Problems 4001 to 4100

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


#	ODE	CAS classiﬁcation	Solved?	time (sec)

4001	$x y^{''} - (x - 1) y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.707

4002	$(2 x^{2} + 1) y^{''} + 7 y^{'} x + 2 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.511

4003	$4 y^{''} + y^{'} x + 4 y = 0$ i.c.	[_Lienard]	✓	0.380

4004	$y^{''} + 2 x^{2} y^{'} + x y = 2 \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.556

4005	$y^{''} + y^{'} x - 4 y = 6 e^{x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.449

4006	$y^{''} + \frac{y^{'}}{1 - x} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.534

4007	$x^{2} y^{''} + \frac{x y^{'}}{{(- x^{2} + 1)}^{2}} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.643

4008	${(x - 2)}^{2} y^{''} + (x - 2) e^{x} y^{'} + \frac{4 y}{x} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.961

4009	$y^{''} + \frac{2 y^{'}}{x (x - 3)} - \frac{y}{x^{3} (3 + x)} = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.174

4010	$x^{2} y^{''} + x (1 - x) y^{'} - 7 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.859

4011	$4 x^{2} y^{''} + x e^{x} y^{'} - y = 0$	[[_Emden, _Fowler]]	✓	0.922

4012	$4 x y^{''} - y^{'} x + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.379

4013	$x^{2} y^{''} - x \cos (x) y^{'} + 5 e^{2 x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.885

4014	$4 x^{2} y^{''} + 3 y^{'} x + x y = 0$	[[_Emden, _Fowler]]	✓	0.729

4015	$6 x^{2} y^{''} + x (1 + 18 x) y^{'} + (1 + 12 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.803

4016	$x^{2} y^{''} + y^{'} x - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.888

4017	$2 x y^{''} + y^{'} - 2 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.580

4018	$3 x^{2} y^{''} - x (8 + x) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.828

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]]	✓	0.969

4021	$3 x^{2} y^{''} + x (3 x + 7) y^{'} + (1 + 6 x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.802

4022	$x^{2} y^{''} + y^{'} x + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.631

4023	$3 x^{2} y^{''} + x (3 x^{2} + 1) y^{'} - 2 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.929

4024	$4 x^{2} y^{''} - 4 x^{2} y^{'} + (2 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.749

4025	$x^{2} y^{''} + x (3 - 2 x) y^{'} + (- 2 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.692

4026	$x^{2} y^{''} - (3 + x) x y^{'} + (4 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.735

4027	$x^{2} y^{''} + x (3 - x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.708

4028	$x^{2} y^{''} + y^{'} x - (x + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.411

4029	$x^{2} y^{''} - (- x^{2} + x) y^{'} + (x^{3} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.725

4030	$x^{2} y^{''} - (2 \sqrt{5} - 1) x y^{'} + (\frac{19}{4} - 3 x^{2}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.072

4031	$x^{2} y^{''} + (- 2 x^{5} + 9 x) y^{'} + (10 x^{4} + 5 x^{2} + 25) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.692

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]]	✓	1.651

4033	$x^{2} y^{''} + x^{2} y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.361

4034	$x^{2} y^{''} + x (x - 3) y^{'} + (4 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.688

4035	$4 x^{2} y^{''} + 2 x^{2} y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.711

4036	$x^{2} y^{''} + x \cos (x) y^{'} - 2 y e^{x} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.070

4037	$x^{2} y^{''} + x^{2} y^{'} - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.780

4038	$x^{2} y^{''} + 2 x^{2} y^{'} + (x - \frac{3}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.772

4039	$x^{2} y^{''} + y^{'} x + (2 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.373

4040	$x^{2} y^{''} + x^{3} y^{'} - (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.409

4041	$x^{2} (x^{2} + 1) y^{''} + 7 x e^{x} y^{'} + 9 (1 + \tan (x)) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.455

4042	$x^{2} (x + 1) y^{''} + x^{2} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.877

4043	$x^{2} y^{''} + 3 y^{'} x + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.702

4044	$x y^{''} - y = 0$	[[_Emden, _Fowler]]	✓	1.147

4045	$x^{2} y^{''} + x (x^{2} + 6) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.727

4046	$x^{2} y^{''} + x (1 - x) y^{'} - y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.722

4047	$4 x^{2} y^{''} + (1 - 4 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.649

4048	$x y^{''} + y^{'} - 2 y = 0$	[[_Emden, _Fowler]]	✓	0.614

4049	$x^{2} y^{''} + y^{'} x - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.369

4050	$x^{2} y^{''} - (3 + x) x y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.681

4051	$x^{2} y^{''} - x^{2} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.732

4052	$x^{2} y^{''} - x^{2} y^{'} - (3 x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.342

4053	$x^{2} y^{''} + x (5 - x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.745

4054	$4 x^{2} y^{''} + 4 x (1 - x) y^{'} + (2 x - 9) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.849

4055	$x^{2} y^{''} + 2 x (x + 2) y^{'} + 2 (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.421

4056	$x^{2} y^{''} - x (1 - x) y^{'} + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.685

4057	$4 x^{2} y^{''} + 4 x (2 x + 1) y^{'} + (4 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.838

4058	$4 x^{2} y^{''} - (4 x + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.312

4059	$x y^{''} - y^{'} x + y = 0$	[_Laguerre, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.318

4060	$x^{2} y^{''} + x (x + 4) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.347

4061	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{9}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.673

4062	$x y^{''} - y^{'} + x y = 0$	[_Lienard]	✓	1.263

4063	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.342

4064	$y^{''} - x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.366

4065	$(- x^{2} + 1) y^{''} - 6 y^{'} x - 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.506

4066	$x y^{''} + y^{'} + 2 y = 0$	[[_Emden, _Fowler]]	✓	0.578

4067	$x y^{''} + 2 y^{'} + x y = 0$	[_Lienard]	✓	0.600

4068	$2 x y^{''} + 5 (- 2 x + 1) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.699

4069	$x y^{''} + y^{'} + x y = 0$	[_Lienard]	✓	0.504

4070	$(4 x^{2} + 1) y^{''} - 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.436

4071	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.714

4072	$4 x y^{''} + 3 y^{'} + 3 y = 0$	[[_Emden, _Fowler]]	✓	0.703

4073	$x^{2} y^{''} + \frac{3 y^{'} x}{2} - \frac{(x + 1) y}{2} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.798

4074	$x^{2} y^{''} - x (2 - x) y^{'} + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.345

4075	$x^{2} y^{''} - 3 y^{'} x + 4 (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.663

4076	$y^{''} + (1 - \frac{3}{4 x^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.093

4077	$5 x y + 4 y^{2} + 1 + (x^{2} + 2 x y) y^{'} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.629

4078	$2 x \tan (y) + (x - x^{2} \tan (y)) y^{'} = 0$	[[_1st_order, _with_exponential_symmetries]]	✓	2.074

4079	$y^{2} (x^{2} + 1) + y + (1 + 2 x y) y^{'} = 0$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.820

4080	$4 x y^{2} + 6 y + (5 x^{2} y + 8 x) y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	2.198

4081	$5 x + 2 y + 1 + (2 x + y + 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.823

4082	$3 x - y + 1 - (6 x - 2 y - 3) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.130

4083	$x - 2 y - 3 + (2 x + y - 1) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.542

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‘]]	✓	3.779

4085	$3 x - y - 6 + (x + y + 2) y^{'} = 0$ i.c.	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	21.910

4086	$2 x + 3 y + 1 + (4 x + 6 y + 1) y^{'} = 0$ i.c.	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.156

4087	$y = y^{'} + \frac{{y^{'}}^{2}}{2}$	[_quadrature]	✓	0.525

4088	${(y - y^{'} x)}^{2} = 1 + {y^{'}}^{2}$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	0.657

4089	$y - x = {y^{'}}^{2} (1 - \frac{2 y^{'}}{3})$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	0.603

4090	$x^{2} y^{'} = x (- 1 + y) + {(- 1 + y)}^{2}$	[[_homogeneous, ‘class C‘], _rational, _Riccati]	✓	1.720

4091	$y^{'} = e^{- x}$	[_quadrature]	✓	0.333

4092	$y^{'} = 1 - x^{5} + \sqrt{x}$	[_quadrature]	✓	0.295

4093	$3 y - 2 x + (3 x - 2) y^{'} = 0$	[_linear]	✓	1.867

4094	$x^{2} + x - 1 + (2 x y + y) y^{'} = 0$	[_separable]	✓	2.038

4095	$e^{2 y} + (x + 1) y^{'} = 0$	[_separable]	✓	1.900

4096	$(x + 1) y^{'} - x^{2} y^{2} = 0$	[_separable]	✓	1.366

4097	$y^{'} = \frac{y - 2 x}{x}$	[_linear]	✓	1.364

4098	$x^{3} + y^{3} - x y^{2} y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	11.438

4099	$y^{'} + y = 0$	[_quadrature]	✓	0.735

4100	$y^{'} + y = x^{2} + 2$	[[_linear, ‘class A‘]]	✓	1.072

2.2.41 Problems 4001 to 4100