2.2.86 Problems 8501 to 8600

Table 2.173: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

8501

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

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

0.313

8502

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

[[_2nd_order, _missing_y]]

1.823

8503

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

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

0.472

8504

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

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

0.575

8505

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

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

1.371

8506

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

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

7.792

8507

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

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

6.210

8508

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

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

6.227

8509

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

[[_2nd_order, _missing_y]]

0.799

8510

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

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

0.359

8511

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

[[_2nd_order, _missing_y]]

0.273

8512

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

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

1.189

8513

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

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

0.184

8514

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

1.382

8515

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

[[_2nd_order, _missing_x]]

2.299

8516

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

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

0.996

8517

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

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

2.001

8518

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

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

11.123

8519

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

[[_2nd_order, _missing_y]]

0.510

8520

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

[[_2nd_order, _missing_y]]

0.416

8521

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

[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]

0.600

8522

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

[[_2nd_order, _missing_y]]

0.408

8523

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

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]

0.756

8524

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

[[_2nd_order, _missing_y]]

1.798

8525

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

[[_2nd_order, _missing_y]]

0.454

8526

\[ {}{y^{\prime \prime }}^{3} = 12 y^{\prime } \left (x y^{\prime \prime }-2 y^{\prime }\right ) \]

[[_2nd_order, _missing_y]]

11.020

8527

\[ {}3 y y^{\prime } y^{\prime \prime } = {y^{\prime }}^{3}-1 \]

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

1.464

8528

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

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

1.202

8529

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

[[_2nd_order, _linear, _nonhomogeneous]]

2.847

8530

\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = {\mathrm e}^{x} \]

[[_2nd_order, _with_linear_symmetries]]

1.095

8531

\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 x^{2} \]

[[_2nd_order, _with_linear_symmetries]]

1.122

8532

\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = x^{2}+2 x +1 \]

[[_2nd_order, _with_linear_symmetries]]

1.141

8533

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

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

2.808

8534

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

[_quadrature]

0.470

8535

\[ {}9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

69.998

8536

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

[[_1st_order, _with_linear_symmetries], _rational]

3.358

8537

\[ {}x^{6} {y^{\prime }}^{2}-2 x y^{\prime }-4 y = 0 \]

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

1.784

8538

\[ {}5 {y^{\prime }}^{2}+6 x y^{\prime }-2 y = 0 \]

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.464

8539

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

[_quadrature]

0.589

8540

\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \]

[[_homogeneous, ‘class G‘]]

3.873

8541

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

[[_1st_order, _with_linear_symmetries]]

113.043

8542

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

2.441

8543

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

1.623

8544

\[ {}16 x {y^{\prime }}^{2}+8 y y^{\prime }+y^{6} = 0 \]

[[_homogeneous, ‘class G‘]]

2.660

8545

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

[_quadrature]

0.412

8546

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.177

8547

\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-1 = 0 \]

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

3.215

8548

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

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

0.663

8549

\[ {}x^{6} {y^{\prime }}^{2} = 16 y+8 x y^{\prime } \]

[[_homogeneous, ‘class G‘]]

1.771

8550

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

[_linear]

1.448

8551

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.847

8552

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.520

8553

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

[_quadrature]

0.484

8554

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

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

1.865

8555

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

[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

0.604

8556

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

[[_1st_order, _with_linear_symmetries]]

144.346

8557

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

[[_2nd_order, _missing_x]]

0.454

8558

\[ {}y^{\prime \prime }-9 y = 0 \]

[[_2nd_order, _missing_x]]

0.450

8559

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.392

8560

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.418

8561

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.418

8562

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

[[_2nd_order, _with_linear_symmetries]]

0.329

8563

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+10 x y^{\prime }+20 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.517

8564

\[ {}\left (x^{2}+4\right ) y^{\prime \prime }+2 x y^{\prime }-12 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.442

8565

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

[[_2nd_order, _with_linear_symmetries]]

0.403

8566

\[ {}y^{\prime \prime }+2 x y^{\prime }+5 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.390

8567

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.520

8568

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

[[_2nd_order, _with_linear_symmetries]]

0.499

8569

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

[[_Emden, _Fowler]]

0.330

8570

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

[_Gegenbauer]

0.437

8571

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

[[_2nd_order, _with_linear_symmetries]]

0.402

8572

\[ {}y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y = 0 \]

[[_3rd_order, _exact, _linear, _homogeneous]]

0.818

8573

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.437

8574

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.392

8575

\[ {}y^{\prime \prime }+3 x y^{\prime }+7 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.401

8576

\[ {}2 y^{\prime \prime }+9 x y^{\prime }-36 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.417

8577

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

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

0.419

8578

\[ {}\left (x^{2}+4\right ) y^{\prime \prime }+3 x y^{\prime }-8 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.422

8579

\[ {}\left (9 x^{2}+1\right ) y^{\prime \prime }-18 y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.405

8580

\[ {}\left (3 x^{2}+1\right ) y^{\prime \prime }+13 x y^{\prime }+7 y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.519

8581

\[ {}\left (2 x^{2}+1\right ) y^{\prime \prime }+11 x y^{\prime }+9 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.524

8582

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

[[_2nd_order, _with_linear_symmetries]]

0.429

8583

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

[[_2nd_order, _with_linear_symmetries]]

0.359

8584

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

[[_2nd_order, _with_linear_symmetries]]

0.365

8585

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.772

8586

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

[[_2nd_order, _with_linear_symmetries]]

0.724

8587

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

[[_2nd_order, _with_linear_symmetries]]

0.724

8588

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

[[_Emden, _Fowler]]

0.797

8589

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

[[_2nd_order, _with_linear_symmetries]]

0.842

8590

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

[[_2nd_order, _with_linear_symmetries]]

0.840

8591

\[ {}8 x^{2} y^{\prime \prime }+10 x y^{\prime }-\left (x +1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.869

8592

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

[[_2nd_order, _with_linear_symmetries]]

0.687

8593

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

[[_2nd_order, _with_linear_symmetries]]

0.802

8594

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

[[_2nd_order, _exact, _linear, _homogeneous]]

0.666

8595

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

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

0.961

8596

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

[[_2nd_order, _with_linear_symmetries]]

0.854

8597

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

[[_2nd_order, _with_linear_symmetries]]

0.823

8598

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

[[_2nd_order, _with_linear_symmetries]]

0.890

8599

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

[[_2nd_order, _with_linear_symmetries]]

0.868

8600

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

[[_2nd_order, _with_linear_symmetries]]

0.908