Problems 6101 to 6200

Table 2.125: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

6101	$(1 + y) y^{'} = y$ i.c.	[_quadrature]	✓	1.477

6102	$y^{'} - x y = x$ i.c.	[_separable]	✓	1.420

6103	$2 y^{'} = 3 {(- 2 + y)}^{1 / 3}$ i.c.	[_quadrature]	✓	1.145

6104	$(x + x y) y^{'} + y = 0$ i.c.	[_separable]	✓	1.965

6105	$y^{'} + y = e^{x}$	[[_linear, ‘class A‘]]	✓	0.188

6106	$x^{2} y^{'} + 3 x y = 1$	[_linear]	✓	0.161

6107	$y^{'} + 2 x y - x e^{- x^{2}} = 0$	[_linear]	✓	0.215

6108	$2 y^{'} x + y = 2 x^{5 / 2}$	[_linear]	✓	0.193

6109	$\cos (x) y^{'} + y = \cos {(x)}^{2}$	[_linear]	✓	0.446

6110	$y^{'} + \frac{y}{\sqrt{x^{2} + 1}} = \frac{1}{x + \sqrt{x^{2} + 1}}$	[_linear]	✓	0.199

6111	$(e^{x} + 1) y^{'} + 2 y e^{x} = (e^{x} + 1) e^{x}$	[_linear]	✓	0.223

6112	$x \ln (x) y^{'} + y = \ln (x)$	[_linear]	✓	0.145

6113	$(- x^{2} + 1) y^{'} = x y + 2 x \sqrt{- x^{2} + 1}$	[_linear]	✓	0.241

6114	$y^{'} + y \tanh (x) = 2 e^{x}$	[_linear]	✓	0.294

6115	$y^{'} + \cos (x) y = \sin (2 x)$	[_linear]	✓	0.286

6116	$x^{'} = \cos (y) - x \tan (y)$	[_linear]	✓	0.267

6117	$x^{'} + x - e^{y} = 0$	[[_linear, ‘class A‘]]	✓	0.214

6118	$x^{'} = \frac{3 y^{2 / 3} - x}{3 y}$	[_linear]	✓	0.181

6119	$y^{'} + y = x y^{2 / 3}$	[_Bernoulli]	✓	1.446

6120	$y^{'} + \frac{y}{x} = 2 x^{3 / 2} \sqrt{y}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	9.165

6121	$3 x y^{2} y^{'} + 3 y^{3} = 1$	[_separable]	✓	3.106

6122	$2 x e^{3 y} + e^{x} + (3 x^{2} e^{3 y} - y^{2}) y^{'} = 0$	[_exact]	✓	1.958

6123	$(x - y) y^{'} + x + y + 1 = 0$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.334

6124	$\cos (x) \cos (y) + \sin {(x)}^{2} - (\sin (x) \sin (y) + \cos {(y)}^{2}) y^{'} = 0$	unknown	✓	43.567

6125	$x^{2} y^{'} + y^{2} - x y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.760

6126	$y y^{'} = \sqrt{x^{2} + y^{2}} - x$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	6.226

6127	$x y + (y^{2} - x^{2}) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.439

6128	$y^{2} - x y + (x^{2} + x y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	3.462

6129	$y^{'} = \cos (x + y)$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	2.080

6130	$y^{'} = \frac{y}{x} - \tan (\frac{y}{x})$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	3.361

6131	$(x - 1) y^{'} + y - \frac{1}{x^{2}} + \frac{2}{x^{3}} = 0$	[_linear]	✓	2.784

6132	$y^{'} = x y^{2} - \frac{2 y}{x} - \frac{1}{x^{3}}$	[[_homogeneous, ‘class G‘], _rational, _Riccati]	✓	1.832

6133	$y^{'} = \frac{2 y^{2}}{x} + \frac{y}{x} - 2 x$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	1.822

6134	$y^{'} = y^{2} e^{- x} + y - e^{x}$	[[_1st_order, _with_linear_symmetries], _Riccati]	✓	1.929

6135	$y^{''} + y^{'} - 2 y = 0$	[[_2nd_order, _missing_x]]	✓	0.354

6136	$y^{''} - 4 y^{'} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓	0.452

6137	$y^{''} + 9 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓	1.120

6138	$y^{''} + 2 y^{'} + 2 y = 0$	[[_2nd_order, _missing_x]]	✓	0.482

6139	$y^{''} - 2 y^{'} + 6 y = 0$	[[_2nd_order, _missing_x]]	✓	0.556

6140	$y^{''} + 16 y = 0$	[[_2nd_order, _missing_x]]	✓	1.455

6141	$y^{''} - 5 y^{'} + 6 y = 0$	[[_2nd_order, _missing_x]]	✓	0.332

6142	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓	1.102

6143	$y^{''} - 4 y^{'} + 13 y = 0$	[[_2nd_order, _missing_x]]	✓	0.546

6144	$2 y^{''} + y^{'} - y = 0$	[[_2nd_order, _missing_x]]	✓	0.353

6145	$y^{''} + (1 + 2 i) y^{'} + (- 1 + i) y = 0$	[[_2nd_order, _missing_x]]	✓	0.330

6146	$y^{''} + (1 + 2 i) y^{'} + (- 1 + i) y = 0$	[[_2nd_order, _missing_x]]	✓	0.317

6147	$y^{'''} + y = 0$	[[_3rd_order, _missing_x]]	✓	0.086

6148	$y^{'''} + y^{''} - 6 y^{'} = 0$	[[_3rd_order, _missing_x]]	✓	0.088

6149	$y^{'''} + 3 y^{''} - 9 y^{'} - 5 y = 0$	[[_3rd_order, _missing_x]]	✓	0.144

6150	$y^{''''} + 4 y = 0$	[[_high_order, _missing_x]]	✓	0.093

6151	$y^{''} - 4 y^{'} = 10$	[[_2nd_order, _missing_x]]	✓	1.190

6152	$y^{''} - 4 y^{'} + 4 y = 16$	[[_2nd_order, _missing_x]]	✓	0.559

6153	$y^{''} + y^{'} - 2 y = e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.492

6154	$y^{''} - 2 y^{'} - 3 y = 24 e^{- 3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.455

6155	$y^{''} + y = 2 e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.651

6156	$y^{''} + 6 y^{'} + 9 y = 12 e^{- x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.531

6157	$y^{''} - y^{'} - 2 y = 3 e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.604

6158	$y^{''} - 16 y = 40 e^{4 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.616

6159	$y^{''} + 2 y^{'} + y = 2 e^{- x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.554

6160	$y^{''} - 6 y^{'} + 9 y = 6 e^{3 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.574

6161	$y^{''} + 2 y^{'} + 10 y = 100 \cos (4 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.739

6162	$y^{''} + 4 y^{'} + 12 y = 80 \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.728

6163	$y^{''} - 2 y^{'} + y = 2 \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.665

6164	$y^{''} + 8 y^{'} + 25 y = 120 \sin (5 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.691

6165	$5 y^{''} + 12 y^{'} + 20 y = 120 \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.668

6166	$y^{''} + 9 y = 30 \sin (3 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.742

6167	$y^{''} + 16 y = 16 \cos (4 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.776

6168	$y^{''} + 2 y^{'} + 17 y = 60 e^{- 4 x} \sin (5 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.815

6169	$4 y^{''} + 4 y^{'} + 5 y = 40 e^{- \frac{3 x}{2}} \sin (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.731

6170	$y^{''} + 4 y^{'} + 8 y = 30 e^{- \frac{x}{2}} \cos (\frac{5 x}{2})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.725

6171	$5 y^{''} + 6 y^{'} + 2 y = x^{2} + 6 x$	[[_2nd_order, _with_linear_symmetries]]	✓	0.664

6172	$2 y^{''} + y^{'} = 2 x$	[[_2nd_order, _missing_y]]	✓	1.065

6173	$y^{''} + y = 2 x e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.655

6174	$y^{''} - 6 y^{'} + 9 y = 12 x e^{3 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.577

6175	$y^{''} - 2 y^{'} - 3 y = 16 x^{2} e^{- x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.593

6176	$y^{''} + y = 8 x \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.731

6177	$y^{''} + y = x^{3} - 1 + 2 \cos (x) + (2 - 4 x) e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.513

6178	$y^{''} - 5 y^{'} + 6 y = 2 e^{x} + 6 x - 5$	[[_2nd_order, _with_linear_symmetries]]	✓	0.510

6179	$y^{''} - y = \sinh (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.812

6180	$y^{''} + y = 2 \sin (x) + 4 x \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.867

6181	$y^{''} + 2 y^{'} + y = 4 e^{x} + (1 - x) (- 1 + e^{2 x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.778

6182	$y^{''} - 2 y^{'} = 9 x e^{- x} - 6 x^{2} + 4 e^{2 x}$	[[_2nd_order, _missing_y]]	✓	1.516

6183	$y^{''} + y y^{'} = 0$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.935

6184	$y^{''} + y y^{'} = 0$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.651

6185	$y^{''} + y y^{'} = 0$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.456

6186	$y^{''} + y y^{'} = 0$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	1.361

6187	$y^{''} + 2 y^{'} x = 0$	[[_2nd_order, _missing_y]]	✓	0.614

6188	$2 y y^{''} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.542

6189	$x y^{''} = {y^{'}}^{3} + y^{'}$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	0.672

6190	${y^{''}}^{2} = k^{2} (1 + {y^{'}}^{2})$	[[_2nd_order, _missing_x]]	✓	3.738

6191	$k = \frac{y^{''}}{{(y^{'} + 1)}^{3 / 2}}$	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear]]	✓	2.248

6192	$x^{2} y^{''} + 3 y^{'} x - 3 y = 0$	[[_Emden, _Fowler]]	✓	0.898

6193	$x^{2} y^{''} + y^{'} x - 4 y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.837

6194	$x^{2} y^{''} + 7 y^{'} x + 9 y = 0$	[[_Emden, _Fowler]]	✓	0.898

6195	$x^{2} y^{''} - y^{'} x + 6 y = 0$	[[_Emden, _Fowler]]	✓	1.174

6196	$x^{2} y^{''} + y^{'} x - 16 y = 8 x^{4}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.735

6197	$x^{2} y^{''} + y^{'} x - y = x - \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓	2.268

6198	$x^{2} y^{''} - 5 y^{'} x + 9 y = 2 x^{3}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.493

6199	$x^{2} y^{''} - 3 y^{'} x + 4 y = 6 x^{2} \ln (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.452

6200	$x^{2} y^{''} + y = 3 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	1.117

2.2.62 Problems 6101 to 6200