2.2.137 Problems 13601 to 13700

Table 2.275: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

13601

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

[[_2nd_order, _missing_x]]

2.996

13602

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

[[_2nd_order, _missing_x]]

3.290

13603

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

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

1.507

13604

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

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

1.598

13605

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

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

1.876

13606

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

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

2.500

13607

\[ {}\left [\begin {array}{c} x^{\prime }=x+3 y \\ y^{\prime }=3 x+y \end {array}\right ] \]

system_of_ODEs

0.437

13608

\[ {}\left [\begin {array}{c} x^{\prime }=3 x+2 y \\ y^{\prime }=x+2 y \end {array}\right ] \]

system_of_ODEs

0.449

13609

\[ {}\left [\begin {array}{c} x^{\prime }=3 x+4 y \\ y^{\prime }=3 x+2 y \end {array}\right ] \]

system_of_ODEs

0.461

13610

\[ {}\left [\begin {array}{c} x^{\prime }=2 x+5 y \\ y^{\prime }=x-2 y \end {array}\right ] \]

system_of_ODEs

0.454

13611

\[ {}\left [\begin {array}{c} x^{\prime }=2 x-4 y \\ y^{\prime }=2 x-2 y \end {array}\right ] \]

system_of_ODEs

0.527

13612

\[ {}\left [\begin {array}{c} x^{\prime }=x-2 y \\ y^{\prime }=4 x+5 y \end {array}\right ] \]

system_of_ODEs

0.566

13613

\[ {}\left [\begin {array}{c} x^{\prime }=x-y \\ y^{\prime }=x+5 y \end {array}\right ] \]

system_of_ODEs

0.603

13614

\[ {}\left [\begin {array}{c} x^{\prime }=x+7 y \\ y^{\prime }=3 x+5 y \end {array}\right ] \]

system_of_ODEs

0.465

13615

\[ {}\left [\begin {array}{c} x^{\prime }=x+y \\ y^{\prime }=3 x-y \end {array}\right ] \]

system_of_ODEs

0.470

13616

\[ {}\left [\begin {array}{c} x^{\prime }=a x+b y \\ y^{\prime }=c x+d y \end {array}\right ] \]

system_of_ODEs

0.797

13617

\[ {}\left [\begin {array}{c} x^{\prime }=4 x-4 y-x \left (x^{2}+y^{2}\right ) \\ y^{\prime }=4 x+4 y-y \left (x^{2}+y^{2}\right ) \end {array}\right ] \]

system_of_ODEs

0.037

13618

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

system_of_ODEs

0.050

13619

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

[[_2nd_order, _missing_x]]

2.490

13620

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

[[_2nd_order, _missing_x]]

1.785

13621

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

[[_2nd_order, _missing_x]]

3.659

13622

\[ {}x^{\prime \prime }+\left (5 x^{4}-6 x^{2}\right ) x^{\prime }+x^{3} = 0 \]

[[_2nd_order, _missing_x]]

3.215

13623

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

[[_2nd_order, _missing_x]]

2.081

13624

\[ {}\left [\begin {array}{c} x^{\prime }=x-x^{2} \\ y^{\prime }=2 y-y^{2} \end {array}\right ] \]

system_of_ODEs

0.031

13625

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

[_quadrature]

0.599

13626

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

[_quadrature]

0.526

13627

\[ {}u^{\prime } = 4 t \ln \left (t \right ) \]

[_quadrature]

0.489

13628

\[ {}z^{\prime } = x \,{\mathrm e}^{-2 x} \]

[_quadrature]

0.480

13629

\[ {}T^{\prime } = {\mathrm e}^{-t} \sin \left (2 t \right ) \]

[_quadrature]

0.628

13630

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

[_quadrature]

0.918

13631

\[ {}y^{\prime } = x -\frac {1}{3} x^{3} \]
i.c.

[_quadrature]

0.713

13632

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

[_quadrature]

0.870

13633

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

[_quadrature]

0.762

13634

\[ {}x^{\prime } {\mathrm e}^{3 t}+3 x \,{\mathrm e}^{3 t} = {\mathrm e}^{-t} \]
i.c.

[[_linear, ‘class A‘]]

1.868

13635

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

[_quadrature]

1.397

13636

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

[_quadrature]

2.309

13637

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

[_quadrature]

7.834

13638

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

[_quadrature]

4.442

13639

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

[_quadrature]

1.817

13640

\[ {}x^{\prime } = t^{3} \left (-x+1\right ) \]
i.c.

[_separable]

1.813

13641

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

[_separable]

3.932

13642

\[ {}x^{\prime } = t^{2} x \]

[_separable]

1.656

13643

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

[_quadrature]

1.997

13644

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

[_separable]

1.897

13645

\[ {}x^{\prime }+p x = q \]

[_quadrature]

0.831

13646

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

[_separable]

1.190

13647

\[ {}i^{\prime } = p \left (t \right ) i \]

[_separable]

0.833

13648

\[ {}x^{\prime } = \lambda x \]

[_quadrature]

0.802

13649

\[ {}m v^{\prime } = -m g +k v^{2} \]

[_quadrature]

2.343

13650

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

[_quadrature]

43.819

13651

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

[_quadrature]

167.463

13652

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

[_linear]

1.573

13653

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

[_separable]

2.009

13654

\[ {}z^{\prime } = z \tan \left (y \right )+\sin \left (y \right ) \]

[_linear]

1.880

13655

\[ {}y^{\prime }+{\mathrm e}^{-x} y = 1 \]
i.c.

[_linear]

1.343

13656

\[ {}x^{\prime }+x \tanh \left (t \right ) = 3 \]

[_linear]

1.459

13657

\[ {}y^{\prime }+2 y \cot \left (x \right ) = 5 \]
i.c.

[_linear]

1.870

13658

\[ {}x^{\prime }+5 x = t \]

[[_linear, ‘class A‘]]

1.294

13659

\[ {}x^{\prime }+\left (a +\frac {1}{t}\right ) x = b \]
i.c.

[_linear]

1.151

13660

\[ {}T^{\prime } = -k \left (T-\mu -a \cos \left (\omega \left (t -\phi \right )\right )\right ) \]

[[_linear, ‘class A‘]]

1.755

13661

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

[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]]

5.750

13662

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

[_linear]

1.865

13663

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

[_exact]

27.749

13664

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

[_exact]

2.715

13665

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

[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]]

3.303

13666

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

[_separable]

1.108

13667

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

[_separable]

1.612

13668

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

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

2.648

13669

\[ {}x^{\prime } = \frac {x^{2}+t \sqrt {x^{2}+t^{2}}}{t x} \]

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

10.556

13670

\[ {}x^{\prime } = k x-x^{2} \]

[_quadrature]

3.430

13671

\[ {}x^{\prime \prime }-3 x^{\prime }+2 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1.184

13672

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

[[_2nd_order, _missing_x]]

1.319

13673

\[ {}z^{\prime \prime }-4 z^{\prime }+13 z = 0 \]
i.c.

[[_2nd_order, _missing_x]]

3.263

13674

\[ {}y^{\prime \prime }+y^{\prime }-6 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

2.321

13675

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

[[_2nd_order, _missing_x]]

1.115

13676

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

[[_2nd_order, _missing_x]]

2.194

13677

\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

3.145

13678

\[ {}2 z^{\prime \prime }+7 z^{\prime }-4 z = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1.482

13679

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

[[_2nd_order, _missing_x]]

1.395

13680

\[ {}x^{\prime \prime }+6 x^{\prime }+10 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

2.392

13681

\[ {}4 x^{\prime \prime }-20 x^{\prime }+21 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1.512

13682

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

[[_2nd_order, _missing_x]]

1.470

13683

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

[[_2nd_order, _missing_x]]

2.633

13684

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

[[_2nd_order, _missing_x]]

1.355

13685

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

[[_2nd_order, _missing_x]]

1.601

13686

\[ {}x^{\prime \prime }-4 x = t^{2} \]

[[_2nd_order, _with_linear_symmetries]]

1.180

13687

\[ {}x^{\prime \prime }-4 x^{\prime } = t^{2} \]

[[_2nd_order, _missing_y]]

2.258

13688

\[ {}x^{\prime \prime }+x^{\prime }-2 x = 3 \,{\mathrm e}^{-t} \]

[[_2nd_order, _with_linear_symmetries]]

1.151

13689

\[ {}x^{\prime \prime }+x^{\prime }-2 x = {\mathrm e}^{t} \]

[[_2nd_order, _with_linear_symmetries]]

1.184

13690

\[ {}x^{\prime \prime }+2 x^{\prime }+x = {\mathrm e}^{-t} \]

[[_2nd_order, _with_linear_symmetries]]

1.125

13691

\[ {}x^{\prime \prime }+\omega ^{2} x = \sin \left (\alpha t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

2.364

13692

\[ {}x^{\prime \prime }+\omega ^{2} x = \sin \left (\omega t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

3.414

13693

\[ {}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t} \]

[[_2nd_order, _with_linear_symmetries]]

8.211

13694

\[ {}x^{\prime \prime }+2 x^{\prime }+10 x = {\mathrm e}^{-t} \cos \left (3 t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

20.671

13695

\[ {}x^{\prime \prime }+6 x^{\prime }+10 x = {\mathrm e}^{-2 t} \cos \left (t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

13.532

13696

\[ {}x^{\prime \prime }+4 x^{\prime }+4 x = {\mathrm e}^{2 t} \]

[[_2nd_order, _with_linear_symmetries]]

1.227

13697

\[ {}x^{\prime \prime }+x^{\prime }-2 x = 12 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{t} \]

[[_2nd_order, _linear, _nonhomogeneous]]

1.300

13698

\[ {}x^{\prime \prime }+4 x = 289 t \,{\mathrm e}^{t} \sin \left (2 t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

3.970

13699

\[ {}x^{\prime \prime }+\omega ^{2} x = \cos \left (\alpha t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

2.528

13700

\[ {}x^{\prime \prime }+\omega ^{2} x = \cos \left (\omega t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

3.486