2.2.43 Problems 4201 to 4300

Table 2.87: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

4201

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

[_linear]

1.832

4202

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

[_linear]

2.214

4203

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

[_linear]

2.131

4204

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

[_linear]

2.726

4205

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

[_linear]

2.022

4206

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

[_linear]

3.125

4207

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

[_linear]

1.995

4208

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

[_linear]

3.104

4209

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

[_linear]

1.806

4210

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

[_linear]

1.964

4211

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

[_linear]

2.197

4212

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

[_linear]

3.120

4213

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

[_separable]

2.297

4214

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

[_separable]

2.140

4215

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

[_separable]

1.814

4216

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

[_separable]

2.352

4217

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

[_separable]

1.381

4218

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

[_quadrature]

1.470

4219

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

[_separable]

1.597

4220

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

[_separable]

2.006

4221

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

[_separable]

1.698

4222

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

[_separable]

1.726

4223

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

[_separable]

3.608

4224

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

[_separable]

2.799

4225

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

[_separable]

2.450

4226

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

[_separable]

2.252

4227

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

[_separable]

1.954

4228

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

[_separable]

2.125

4229

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

[_quadrature]

0.991

4230

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

[_separable]

3.590

4231

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

[_separable]

3.017

4232

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

[_separable]

2.355

4233

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

[_separable]

1.755

4234

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

[_separable]

2.088

4235

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

[_separable]

2.350

4236

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

[_separable]

1.700

4237

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

[_separable]

3.712

4238

\[ {}x y y^{\prime } = \sqrt {y^{2}-9} \]
i.c.

[_separable]

3.695

4239

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

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

3.148

4240

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

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

62.476

4241

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

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

5.033

4242

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

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

2.891

4243

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

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

11.056

4244

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

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

5.742

4245

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

[[_homogeneous, ‘class D‘]]

1.949

4246

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

[[_homogeneous, ‘class C‘], _Riccati]

1.540

4247

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

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

6.210

4248

\[ {}y^{\prime } = \frac {x +y+4}{x -y-6} \]

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

36.808

4249

\[ {}y^{\prime } = \frac {x +y+4}{x +y-6} \]

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

1.814

4250

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

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

1.673

4251

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

[‘y=_G(x,y’)‘]

40.543

4252

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

[_exact, _rational]

1.289

4253

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

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

1.609

4254

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

[_separable]

2.252

4255

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

[_separable]

3.536

4256

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

[_exact]

29.115

4257

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

[_separable]

1.736

4258

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

[_separable]

1.777

4259

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

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

39.178

4260

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

[_exact, _rational, _Riccati]

1.671

4261

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

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

34.222

4262

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

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

1.288

4263

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

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

11.630

4264

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

[[_1st_order, _with_linear_symmetries], _rational]

1.788

4265

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

[_separable]

2.314

4266

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

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

1.938

4267

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

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

39.176

4268

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

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

1.502

4269

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

[_linear]

1.819

4270

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

[_linear]

1.800

4271

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

[_linear]

1.805

4272

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

[[_linear, ‘class A‘]]

2.895

4273

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

[_linear]

1.908

4274

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

[_linear]

1.733

4275

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

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

1.651

4276

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

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

36.956

4277

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

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

9.378

4278

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

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

2.683

4279

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

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

2.994

4280

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

[_linear]

1.417

4281

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

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

40.695

4282

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

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

2.067

4283

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

[_linear]

1.661

4284

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

[_linear]

1.501

4285

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

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

1.934

4286

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

[[_1st_order, _with_linear_symmetries], _exact]

4.427

4287

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

[_exact]

35.586

4288

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

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

2.138

4289

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

[_linear]

1.695

4290

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

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

77.912

4291

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

[_linear]

1.791

4292

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

[_exact]

39.335

4293

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

[_exact]

1.931

4294

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

[‘y=_G(x,y’)‘]

1.870

4295

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

[_separable]

2.279

4296

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

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

1.181

4297

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

[_linear]

1.329

4298

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

[_exact]

40.373

4299

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

[_exact]

37.272

4300

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

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

553.295