2.2.41 Problems 4001 to 4100

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

#

ODE

CAS classification

Solved?

time (sec)

4001

\[ {}x y^{\prime \prime }-\left (-1+x \right ) y^{\prime }-x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.542

4002

\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y = 0 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.388

4003

\[ {}4 y^{\prime \prime }+x y^{\prime }+4 y = 0 \]
i.c.

[_Lienard]

0.328

4004

\[ {}y^{\prime \prime }+2 x^{2} y^{\prime }+x y = 2 \cos \left (x \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

0.468

4005

\[ {}y^{\prime \prime }+x y^{\prime }-4 y = 6 \,{\mathrm e}^{x} \]

[[_2nd_order, _linear, _nonhomogeneous]]

0.403

4006

\[ {}y^{\prime \prime }+\frac {y^{\prime }}{1-x}+x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.428

4007

\[ {}x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{\left (-x^{2}+1\right )^{2}}+y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.538

4008

\[ {}\left (-2+x \right )^{2} y^{\prime \prime }+\left (-2+x \right ) {\mathrm e}^{x} y^{\prime }+\frac {4 y}{x} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.935

4009

\[ {}y^{\prime \prime }+\frac {2 y^{\prime }}{x \left (x -3\right )}-\frac {y}{x^{3} \left (x +3\right )} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.110

4010

\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-7 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.864

4011

\[ {}4 x^{2} y^{\prime \prime }+x \,{\mathrm e}^{x} y^{\prime }-y = 0 \]

[[_Emden, _Fowler]]

0.835

4012

\[ {}4 x y^{\prime \prime }-x y^{\prime }+2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.329

4013

\[ {}x^{2} y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+5 \,{\mathrm e}^{2 x} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.749

4014

\[ {}4 x^{2} y^{\prime \prime }+3 x y^{\prime }+x y = 0 \]

[[_Emden, _Fowler]]

0.670

4015

\[ {}6 x^{2} y^{\prime \prime }+x \left (1+18 x \right ) y^{\prime }+\left (1+12 x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.744

4016

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.819

4017

\[ {}2 x y^{\prime \prime }+y^{\prime }-2 x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.520

4018

\[ {}3 x^{2} y^{\prime \prime }-x \left (x +8\right ) y^{\prime }+6 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.754

4019

\[ {}2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.777

4020

\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-\left (5+x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.880

4021

\[ {}3 x^{2} y^{\prime \prime }+x \left (7+3 x \right ) y^{\prime }+\left (1+6 x \right ) y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.724

4022

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (1-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.581

4023

\[ {}3 x^{2} y^{\prime \prime }+x \left (3 x^{2}+1\right ) y^{\prime }-2 x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.742

4024

\[ {}4 x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+\left (2 x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.687

4025

\[ {}x^{2} y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (1-2 x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.680

4026

\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.685

4027

\[ {}x^{2} y^{\prime \prime }+x \left (3-x \right ) y^{\prime }+y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.659

4028

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +4\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.284

4029

\[ {}x^{2} y^{\prime \prime }-\left (-x^{2}+x \right ) y^{\prime }+\left (x^{3}+1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.633

4030

\[ {}x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.955

4031

\[ {}x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.593

4032

\[ {}x^{2} y^{\prime \prime }+\left (4 x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}\right ) y^{\prime }-\frac {7 y}{4} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.712

4033

\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }+x y = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

1.260

4034

\[ {}x^{2} y^{\prime \prime }+x \left (x -3\right ) y^{\prime }+\left (4-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.675

4035

\[ {}4 x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.634

4036

\[ {}x^{2} y^{\prime \prime }+x \cos \left (x \right ) y^{\prime }-2 y \,{\mathrm e}^{x} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.976

4037

\[ {}x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\left (x +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.746

4038

\[ {}x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.743

4039

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (2 x -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.297

4040

\[ {}x^{2} y^{\prime \prime }+x^{3} y^{\prime }-\left (x +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.408

4041

\[ {}x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+7 x \,{\mathrm e}^{x} y^{\prime }+9 \left (1+\tan \left (x \right )\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.233

4042

\[ {}x^{2} \left (x +1\right ) y^{\prime \prime }+x^{2} y^{\prime }-2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.759

4043

\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (1-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.670

4044

\[ {}x y^{\prime \prime }-y = 0 \]

[[_Emden, _Fowler]]

1.040

4045

\[ {}x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.685

4046

\[ {}x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.701

4047

\[ {}4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.656

4048

\[ {}x y^{\prime \prime }+y^{\prime }-2 y = 0 \]

[[_Emden, _Fowler]]

0.510

4049

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.295

4050

\[ {}x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.717

4051

\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.708

4052

\[ {}x^{2} y^{\prime \prime }-x^{2} y^{\prime }-\left (3 x +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.457

4053

\[ {}x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.681

4054

\[ {}4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.758

4055

\[ {}x^{2} y^{\prime \prime }+2 x \left (x +2\right ) y^{\prime }+2 \left (x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.388

4056

\[ {}x^{2} y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+\left (1-x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.653

4057

\[ {}4 x^{2} y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }+\left (4 x -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.751

4058

\[ {}4 x^{2} y^{\prime \prime }-\left (3+4 x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.243

4059

\[ {}x y^{\prime \prime }-x y^{\prime }+y = 0 \]

[_Laguerre, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

1.159

4060

\[ {}x^{2} y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (x +2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.383

4061

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.661

4062

\[ {}x y^{\prime \prime }-y^{\prime }+x y = 0 \]

[_Lienard]

1.080

4063

\[ {}y^{\prime \prime }+x y = 0 \]

[[_Emden, _Fowler]]

0.233

4064

\[ {}y^{\prime \prime }-x^{2} y = 0 \]

[[_Emden, _Fowler]]

0.288

4065

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-6 x y^{\prime }-4 y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.395

4066

\[ {}x y^{\prime \prime }+y^{\prime }+2 y = 0 \]

[[_Emden, _Fowler]]

0.504

4067

\[ {}x y^{\prime \prime }+2 y^{\prime }+x y = 0 \]

[_Lienard]

0.592

4068

\[ {}2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.715

4069

\[ {}x y^{\prime \prime }+y^{\prime }+x y = 0 \]

[_Lienard]

0.423

4070

\[ {}\left (4 x^{2}+1\right ) y^{\prime \prime }-8 y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.349

4071

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.654

4072

\[ {}4 x y^{\prime \prime }+3 y^{\prime }+3 y = 0 \]

[[_Emden, _Fowler]]

0.658

4073

\[ {}x^{2} y^{\prime \prime }+\frac {3 x y^{\prime }}{2}-\frac {\left (x +1\right ) y}{2} = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.724

4074

\[ {}x^{2} y^{\prime \prime }-x \left (2-x \right ) y^{\prime }+\left (x^{2}+2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.395

4075

\[ {}x^{2} y^{\prime \prime }-3 x y^{\prime }+4 \left (x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.650

4076

\[ {}y^{\prime \prime }+\left (1-\frac {3}{4 x^{2}}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.153

4077

\[ {}5 x y+4 y^{2}+1+\left (x^{2}+2 x y\right ) y^{\prime } = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1.662

4078

\[ {}2 x \tan \left (y\right )+\left (x -x^{2} \tan \left (y\right )\right ) y^{\prime } = 0 \]

[[_1st_order, _with_exponential_symmetries]]

1.856

4079

\[ {}y^{2} \left (x^{2}+1\right )+y+\left (2 x y+1\right ) y^{\prime } = 0 \]

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1.646

4080

\[ {}4 x y^{2}+6 y+\left (5 x^{2} y+8 x \right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

2.462

4081

\[ {}5 x +2 y+1+\left (2 x +y+1\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3.129

4082

\[ {}3 x -y+1-\left (6 x -2 y-3\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1.964

4083

\[ {}x -2 y-3+\left (2 x +y-1\right ) y^{\prime } = 0 \]

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

36.871

4084

\[ {}6 x +4 y+1+\left (4 x +2 y+2\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

4.363

4085

\[ {}3 x -y-6+\left (x +y+2\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

10.793

4086

\[ {}2 x +3 y+1+\left (4 x +6 y+1\right ) y^{\prime } = 0 \]
i.c.

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

2.385

4087

\[ {}y = y^{\prime }+\frac {{y^{\prime }}^{2}}{2} \]

[_quadrature]

0.401

4088

\[ {}\left (y-x y^{\prime }\right )^{2} = 1+{y^{\prime }}^{2} \]

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

0.572

4089

\[ {}y-x = {y^{\prime }}^{2} \left (1-\frac {2 y^{\prime }}{3}\right ) \]

[[_homogeneous, ‘class C‘], _dAlembert]

1.159

4090

\[ {}x^{2} y^{\prime } = x \left (y-1\right )+\left (y-1\right )^{2} \]

[[_homogeneous, ‘class C‘], _rational, _Riccati]

1.832

4091

\[ {}y^{\prime } = {\mathrm e}^{-x} \]

[_quadrature]

0.477

4092

\[ {}y^{\prime } = 1-x^{5}+\sqrt {x} \]

[_quadrature]

0.268

4093

\[ {}3 y-2 x +\left (3 x -2\right ) y^{\prime } = 0 \]

[_linear]

2.045

4094

\[ {}x^{2}+x -1+\left (2 x y+y\right ) y^{\prime } = 0 \]

[_separable]

2.241

4095

\[ {}{\mathrm e}^{2 y}+\left (x +1\right ) y^{\prime } = 0 \]

[_separable]

1.674

4096

\[ {}\left (x +1\right ) y^{\prime }-y^{2} x^{2} = 0 \]

[_separable]

1.877

4097

\[ {}y^{\prime } = \frac {y-2 x}{x} \]

[_linear]

1.730

4098

\[ {}x^{3}+y^{3}-x y^{2} y^{\prime } = 0 \]

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

11.654

4099

\[ {}y^{\prime }+y = 0 \]

[_quadrature]

1.595

4100

\[ {}y^{\prime }+y = x^{2}+2 \]

[[_linear, ‘class A‘]]

1.339