Problems 2001 to 2100

Table 2.43: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

2001	$x^{2} (- x^{2} + 2) y^{''} - 2 x (2 x^{2} + 1) y^{'} + (- 2 x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.641

2002	$x^{2} y^{''} - x (5 - x) y^{'} + (9 - 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.668

2003	$x^{2} y^{''} - x (1 - x) y^{'} + (- x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.802

2004	$x^{2} (2 x^{2} + x + 1) y^{''} + x (7 x^{2} + 6 x + 3) y^{'} + (- 3 x^{2} + 6 x + 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.895

2005	$x^{2} (x^{2} + 2 x + 1) y^{''} + x (4 x^{2} + 3 x + 1) y^{'} - x (1 - 2 x) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.799

2006	$4 x^{2} (x^{2} + x + 1) y^{''} + 12 x^{2} (x + 1) y^{'} + (3 x^{2} + 3 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.904

2007	$x^{2} (x^{2} + x + 1) y^{''} - x (- 2 x^{2} - 4 x + 1) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.876

2008	$9 x^{2} y^{''} + 3 x (- 2 x^{2} + 3 x + 5) y^{'} + (- 14 x^{2} + 12 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.845

2009	$x^{2} y^{''} + x (x^{2} + x + 1) y^{'} + x (2 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.813

2010	$x^{2} (2 x + 1) y^{''} + x (3 x^{2} + 14 x + 5) y^{'} + (12 x^{2} + 18 x + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.809

2011	$4 x^{2} y^{''} + 2 x (x^{2} + x + 4) y^{'} + (3 x^{2} + 5 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.929

2012	$16 x^{2} y^{''} + 4 x (2 x^{2} + x + 6) y^{'} + (18 x^{2} + 5 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.842

2013	$9 x^{2} (x + 1) y^{''} + 3 x (- x^{2} + 11 x + 5) y^{'} + (- 7 x^{2} + 16 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.864

2014	$4 x^{2} y^{''} + (1 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.589

2015	$36 x^{2} (1 - 2 x) y^{''} + 24 x (1 - 9 x) y^{'} + (1 - 70 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.720

2016	$x^{2} (x + 1) y^{''} - x (3 - x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.657

2017	$x^{2} (1 - 2 x) y^{''} - x (5 - 4 x) y^{'} + (9 - 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.671

2018	$25 x^{2} y^{''} + x (15 + x) y^{'} + (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.704

2019	$2 x^{2} (x + 2) y^{''} + x^{2} y^{'} + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.713

2020	$x^{2} (9 + 4 x) y^{''} + 3 x y^{'} + (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.749

2021	$x^{2} y^{''} - x (3 - 2 x) y^{'} + (4 + 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.713

2022	$x^{2} (1 - 4 x) y^{''} + 3 x (1 - 6 x) y^{'} + (1 - 12 x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.734

2023	$x^{2} (2 x + 1) y^{''} + x (3 + 5 x) y^{'} + (1 - 2 x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.727

2024	$2 x^{2} (x + 1) y^{''} - x (6 - x) y^{'} + (8 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.728

2025	$x^{2} (2 x + 1) y^{''} + x (5 + 9 x) y^{'} + (4 + 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.844

2026	$x^{2} (1 - 2 x) y^{''} - x (5 + 4 x) y^{'} + (9 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.823

2027	$x^{2} (1 + 4 x) y^{''} - x (1 - 4 x) y^{'} + (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.903

2028	$x^{2} (x + 1) y^{''} + x (2 x + 1) y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.822

2029	$x^{2} (1 - x) y^{''} + x (7 + x) y^{'} + (9 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.841

2030	$x^{2} y^{''} - x (- x^{2} + 1) y^{'} + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.541

2031	$x^{2} (x^{2} + 1) y^{''} - 3 x (- x^{2} + 1) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.547

2032	$4 x^{2} y^{''} + 2 x^{3} y^{'} + (3 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.539

2033	$x^{2} (x^{2} + 1) y^{''} - x (- 2 x^{2} + 1) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.605

2034	$2 x^{2} (x^{2} + 2) y^{''} + 7 x^{3} y^{'} + (3 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.654

2035	$x^{2} (x^{2} + 1) y^{''} - x (- 4 x^{2} + 1) y^{'} + (2 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.578

2036	$4 x^{2} (x^{2} + 4) y^{''} + 3 x (3 x^{2} + 8) y^{'} + (- 9 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.646

2037	$3 x^{2} (x^{2} + 3) y^{''} + x (11 x^{2} + 3) y^{'} + (5 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.656

2038	$4 x^{2} (4 x^{2} + 1) y^{''} + 32 x^{3} y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.638

2039	$9 x^{2} y^{''} - 3 x (- 2 x^{2} + 7) y^{'} + (2 x^{2} + 25) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.566

2040	$x^{2} (2 x^{2} + 1) y^{''} + x (7 x^{2} + 3) y^{'} + (- 3 x^{2} + 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.592

2041	$x^{2} (x^{2} + 1) y^{''} + x (8 x^{2} + 3) y^{'} + (12 x^{2} + 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.639

2042	$x^{2} y^{''} - x (- x^{2} + 1) y^{'} + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.537

2043	$x^{2} (- 2 x^{2} + 1) y^{''} + x (- 9 x^{2} + 5) y^{'} + (- 3 x^{2} + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.674

2044	$x^{2} (x^{2} + 2) y^{''} + x (- x^{2} + 14) y^{'} + 2 (x^{2} + 9) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.659

2045	$x^{2} (x^{2} + 1) y^{''} + x (7 x^{2} + 3) y^{'} + (8 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.574

2046	$x^{2} (1 - 2 x) y^{''} + 3 x y^{'} + (1 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.718

2047	$x (x + 1) y^{''} + (1 - x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.548

2048	$x^{2} (1 - x) y^{''} + x (3 - 2 x) y^{'} + (2 x + 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.686

2049	$4 x^{2} (x + 1) y^{''} + 4 x^{2} y^{'} + (1 - 5 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.796

2050	$x^{2} (1 - x) y^{''} - x (3 - 5 x) y^{'} + (4 - 5 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.708

2051	$x^{2} (x^{2} + 1) y^{''} - x (9 x^{2} + 1) y^{'} + (25 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.652

2052	$9 x^{2} y^{''} + 3 x (- x^{2} + 1) y^{'} + (7 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.586

2053	$x (x^{2} + 1) y^{''} + (- x^{2} + 1) y^{'} - 8 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.484

2054	$4 x^{2} y^{''} + 2 x (- x^{2} + 4) y^{'} + (7 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.638

2055	$4 x^{2} (x + 1) y^{''} + 8 x^{2} y^{'} + (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.612

2056	$9 x^{2} (x + 3) y^{''} + 3 x (3 + 7 x) y^{'} + (3 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.562

2057	$x^{2} (- x^{2} + 2) y^{''} - x (3 x^{2} + 2) y^{'} + (- x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.551

2058	$16 x^{2} (x^{2} + 1) y^{''} + 8 x (9 x^{2} + 1) y^{'} + (49 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.547

2059	$x^{2} (4 + 3 x) y^{''} - x (4 - 3 x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.534

2060	$4 x^{2} (x^{2} + 3 x + 1) y^{''} + 8 x^{2} (2 x + 3) y^{'} + (9 x^{2} + 3 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.673

2061	$x^{2} {(1 - x)}^{2} y^{''} - x (- 3 x^{2} + 2 x + 1) y^{'} + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.635

2062	$9 x^{2} (x^{2} + x + 1) y^{''} + 3 x (13 x^{2} + 7 x + 1) y^{'} + (25 x^{2} + 4 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.668

2063	$2 x^{2} (x + 2) y^{''} - x (4 - 7 x) y^{'} - (5 - 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.664

2064	$x^{2} (1 - 2 x) y^{''} + x (8 - 9 x) y^{'} + (6 - 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.842

2065	$x^{2} (x^{2} + 1) y^{''} + x (10 x^{2} + 3) y^{'} - (- 14 x^{2} + 15) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.648

2066	$x^{2} (- 2 x^{2} + 1) y^{''} + x (- 13 x^{2} + 7) y^{'} - 14 x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.711

2067	$x^{2} y^{''} - 3 x y^{'} + (3 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.313

2068	$x y^{''} + y = 0$	[[_Emden, _Fowler]]	✓	1.031

2069	$4 x^{2} (x + 1) y^{''} + 4 x (2 x + 1) y^{'} - (1 + 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.457

2070	$x (x + 1) y^{''} + x y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.316

2071	$2 x^{2} (3 x + 2) y^{''} + x (4 + 21 x) y^{'} - (1 - 9 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.519

2072	$x^{2} y^{''} + x (x + 2) y^{'} - (2 - 3 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.370

2073	$4 x^{2} y^{''} + 4 x y^{'} - (9 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.296

2074	$x^{2} y^{''} + 10 x y^{'} + (14 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.309

2075	$4 x^{2} (x + 1) y^{''} + 4 x (3 + 8 x) y^{'} - (5 - 49 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.573

2076	$x^{2} (x + 1) y^{''} - x (3 + 10 x) y^{'} + 30 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.414

2077	$x^{2} y^{''} + x (x + 1) y^{'} - 3 (x + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.374

2078	$x^{2} y^{''} + x (1 - 2 x) y^{'} - (x + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.387

2079	$x (x + 1) y^{''} - 4 y^{'} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.692

2080	$x^{2} (2 x + 1) y^{''} + x (9 + 13 x) y^{'} + (7 + 5 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.965

2081	$4 x^{2} (2 x + 1) y^{''} - 2 x (4 - x) y^{'} - (7 + 5 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.854

2082	$3 x^{2} (x + 3) y^{''} - x (15 + x) y^{'} - 20 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.818

2083	$x^{2} (x + 1) y^{''} + x (1 - 10 x) y^{'} - (9 - 10 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.901

2084	$x^{2} (x + 1) y^{''} + 3 x^{2} y^{'} - (6 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.820

2085	$x^{2} (2 x + 1) y^{''} - 2 x (3 + 14 x) y^{'} + (6 + 100 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.810

2086	$x^{2} (x + 1) y^{''} - x (6 + 11 x) y^{'} + (6 + 32 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.808

2087	$4 x^{2} (x + 1) y^{''} + 4 x (1 + 4 x) y^{'} - (49 + 27 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.895

2088	$x^{2} (2 x + 1) y^{''} - x (9 + 8 x) y^{'} - 12 x y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.243

2089	$x^{2} (x^{2} + 1) y^{''} - x (- 2 x^{2} + 7) y^{'} + 12 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.405

2090	$x^{2} y^{''} - x (- x^{2} + 7) y^{'} + 12 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.317

2091	$x y^{''} - 5 y^{'} + x y = 0$	[_Lienard]	✓	1.147

2092	$x^{2} y^{''} + x (2 x^{2} + 1) y^{'} - (- 10 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.317

2093	$x^{2} y^{''} - x y^{'} - (- x^{2} + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.171

2094	$4 x^{2} y^{''} + 2 x (x^{2} + 8) y^{'} + (3 x^{2} + 5) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.420

2095	$x^{2} y^{''} + x (x^{2} + 1) y^{'} - (- 3 x^{2} + 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	1.307

2096	$x^{2} y^{''} + x (- 2 x^{2} + 1) y^{'} - 4 (2 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.324

2097	$4 x^{2} y^{''} + 8 x y^{'} - (- x^{2} + 35) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.302

2098	$9 x^{2} y^{''} - 3 x (2 x^{2} + 11) y^{'} + (10 x^{2} + 13) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.533

2099	$x^{2} y^{''} + x (- 2 x^{2} + 1) y^{'} - 4 (- x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.350

2100	$x^{2} y^{''} + x (- 3 x^{2} + 1) y^{'} - 4 (- 3 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.353

2.2.21 Problems 2001 to 2100