Problems 4001 to 4100

Table 2.83: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

4001	$x y^{''} - (- 1 + x) y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.542

4002	$(2 x^{2} + 1) y^{''} + 7 x y^{'} + 2 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.388

4003	$4 y^{''} + x y^{'} + 4 y = 0$ i.c.	[_Lienard]	✓	0.328

4004	$y^{''} + 2 x^{2} y^{'} + x y = 2 \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.468

4005	$y^{''} + x y^{'} - 4 y = 6 e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.403

4006	$y^{''} + \frac{y^{'}}{1 - x} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.428

4007	$x^{2} y^{''} + \frac{x y^{'}}{{(- x^{2} + 1)}^{2}} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.538

4008	${(- 2 + x)}^{2} y^{''} + (- 2 + x) e^{x} y^{'} + \frac{4 y}{x} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.935

4009	$y^{''} + \frac{2 y^{'}}{x (x - 3)} - \frac{y}{x^{3} (x + 3)} = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.110

4010	$x^{2} y^{''} + x (1 - x) y^{'} - 7 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.864

4011	$4 x^{2} y^{''} + x e^{x} y^{'} - y = 0$	[[_Emden, _Fowler]]	✓	0.835

4012	$4 x y^{''} - x y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.329

4013	$x^{2} y^{''} - x \cos (x) y^{'} + 5 e^{2 x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.749

4014	$4 x^{2} y^{''} + 3 x y^{'} + x y = 0$	[[_Emden, _Fowler]]	✓	0.670

4015	$6 x^{2} y^{''} + x (1 + 18 x) y^{'} + (1 + 12 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.744

4016	$x^{2} y^{''} + x y^{'} - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.819

4017	$2 x y^{''} + y^{'} - 2 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.520

4018	$3 x^{2} y^{''} - x (x + 8) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.754

4019	$2 x^{2} y^{''} - x (2 x + 1) y^{'} + 2 (4 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.777

4020	$x^{2} y^{''} + x (1 - x) y^{'} - (5 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.880

4021	$3 x^{2} y^{''} + x (7 + 3 x) y^{'} + (1 + 6 x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.724

4022	$x^{2} y^{''} + x y^{'} + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.581

4023	$3 x^{2} y^{''} + x (3 x^{2} + 1) y^{'} - 2 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.742

4024	$4 x^{2} y^{''} - 4 x^{2} y^{'} + (2 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.687

4025	$x^{2} y^{''} + x (3 - 2 x) y^{'} + (1 - 2 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.680

4026	$x^{2} y^{''} - x (x + 3) y^{'} + (4 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.685

4027	$x^{2} y^{''} + x (3 - x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.659

4028	$x^{2} y^{''} + x y^{'} - (x + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.284

4029	$x^{2} y^{''} - (- x^{2} + x) y^{'} + (x^{3} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.633

4030	$x^{2} y^{''} - (2 \sqrt{5} - 1) x y^{'} + (\frac{19}{4} - 3 x^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.955

4031	$x^{2} y^{''} + (- 2 x^{5} + 9 x) y^{'} + (10 x^{4} + 5 x^{2} + 25) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.593

4032	$x^{2} y^{''} + (4 x + \frac{1}{2} x^{2} - \frac{1}{3} x^{3}) y^{'} - \frac{7 y}{4} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.712

4033	$x^{2} y^{''} + x^{2} y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.260

4034	$x^{2} y^{''} + x (x - 3) y^{'} + (4 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.675

4035	$4 x^{2} y^{''} + 2 x^{2} y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.634

4036	$x^{2} y^{''} + x \cos (x) y^{'} - 2 y e^{x} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.976

4037	$x^{2} y^{''} + x^{2} y^{'} - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.746

4038	$x^{2} y^{''} + 2 x^{2} y^{'} + (x - \frac{3}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.743

4039	$x^{2} y^{''} + x y^{'} + (2 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.297

4040	$x^{2} y^{''} + x^{3} y^{'} - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.408

4041	$x^{2} (x^{2} + 1) y^{''} + 7 x e^{x} y^{'} + 9 (1 + \tan (x)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.233

4042	$x^{2} (x + 1) y^{''} + x^{2} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.759

4043	$x^{2} y^{''} + 3 x y^{'} + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.670

4044	$x y^{''} - y = 0$	[[_Emden, _Fowler]]	✓	1.040

4045	$x^{2} y^{''} + x (x^{2} + 6) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.685

4046	$x^{2} y^{''} + x (1 - x) y^{'} - y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.701

4047	$4 x^{2} y^{''} + (1 - 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.656

4048	$x y^{''} + y^{'} - 2 y = 0$	[[_Emden, _Fowler]]	✓	0.510

4049	$x^{2} y^{''} + x y^{'} - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.295

4050	$x^{2} y^{''} - x (x + 3) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.717

4051	$x^{2} y^{''} - x^{2} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.708

4052	$x^{2} y^{''} - x^{2} y^{'} - (3 x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.457

4053	$x^{2} y^{''} + x (5 - x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.681

4054	$4 x^{2} y^{''} + 4 x (1 - x) y^{'} + (2 x - 9) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.758

4055	$x^{2} y^{''} + 2 x (x + 2) y^{'} + 2 (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.388

4056	$x^{2} y^{''} - x (1 - x) y^{'} + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.653

4057	$4 x^{2} y^{''} + 4 x (2 x + 1) y^{'} + (4 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.751

4058	$4 x^{2} y^{''} - (3 + 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.243

4059	$x y^{''} - x y^{'} + y = 0$	[_Laguerre, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.159

4060	$x^{2} y^{''} + x (x + 4) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.383

4061	$x^{2} y^{''} + x y^{'} + (x^{2} - \frac{9}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.661

4062	$x y^{''} - y^{'} + x y = 0$	[_Lienard]	✓	1.080

4063	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.233

4064	$y^{''} - x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.288

4065	$(- x^{2} + 1) y^{''} - 6 x y^{'} - 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.395

4066	$x y^{''} + y^{'} + 2 y = 0$	[[_Emden, _Fowler]]	✓	0.504

4067	$x y^{''} + 2 y^{'} + x y = 0$	[_Lienard]	✓	0.592

4068	$2 x y^{''} + 5 (1 - 2 x) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.715

4069	$x y^{''} + y^{'} + x y = 0$	[_Lienard]	✓	0.423

4070	$(4 x^{2} + 1) y^{''} - 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.349

4071	$x^{2} y^{''} + x y^{'} + (x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.654

4072	$4 x y^{''} + 3 y^{'} + 3 y = 0$	[[_Emden, _Fowler]]	✓	0.658

4073	$x^{2} y^{''} + \frac{3 x y^{'}}{2} - \frac{(x + 1) y}{2} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.724

4074	$x^{2} y^{''} - x (2 - x) y^{'} + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.395

4075	$x^{2} y^{''} - 3 x y^{'} + 4 (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.650

4076	$y^{''} + (1 - \frac{3}{4 x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.153

4077	$5 x y + 4 y^{2} + 1 + (x^{2} + 2 x y) y^{'} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.662

4078	$2 x \tan (y) + (x - x^{2} \tan (y)) y^{'} = 0$	[[_1st_order, _with_exponential_symmetries]]	✓	1.856

4079	$y^{2} (x^{2} + 1) + y + (2 x y + 1) y^{'} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.646

4080	$4 x y^{2} + 6 y + (5 x^{2} y + 8 x) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.462

4081	$5 x + 2 y + 1 + (2 x + y + 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.129

4082	$3 x - y + 1 - (6 x - 2 y - 3) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.964

4083	$x - 2 y - 3 + (2 x + y - 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	36.871

4084	$6 x + 4 y + 1 + (4 x + 2 y + 2) y^{'} = 0$ i.c.	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	4.363

4085	$3 x - y - 6 + (x + y + 2) y^{'} = 0$ i.c.	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	10.793

4086	$2 x + 3 y + 1 + (4 x + 6 y + 1) y^{'} = 0$ i.c.	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.385

4087	$y = y^{'} + \frac{{y^{'}}^{2}}{2}$	[_quadrature]	✓	0.401

4088	${(y - x y^{'})}^{2} = 1 + {y^{'}}^{2}$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.572

4089	$y - x = {y^{'}}^{2} (1 - \frac{2 y^{'}}{3})$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	1.159

4090	$x^{2} y^{'} = x (y - 1) + {(y - 1)}^{2}$	[[_homogeneous, ‘class C‘], _rational, _Riccati]	✓	1.832

4091	$y^{'} = e^{- x}$	[_quadrature]	✓	0.477

4092	$y^{'} = 1 - x^{5} + \sqrt{x}$	[_quadrature]	✓	0.268

4093	$3 y - 2 x + (3 x - 2) y^{'} = 0$	[_linear]	✓	2.045

4094	$x^{2} + x - 1 + (2 x y + y) y^{'} = 0$	[_separable]	✓	2.241

4095	$e^{2 y} + (x + 1) y^{'} = 0$	[_separable]	✓	1.674

4096	$(x + 1) y^{'} - y^{2} x^{2} = 0$	[_separable]	✓	1.877

4097	$y^{'} = \frac{y - 2 x}{x}$	[_linear]	✓	1.730

4098	$x^{3} + y^{3} - x y^{2} y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	11.654

4099	$y^{'} + y = 0$	[_quadrature]	✓	1.595

4100	$y^{'} + y = x^{2} + 2$	[[_linear, ‘class A‘]]	✓	1.339

2.2.41 Problems 4001 to 4100