2.2.54 Problems 5301 to 5400

Table 2.109: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

5301

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

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

19.604

5302

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

[_rational]

100.476

5303

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

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

7.046

5304

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

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

15.757

5305

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

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

2.184

5306

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

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

2.289

5307

\[ {}x^{3} \left (1+5 x^{3} y^{7}\right ) y^{\prime }+\left (3 x^{5} y^{5}-1\right ) y^{3} = 0 \]

[_rational]

1.833

5308

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

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

4.855

5309

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

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

1.615

5310

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

[_Bernoulli]

4.944

5311

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

[_separable]

2.034

5312

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

[_separable]

2.270

5313

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

[_separable]

66.038

5314

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

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

1.569

5315

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

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

33.194

5316

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

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

24.385

5317

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

[_separable]

2.950

5318

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

[_separable]

2.902

5319

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

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

18.819

5320

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

[‘y=_G(x,y’)‘]

3.949

5321

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

[[_homogeneous, ‘class G‘], _dAlembert]

193.199

5322

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

42.096

5323

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

[[_1st_order, _with_linear_symmetries]]

3.125

5324

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

unknown

39.572

5325

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

[_exact]

37.986

5326

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

[NONE]

43.296

5327

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

[[_1st_order, _with_linear_symmetries]]

1.942

5328

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

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

7.309

5329

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

[_exact]

1.727

5330

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

[[_1st_order, _with_linear_symmetries]]

1.707

5331

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

[_exact]

36.172

5332

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

[_separable]

8.996

5333

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

[_quadrature]

0.433

5334

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

[_quadrature]

0.358

5335

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

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

0.705

5336

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

[[_homogeneous, ‘class G‘]]

1.507

5337

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

[[_homogeneous, ‘class G‘]]

1.570

5338

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

[[_homogeneous, ‘class G‘]]

1.592

5339

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

[[_homogeneous, ‘class G‘]]

1.710

5340

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

[_quadrature]

0.862

5341

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

[_quadrature]

0.685

5342

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

[_quadrature]

0.843

5343

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

[_quadrature]

0.957

5344

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

[_quadrature]

1.176

5345

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

[_separable]

2.446

5346

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

[_quadrature]

15.083

5347

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

[_quadrature]

70.980

5348

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

[_quadrature]

0.783

5349

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

[_quadrature]

0.868

5350

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

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

1.104

5351

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

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

1.092

5352

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

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

4.507

5353

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

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

2.095

5354

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

[_separable]

1.577

5355

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

[‘y=_G(x,y’)‘]

12.788

5356

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

[_quadrature]

0.187

5357

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

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

0.444

5358

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

[_quadrature]

11.882

5359

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

[_quadrature]

0.352

5360

\[ {}{y^{\prime }}^{2}-7 y^{\prime }+12 = 0 \]

[_quadrature]

0.332

5361

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

[_quadrature]

0.190

5362

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

[_quadrature]

0.201

5363

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

[_quadrature]

0.495

5364

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

[_quadrature]

0.268

5365

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.414

5366

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.405

5367

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.452

5368

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.500

5369

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.428

5370

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.424

5371

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.479

5372

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.462

5373

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

[_quadrature]

0.299

5374

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

[_quadrature]

0.406

5375

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.458

5376

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.447

5377

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.405

5378

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

[_quadrature]

0.210

5379

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.492

5380

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.502

5381

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.442

5382

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

[_quadrature]

0.214

5383

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.477

5384

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

[[_homogeneous, ‘class G‘]]

2.624

5385

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.664

5386

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

[_quadrature]

0.353

5387

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

[[_1st_order, _with_linear_symmetries]]

2.580

5388

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

[[_1st_order, _with_linear_symmetries]]

2.179

5389

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

[[_1st_order, _with_linear_symmetries]]

1.803

5390

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

[_quadrature]

0.388

5391

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

[_quadrature]

0.502

5392

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

[[_1st_order, _with_linear_symmetries]]

3.914

5393

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

[_quadrature]

0.518

5394

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

[_dAlembert]

41.984

5395

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

[_quadrature]

0.421

5396

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

[_quadrature]

0.626

5397

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

[_quadrature]

1.135

5398

\[ {}{y^{\prime }}^{2}-2 \left (1-3 y\right ) y^{\prime }-\left (4-9 y\right ) y = 0 \]

[_quadrature]

38.253

5399

\[ {}{y^{\prime }}^{2}+\left (a +6 y\right ) y^{\prime }+y \left (3 a +b +9 y\right ) = 0 \]

[_quadrature]

39.197

5400

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

[_dAlembert]

1.809