2.2.61 Problems 6001 to 6100

Table 2.123: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

6001

\[ {}r^{\prime \prime } = -\frac {k}{r^{2}} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

66.330

6002

\[ {}y^{\prime \prime } = \frac {3 k y^{2}}{2} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

4.692

6003

\[ {}y^{\prime \prime } = 2 k y^{3} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

5.739

6004

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.860

6005

\[ {}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

2.589

6006

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

0.487

6007

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.526

6008

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

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

0.825

6009

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

[[_2nd_order, _missing_y]]

1.405

6010

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.590

6011

\[ {}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y} \]
i.c.

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

2.502

6012

\[ {}y^{\prime \prime } = 2 y y^{\prime } \]
i.c.

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

13.908

6013

\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \]
i.c.

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

1.305

6014

\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \]
i.c.

[[_2nd_order, _missing_y]]

1.089

6015

\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \]
i.c.

[[_2nd_order, _missing_y]]

1.419

6016

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

[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.187

6017

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

[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

1.177

6018

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

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.124

6019

\[ {}-a y^{3}-\frac {b}{x^{{3}/{2}}}+y^{\prime } = 0 \]

[[_homogeneous, ‘class G‘], _rational, _Abel]

4.980

6020

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

[[_homogeneous, ‘class G‘], _Abel]

2.475

6021

\[ {}y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y-x^{-2 a}+a \,x^{-a -1} = 0 \]

[_Abel]

3.372

6022

\[ {}y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )} = 0 \]

[_Abel]

108.821

6023

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

[_rational, _Abel]

0.889

6024

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

[_rational, _Abel]

2.217

6025

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

[_separable]

1.794

6026

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

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

0.979

6027

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

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

1.876

6028

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

[_quadrature]

0.360

6029

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

[[_linear, ‘class A‘]]

0.767

6030

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

[[_2nd_order, _missing_x]]

205.148

6031

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

[_separable]

1.495

6032

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

[_separable]

1.448

6033

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

[_separable]

1.760

6034

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

[_separable]

4.317

6035

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

[_separable]

1.596

6036

\[ {}y^{\prime }+b^{2} y^{2} = a^{2} \]

[_quadrature]

3.583

6037

\[ {}y^{\prime } = \frac {1+y^{2}}{x^{2}+1} \]

[_separable]

2.274

6038

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

[_separable]

3.194

6039

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

[_separable]

1.662

6040

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

[[_2nd_order, _with_linear_symmetries]]

0.774

6041

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

[[_Emden, _Fowler]]

0.228

6042

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.698

6043

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.644

6044

\[ {}y^{\prime \prime }+a \,x^{2} y = x +1 \]

[[_2nd_order, _linear, _nonhomogeneous]]

0.351

6045

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

[[_2nd_order, _with_linear_symmetries]]

0.153

6046

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

[[_2nd_order, _with_linear_symmetries]]

0.733

6047

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.697

6048

\[ {}\left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.783

6049

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

[[_2nd_order, _with_linear_symmetries]]

0.704

6050

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

[[_2nd_order, _with_linear_symmetries]]

0.695

6051

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

[[_2nd_order, _with_linear_symmetries]]

0.900

6052

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

[[_2nd_order, _with_linear_symmetries]]

0.828

6053

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

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

0.533

6054

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

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

0.518

6055

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

[[_Emden, _Fowler]]

0.497

6056

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

[[_2nd_order, _with_linear_symmetries]]

0.444

6057

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

[[_Emden, _Fowler]]

1.033

6058

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

[[_2nd_order, _with_linear_symmetries]]

0.092

6059

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

[[_2nd_order, _with_linear_symmetries]]

1.380

6060

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

[[_2nd_order, _with_linear_symmetries]]

1.219

6061

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

[[_elliptic, _class_I]]

0.479

6062

\[ {}y^{\prime \prime }+\frac {a y}{x^{{3}/{2}}} = 0 \]

[[_Emden, _Fowler]]

0.147

6063

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

[[_2nd_order, _with_linear_symmetries]]

1.346

6064

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

[[_elliptic, _class_II]]

0.487

6065

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

[_Jacobi]

1.391

6066

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.105

6067

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.125

6068

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

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

0.798

6069

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

[_Jacobi]

0.725

6070

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

[[_2nd_order, _with_linear_symmetries]]

1.336

6071

\[ {}2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y = 0 \]

[_Jacobi]

0.726

6072

\[ {}x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9} = 0 \]

[_Jacobi]

0.737

6073

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.697

6074

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

[[_2nd_order, _with_linear_symmetries]]

0.432

6075

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

[_rational, _Riccati]

1.638

6076

\[ {}u^{\prime \prime }-\frac {a^{2} u}{x^{{2}/{3}}} = 0 \]

[[_Emden, _Fowler]]

0.667

6077

\[ {}u^{\prime \prime }-\frac {2 u^{\prime }}{x}-a^{2} u = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.165

6078

\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}-a^{2} u = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.175

6079

\[ {}u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u = 0 \]

[[_2nd_order, _with_linear_symmetries]]

2.287

6080

\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}-a^{2} u = 0 \]

[[_2nd_order, _with_linear_symmetries]]

3.815

6081

\[ {}u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u = 0 \]

[[_2nd_order, _with_linear_symmetries]]

2.663

6082

\[ {}y^{\prime \prime }-a^{2} y = \frac {6 y}{x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

5.316

6083

\[ {}y^{\prime \prime }+n^{2} y = \frac {6 y}{x^{2}} \]

[[_2nd_order, _with_linear_symmetries]]

3.208

6084

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

[[_2nd_order, _with_linear_symmetries]]

1.106

6085

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

[[_2nd_order, _with_linear_symmetries]]

3.144

6086

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

[[_2nd_order, _with_linear_symmetries]]

1.899

6087

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

[[_2nd_order, _with_linear_symmetries]]

0.959

6088

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

[[_2nd_order, _with_linear_symmetries]]

0.453

6089

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

[[_Emden, _Fowler]]

0.574

6090

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

[[_Emden, _Fowler]]

0.493

6091

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

[[_Emden, _Fowler]]

2.104

6092

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

[_quadrature]

1.528

6093

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

[_separable]

2.029

6094

\[ {}x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime } = 0 \]
i.c.

[_separable]

7.838

6095

\[ {}\sin \left (x \right ) y^{\prime } = y \ln \left (y\right ) \]
i.c.

[_separable]

16.326

6096

\[ {}1+y^{2}+x y y^{\prime } = 0 \]
i.c.

[_separable]

3.221

6097

\[ {}x y y^{\prime }-x y = y \]
i.c.

[_quadrature]

0.962

6098

\[ {}y^{\prime } = \frac {2 x y^{2}+x}{x^{2} y-y} \]
i.c.

[_separable]

2.790

6099

\[ {}y y^{\prime }+x y^{2}-8 x = 0 \]
i.c.

[_separable]

3.589

6100

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

[_separable]

2.492