Problems 6001 to 6100

Table 2.123: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

6001	$r^{''} = - \frac{k}{r^{2}}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	66.330

6002	$y^{''} = \frac{3 k y^{2}}{2}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	4.692

6003	$y^{''} = 2 k y^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	5.739

6004	$y y^{''} + {y^{'}}^{2} - y^{'} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.860

6005	$r^{''} = \frac{h^{2}}{r^{3}} - \frac{k}{r^{2}}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	2.589

6006	$y y^{''} + {y^{'}}^{3} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.487

6007	$y y^{''} - 3 {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.526

6008	$(x^{2} + 1) y^{''} + {y^{'}}^{2} + 1 = 0$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.825

6009	$(x^{2} + 1) y^{''} + 2 x (y^{'} + 1) = 0$	[[_2nd_order, _missing_y]]	✓	1.405

6010	$(y + 1) y^{''} = 3 {y^{'}}^{2}$ i.c.	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.590

6011	$y^{''} = y^{'} e^{y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]	✓	2.502

6012	$y^{''} = 2 y y^{'}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	13.908

6013	$2 y^{''} = e^{y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	1.305

6014	$x^{2} y^{''} + x y^{'} = 1$ i.c.	[[_2nd_order, _missing_y]]	✓	1.089

6015	$x y^{''} - y^{'} = x^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓	1.419

6016	$x y y^{''} - 2 x {y^{'}}^{2} + y y^{'} = 0$	[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.187

6017	$x y y^{''} + x {y^{'}}^{2} - y y^{'} = 0$	[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.177

6018	$x y y^{''} - 2 x {y^{'}}^{2} + (y + 1) y^{'} = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✗	0.124

6019	$- a y^{3} - \frac{b}{x^{3 / 2}} + y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, _Abel]	✓	4.980

6020	$a x y^{3} + b y^{2} + y^{'} = 0$	[[_homogeneous, ‘class G‘], _Abel]	✓	2.475

6021	$y^{'} - x^{a} y^{3} + 3 y^{2} - x^{- a} y - x^{- 2 a} + a x^{- a - 1} = 0$	[_Abel]	✓	3.372

6022	$y^{'} - (y - f (x)) (y - g (x)) (y - \frac{a f (x) + b g (x)}{a + b}) h (x) - \frac{f^{'} (x) (y - g (x))}{f (x) - g (x)} - \frac{g^{'} (x) (y - f (x))}{g (x) - f (x)} = 0$	[_Abel]	✗	108.821

6023	$x^{2} y^{'} + x y^{3} + a y^{2} = 0$	[_rational, _Abel]	✗	0.889

6024	${(a x + b)}^{2} y^{'} + (a x + b) y^{3} + c y^{2} = 0$	[_rational, _Abel]	✗	2.217

6025	$y^{'} + y \tan (x) = 0$	[_separable]	✓	1.794

6026	$x^{2} y^{''} - 2 x y^{'} + 2 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.979

6027	$y {y^{'}}^{2} + 2 x y^{'} - y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	1.876

6028	$(- x^{2} + 1) {y^{'}}^{2} + 1 = 0$	[_quadrature]	✓	0.360

6029	$y^{'} = e^{a x} + a y$	[[_linear, ‘class A‘]]	✓	0.767

6030	${(1 + {y^{'}}^{2})}^{3} = a^{2} {y^{''}}^{2}$	[[_2nd_order, _missing_x]]	✓	205.148

6031	$(x + 1) y + (1 - y) x y^{'} = 0$	[_separable]	✓	1.495

6032	$y^{'} = a y^{2} x$	[_separable]	✓	1.448

6033	$y^{2} + x y^{2} + (x^{2} - x^{2} y) y^{'} = 0$	[_separable]	✓	1.760

6034	$x y (x^{2} + 1) y^{'} = 1 + y^{2}$	[_separable]	✓	4.317

6035	$\frac{x}{y + 1} = \frac{y y^{'}}{x + 1}$	[_separable]	✓	1.596

6036	$y^{'} + b^{2} y^{2} = a^{2}$	[_quadrature]	✓	3.583

6037	$y^{'} = \frac{1 + y^{2}}{x^{2} + 1}$	[_separable]	✓	2.274

6038	$\sin (x) \cos (y) = \cos (x) \sin (y) y^{'}$	[_separable]	✓	3.194

6039	$a x y^{'} + 2 y = x y y^{'}$	[_separable]	✓	1.662

6040	$x y^{''} + (x + n) y^{'} + (n + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.774

6041	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.228

6042	$2 x^{2} y^{''} - x y^{'} + (- x^{2} + 1) y = x^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.698

6043	$x y^{''} + 2 y^{'} + a^{3} x^{2} y = 2$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.644

6044	$y^{''} + a x^{2} y = x + 1$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.351

6045	$x^{4} y^{''} + x y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.153

6046	$x^{2} y^{''} + (2 x^{2} + x) y^{'} - 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.733

6047	$(- x^{2} + x) y^{''} + 3 y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.697

6048	$(4 x^{3} - 14 x^{2} - 2 x) y^{''} - (6 x^{2} - 7 x + 1) y^{'} + (6 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.783

6049	$x^{2} y^{''} + x^{2} y^{'} + (- 2 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.704

6050	$x^{2} y^{''} - x^{2} y^{'} + (- 2 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.695

6051	$x^{2} (1 - 4 x) y^{''} + ((1 - n) x - (6 - 4 n) x^{2}) y^{'} + n (1 - n) x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.900

6052	$x^{2} y^{''} + (x^{2} + x) y^{'} + (x - 9) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.828

6053	$(a^{2} + x^{2}) y^{''} + x y^{'} - n^{2} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.533

6054	$(- x^{2} + 1) y^{''} - x y^{'} + a^{2} y = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.518

6055	$x y^{''} + y^{'} + y = 0$	[[_Emden, _Fowler]]	✓	0.497

6056	$x y^{''} + y^{'} + p x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.444

6057	$x y^{''} + y = 0$	[[_Emden, _Fowler]]	✓	1.033

6058	$x^{3} y^{''} - (2 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.092

6059	$x^{2} y^{''} + x (x + 1) y^{'} + (3 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.380

6060	$(- x^{2} + x) y^{''} - y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.219

6061	$x (- x^{2} + 1) y^{''} + (- 3 x^{2} + 1) y^{'} - x y = 0$	[[_elliptic, _class_I]]	✓	0.479

6062	$y^{''} + \frac{a y}{x^{3 / 2}} = 0$	[[_Emden, _Fowler]]	✗	0.147

6063	$x^{2} y^{''} - (x^{2} + 4 x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.346

6064	$x (- x^{2} + 1) y^{''} + (- x^{2} + 1) y^{'} + x y = 0$	[[_elliptic, _class_II]]	✓	0.487

6065	$4 x (1 - x) y^{''} - 4 y^{'} - y = 0$	[_Jacobi]	✓	1.391

6066	$x^{3} y^{''} + y = x^{3 / 2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✗	0.105

6067	$2 x^{2} y^{''} - (3 x + 2) y^{'} + \frac{(2 x - 1) y}{x} = \sqrt{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✗	0.125

6068	$(- x^{2} + x) y^{''} + 3 y^{'} + 2 y = 3 x^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	0.798

6069	$x (1 - x) y^{''} + (\frac{3}{2} - 2 x) y^{'} - \frac{y}{4} = 0$	[_Jacobi]	✓	0.725

6070	$2 x (1 - x) y^{''} + x y^{'} - y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.336

6071	$2 x (1 - x) y^{''} + (1 - 11 x) y^{'} - 10 y = 0$	[_Jacobi]	✓	0.726

6072	$x (1 - x) y^{''} + \frac{(1 - 2 x) y^{'}}{3} + \frac{20 y}{9} = 0$	[_Jacobi]	✓	0.737

6073	$2 x (1 - x) y^{''} + y^{'} + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.697

6074	$4 y^{''} + \frac{3 (- x^{2} + 2) y}{{(- x^{2} + 1)}^{2}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.432

6075	$y^{'} + y^{2} = \frac{a^{2}}{x^{4}}$	[_rational, _Riccati]	✓	1.638

6076	$u^{''} - \frac{a^{2} u}{x^{2 / 3}} = 0$	[[_Emden, _Fowler]]	✓	0.667

6077	$u^{''} - \frac{2 u^{'}}{x} - a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.165

6078	$u^{''} + \frac{2 u^{'}}{x} - a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.175

6079	$u^{''} + \frac{2 u^{'}}{x} + a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.287

6080	$u^{''} + \frac{4 u^{'}}{x} - a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	3.815

6081	$u^{''} + \frac{4 u^{'}}{x} + a^{2} u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	2.663

6082	$y^{''} - a^{2} y = \frac{6 y}{x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓	5.316

6083	$y^{''} + n^{2} y = \frac{6 y}{x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓	3.208

6084	$x^{2} y^{''} + x y^{'} - (x^{2} + \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.106

6085	$x^{2} y^{''} + x y^{'} + \frac{(- 9 a^{2} + 4 x^{2}) y}{4 a^{2}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	3.144

6086	$x^{2} y^{''} + x y^{'} + (x^{2} - \frac{25}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.899

6087	$y^{''} + q y^{'} = \frac{2 y}{x^{2}}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.959

6088	$y^{''} + e^{2 x} y = n^{2} y$	[[_2nd_order, _with_linear_symmetries]]	✓	0.453

6089	$y^{''} + \frac{y}{4 x} = 0$	[[_Emden, _Fowler]]	✓	0.574

6090	$x y^{''} + y^{'} + y = 0$	[[_Emden, _Fowler]]	✓	0.493

6091	$x y^{''} + 3 y^{'} + 4 x^{3} y = 0$	[[_Emden, _Fowler]]	✓	2.104

6092	$y^{'} = y$	[_quadrature]	✓	1.528

6093	$x y^{'} = y$ i.c.	[_separable]	✓	2.029

6094	$x \sqrt{1 - y^{2}} + y \sqrt{- x^{2} + 1} y^{'} = 0$ i.c.	[_separable]	✓	7.838

6095	$\sin (x) y^{'} = y \ln (y)$ i.c.	[_separable]	✓	16.326

6096	$1 + y^{2} + x y y^{'} = 0$ i.c.	[_separable]	✓	3.221

6097	$x y y^{'} - x y = y$ i.c.	[_quadrature]	✓	0.962

6098	$y^{'} = \frac{2 x y^{2} + x}{x^{2} y - y}$ i.c.	[_separable]	✓	2.790

6099	$y y^{'} + x y^{2} - 8 x = 0$ i.c.	[_separable]	✓	3.589

6100	$y^{'} + 2 x y^{2} = 0$ i.c.	[_separable]	✓	2.492

2.2.61 Problems 6001 to 6100