2.2.133 Problems 13201 to 13300

Table 2.267: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

13201

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

[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

9.736

13202

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

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

1.665

13203

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

[_exact, _rational]

7.228

13204

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

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

46.792

13205

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

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

3.596

13206

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

[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

3.513

13207

\[ {}\frac {1+8 x y^{{2}/{3}}}{x^{{2}/{3}} y^{{1}/{3}}}+\frac {\left (2 x^{{4}/{3}} y^{{2}/{3}}-x^{{1}/{3}}\right ) y^{\prime }}{y^{{4}/{3}}} = 0 \]
i.c.

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

3.853

13208

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

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

2.750

13209

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

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

3.021

13210

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

[[_1st_order, _with_linear_symmetries], _rational]

2.186

13211

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

[_separable]

1.721

13212

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

[_separable]

1.998

13213

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

[_separable]

2.776

13214

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

[_separable]

2.029

13215

\[ {}\tan \left (\theta \right )+2 r \theta ^{\prime } = 0 \]

[_separable]

3.076

13216

\[ {}\left ({\mathrm e}^{v}+1\right ) \cos \left (u \right )+{\mathrm e}^{v} \left (1+\sin \left (u \right )\right ) v^{\prime } = 0 \]

[_separable]

2.688

13217

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

[_separable]

3.069

13218

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

[_linear]

1.592

13219

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

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

118.247

13220

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

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

35.987

13221

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

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

3.869

13222

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

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

5.008

13223

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

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

6.658

13224

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

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

15.913

13225

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

[_separable]

2.889

13226

\[ {}8 \cos \left (y\right )^{2}+\csc \left (x \right )^{2} y^{\prime } = 0 \]
i.c.

[_separable]

2.975

13227

\[ {}\left (3 x +8\right ) \left (y^{2}+4\right )-4 y \left (x^{2}+5 x +6\right ) y^{\prime } = 0 \]
i.c.

[_separable]

3.602

13228

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

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

8.215

13229

\[ {}2 x -5 y+\left (4 x -y\right ) y^{\prime } = 0 \]
i.c.

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

4.524

13230

\[ {}3 x^{2}+9 x y+5 y^{2}-\left (6 x^{2}+4 x y\right ) y^{\prime } = 0 \]
i.c.

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

73.135

13231

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

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

4.528

13232

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

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

4.781

13233

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

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

107.096

13234

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

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

75.517

13235

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

[_linear]

1.635

13236

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

[_linear]

1.530

13237

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

[[_linear, ‘class A‘]]

1.889

13238

\[ {}y^{\prime }+4 x y = 8 x \]

[_separable]

1.519

13239

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

[_separable]

1.535

13240

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

[_separable]

1.904

13241

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

[_linear]

1.555

13242

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

[_linear]

1.652

13243

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

[_linear]

1.259

13244

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

[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

1.294

13245

\[ {}r^{\prime }+r \tan \left (t \right ) = \cos \left (t \right ) \]

[_linear]

1.833

13246

\[ {}\cos \left (t \right ) r^{\prime }+r \sin \left (t \right )-\cos \left (t \right )^{4} = 0 \]

[_linear]

3.352

13247

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

[_linear]

3.227

13248

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

[_linear]

4.539

13249

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

[_separable]

2.566

13250

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

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

4.004

13251

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

[_separable]

3.271

13252

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

[_separable]

2.428

13253

\[ {}x y^{\prime }-2 y = 2 x^{4} \]
i.c.

[_linear]

2.020

13254

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

[_separable]

1.361

13255

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

[_linear]

2.021

13256

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

[_separable]

1.980

13257

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

[_linear]

2.227

13258

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

[[_linear, ‘class A‘]]

1.895

13259

\[ {}y^{\prime }+\frac {y}{2 x} = \frac {x}{y^{3}} \]
i.c.

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

3.730

13260

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

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

17.087

13261

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} 2 & 0\le x <1 \\ 0 & 1\le x \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.788

13262

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} 5 & 0\le x <10 \\ 1 & 10\le x \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

1.229

13263

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} {\mathrm e}^{-x} & 0\le x <2 \\ {\mathrm e}^{-2} & 2\le x \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.911

13264

\[ {}\left (x +2\right ) y^{\prime }+y = \left \{\begin {array}{cc} 2 x & 0\le x <2 \\ 4 & 2\le x \end {array}\right . \]
i.c.

[_linear]

0.739

13265

\[ {}a y^{\prime }+b y = k \,{\mathrm e}^{-\lambda x} \]

[[_linear, ‘class A‘]]

1.233

13266

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

[[_linear, ‘class A‘]]

2.310

13267

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

[‘y=_G(x,y’)‘]

1.837

13268

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

[_separable]

1.877

13269

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

[_Riccati]

2.563

13270

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

[_Riccati]

1.347

13271

\[ {}y^{\prime } = -8 x y^{2}+4 x \left (4 x +1\right ) y-8 x^{3}-4 x^{2}+1 \]

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

2.684

13272

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

[_separable]

1.712

13273

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

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

2.217

13274

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

[_separable]

1.741

13275

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

[_linear]

1.258

13276

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

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

10.650

13277

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

[_separable]

2.260

13278

\[ {}8 x^{3} y-12 x^{3}+\left (x^{4}+1\right ) y^{\prime } = 0 \]

[_separable]

1.918

13279

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

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

4.889

13280

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

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

4.204

13281

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

[_linear]

2.123

13282

\[ {}y^{\prime } = \frac {2 x -7 y}{3 y-8 x} \]

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

8.796

13283

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

[_separable]

5.668

13284

\[ {}\left (x^{3}+1\right ) y^{\prime }+6 x^{2} y = 6 x^{2} \]

[_separable]

1.919

13285

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

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

6.181

13286

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

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

8.273

13287

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

[_separable]

3.576

13288

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

[_exact, _Bernoulli]

3.750

13289

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

[_exact, _rational]

38.005

13290

\[ {}4 x y y^{\prime } = 1+y^{2} \]
i.c.

[_separable]

4.737

13291

\[ {}y^{\prime } = \frac {2 x +7 y}{2 x -2 y} \]
i.c.

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

5.854

13292

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

[_separable]

2.369

13293

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} 1 & 0\le x <2 \\ 0 & 0<x \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.811

13294

\[ {}\left (x +2\right ) y^{\prime }+y = \left \{\begin {array}{cc} 2 x & 0\le x \le 2 \\ 4 & 2<x \end {array}\right . \]
i.c.

[_linear]

0.783

13295

\[ {}x^{2} y^{\prime }+x y = \frac {y^{3}}{x} \]
i.c.

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

144.209

13296

\[ {}5 x y+4 y^{2}+1+\left (2 x y+x^{2}\right ) y^{\prime } = 0 \]

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

1.703

13297

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

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

2.561

13298

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

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

1.444

13299

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

[_rational]

2.394

13300

\[ {}4 x y^{2}+6 y+\left (5 x^{2} y+8 x \right ) y^{\prime } = 0 \]

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

2.453