2.2.195 Problems 19401 to 19500

Table 2.391: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

19401

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

[[_2nd_order, _with_linear_symmetries]]

1.523

19402

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

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

0.128

19403

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

[[_2nd_order, _with_linear_symmetries]]

1.445

19404

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

[[_2nd_order, _with_linear_symmetries]]

4.235

19405

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

[[_2nd_order, _with_linear_symmetries]]

2.416

19406

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

[[_2nd_order, _with_linear_symmetries]]

0.698

19407

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

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

3.023

19408

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

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

1.096

19409

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

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

1.532

19410

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

[[_2nd_order, _linear, _nonhomogeneous]]

1.464

19411

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = 8 x^{3} \]

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

1.596

19412

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

[[_2nd_order, _linear, _nonhomogeneous]]

5.456

19413

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

[[_2nd_order, _with_linear_symmetries]]

0.922

19414

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

[[_2nd_order, _linear, _nonhomogeneous]]

3.803

19415

\[ {}\left (x^{3}-2 x^{2}\right ) y^{\prime \prime }+2 x^{2} y^{\prime }-12 \left (-2+x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.705

19416

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.820

19417

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

[[_2nd_order, _exact, _linear, _homogeneous]]

1.169

19418

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

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

0.082

19419

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

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

1.829

19420

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

[[_2nd_order, _with_linear_symmetries]]

0.243

19421

\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5} \]

[[_2nd_order, _linear, _nonhomogeneous]]

3.203

19422

\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-\left (4 x^{2}-3 x -5\right ) y^{\prime }+\left (4 x^{2}-6 x -5\right ) y = {\mathrm e}^{2 x} \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

1.520

19423

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

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

1.520

19424

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

[[_2nd_order, _linear, _nonhomogeneous]]

2.645

19425

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

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

1.566

19426

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

[[_2nd_order, _with_linear_symmetries]]

1.410

19427

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

[[_2nd_order, _with_linear_symmetries]]

2.230

19428

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

[[_2nd_order, _with_linear_symmetries]]

0.980

19429

\[ {}\left [\begin {array}{c} t x^{\prime }+y=0 \\ t y^{\prime }+x=0 \end {array}\right ] \]

system_of_ODEs

0.031

19430

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

[_separable]

1.623

19431

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

[_separable]

2.041

19432

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

[[_1st_order, _with_linear_symmetries], _rational]

1.524

19433

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

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

25.421

19434

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

[_separable]

38.124

19435

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

[_separable]

3.562

19436

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

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

159.626

19437

\[ {}y^{\prime } = \frac {6 x -2 y-7}{2 x +3 y-6} \]

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

3.570

19438

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

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

1.866

19439

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

[_linear]

2.002

19440

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

[_linear]

1.674

19441

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

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

25.731

19442

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

[_Bernoulli]

1.799

19443

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

[‘y=_G(x,y’)‘]

5.374

19444

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

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

1.400

19445

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

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

14.799

19446

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

[‘y=_G(x,y’)‘]

48.033

19447

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

[_rational]

1.544

19448

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

[_rational]

1.632

19449

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

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

46.013

19450

\[ {}2 y^{\prime \prime }+9 y^{\prime }-18 y = 0 \]

[[_2nd_order, _missing_x]]

0.842

19451

\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-9 y^{\prime \prime }-11 y^{\prime }-4 y = 0 \]

[[_high_order, _missing_x]]

0.056

19452

\[ {}y^{\prime \prime \prime }-8 y = 0 \]

[[_3rd_order, _missing_x]]

0.056

19453

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

[[_3rd_order, _missing_y]]

0.090

19454

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

[[_2nd_order, _linear, _nonhomogeneous]]

2.889

19455

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

[[_3rd_order, _linear, _nonhomogeneous]]

0.132

19456

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

[[_2nd_order, _linear, _nonhomogeneous]]

2.277

19457

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

[[_3rd_order, _missing_y]]

0.438

19458

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

[[_3rd_order, _linear, _nonhomogeneous]]

0.158

19459

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

[[_2nd_order, _linear, _nonhomogeneous]]

1.455

19460

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

[[_3rd_order, _missing_y]]

0.093

19461

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

[[_2nd_order, _linear, _nonhomogeneous]]

1.209

19462

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

[[_2nd_order, _linear, _nonhomogeneous]]

2.675

19463

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

[[_2nd_order, _linear, _nonhomogeneous]]

3.204

19464

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

[[_2nd_order, _linear, _nonhomogeneous]]

1.461

19465

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

[_quadrature]

1.044

19466

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

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

9.335

19467

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

[_quadrature]

0.963

19468

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

[_quadrature]

1.864

19469

\[ {}y-\frac {1}{\sqrt {1+{y^{\prime }}^{2}}} = b \]

[_quadrature]

1.572

19470

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

[_dAlembert]

64.668

19471

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

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

0.804

19472

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

[_quadrature]

0.563

19473

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

[_quadrature]

1.862

19474

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

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

0.819

19475

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.468

19476

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

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

1.584

19477

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

[[_1st_order, _with_linear_symmetries]]

107.864

19478

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

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

1.676

19479

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

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

1.101

19480

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

[[_1st_order, _with_linear_symmetries]]

2.605

19481

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

[_separable]

3.964

19482

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

[_separable]

1.552

19483

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

[_rational]

87.430

19484

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

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

3.422

19485

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

[‘y=_G(x,y’)‘]

22.346

19486

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

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

147.380

19487

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

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

1.012

19488

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

[_quadrature]

0.553

19489

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

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

6.063

19490

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

[_quadrature]

0.367

19491

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

[_quadrature]

0.531

19492

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

[[_1st_order, _with_linear_symmetries]]

10.423

19493

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.422

19494

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

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

3.028

19495

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

1.411

19496

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

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

1.528

19497

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

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

72.468

19498

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

[‘y=_G(x,y’)‘]

67.620

19499

\[ {}x^{4} y^{\prime \prime \prime }+2 x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y = 1 \]

[[_3rd_order, _with_linear_symmetries]]

0.273

19500

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

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

0.920