{2} = 8 y^{'} x + 16 y$	[[_homogeneous, ‘class G‘]]	✓	1.824

8550	$x^{2} {y^{'}}^{2} = {(x - y)}^{2}$	[_linear]	✓	1.053

8551	${(y^{'} + 1)}^{2} (y - y^{'} x) = 1$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	0.864

8552	${y^{'}}^{3} - {y^{'}}^{2} + y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	0.554

8553	$x {y^{'}}^{2} + y (1 - x) y^{'} - y^{2} = 0$	[_quadrature]	✓	0.341

8554	$y {y^{'}}^{2} - (x + y) y^{'} + y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	2.100

8555	$x {y^{'}}^{2} + (k - x - y) y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	0.704

8556	$x {y^{'}}^{3} - 2 y {y^{'}}^{2} + 4 x^{2} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	75.895

8557	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓	0.340

8558	$y^{''} - 9 y = 0$	[[_2nd_order, _missing_x]]	✓	0.355

8559	$y^{''} + 3 y^{'} x + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.469

8560	$(4 x^{2} + 1) y^{''} - 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.509

8561	$(- 4 x^{2} + 1) y^{''} + 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.512

8562	$(x^{2} + 1) y^{''} - 4 y^{'} x + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.390

8563	$(x^{2} + 1) y^{''} + 10 y^{'} x + 20 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.580

8564	$(x^{2} + 4) y^{''} + 2 y^{'} x - 12 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.508

8565	$(x^{2} - 9) y^{''} + 3 y^{'} x - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.489

8566	$y^{''} + 2 y^{'} x + 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.443

8567	$(x^{2} + 4) y^{''} + 6 y^{'} x + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.581

8568	$(2 x^{2} + 1) y^{''} - 5 y^{'} x + 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.493

8569	$y^{''} + x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.366

8570	$(- 4 x^{2} + 1) y^{''} + 6 y^{'} x - 4 y = 0$	[_Gegenbauer]	✓	0.542

8571	$(2 x^{2} + 1) y^{''} + 3 y^{'} x - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.515

8572	$y^{'''} + x^{2} y^{''} + 5 y^{'} x + 3 y = 0$	[[_3rd_order, _exact, _linear, _homogeneous]]	✓	1.175

8573	$y^{''} + y^{'} x + 3 y = x^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.509

8574	$y^{''} + 2 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.463

8575	$y^{''} + 3 y^{'} x + 7 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.452

8576	$2 y^{''} + 9 y^{'} x - 36 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.468

8577	$(x^{2} + 4) y^{''} + y^{'} x - 9 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.515

8578	$(x^{2} + 4) y^{''} + 3 y^{'} x - 8 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.511

8579	$(9 x^{2} + 1) y^{''} - 18 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.504

8580	$(3 x^{2} + 1) y^{''} + 13 y^{'} x + 7 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.589

8581	$(2 x^{2} + 1) y^{''} + 11 y^{'} x + 9 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.568

8582	$y^{''} - 2 (3 + x) y^{'} - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.472

8583	$y^{''} + (x - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.453

8584	$(x^{2} - 2 x + 2) y^{''} - 4 (x - 1) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.414

8585	$2 x (x + 1) y^{''} + 3 (x + 1) y^{'} - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.811

8586	$4 x^{2} y^{''} + 4 y^{'} x + (4 x^{2} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.831

8587	$4 x^{2} y^{''} + 4 y^{'} x - (4 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.790

8588	$4 x y^{''} + 3 y^{'} + 3 y = 0$	[[_Emden, _Fowler]]	✓	0.943

8589	$2 x^{2} (1 - x) y^{''} - x (1 + 7 x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.899

8590	$2 x y^{''} + 5 (- 2 x + 1) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.996

8591	$8 x^{2} y^{''} + 10 y^{'} x - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.971

8592	$2 x y^{''} + (2 - x) y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.744

8593	$2 x (3 + x) y^{''} - 3 (x + 1) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.891

8594	$2 x y^{''} + (- 2 x^{2} + 1) y^{'} - 4 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.696

8595	$x (4 - x) y^{''} + (2 - x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.002

8596	$3 x^{2} y^{''} + y^{'} x - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.973

8597	$2 x y^{''} + (2 x + 1) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.901

8598	$2 x y^{''} + (2 x + 1) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.927

8599	$2 x^{2} y^{''} - 3 x (1 - x) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.037

8600	$2 x^{2} y^{''} + x (4 x - 1) y^{'} + 2 (3 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.951

2.2.86 Problems 8501 to 8600