2.2.123 Problems 12201 to 12300

Table 2.247: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

12201

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

[_Riccati]

187.858

12202

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

[_Riccati]

2.850

12203

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

[_Riccati]

4.590

12204

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

[_Riccati]

7.076

12205

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

[_Riccati]

2.006

12206

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

[_Riccati]

5.192

12207

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

[_Riccati]

3.326

12208

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

[_Riccati]

13.345

12209

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

[_Riccati]

20.863

12210

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

[_Riccati]

6.424

12211

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

[_Riccati]

4.604

12212

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

[_Riccati]

347.549

12213

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

[_Riccati]

306.688

12214

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

[_Riccati]

48.924

12215

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

[_Riccati]

3.858

12216

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

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

5.062

12217

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

[_Riccati]

3.492

12218

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

[_Riccati]

3.199

12219

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

[_Riccati]

11.829

12220

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

[_Riccati]

54.785

12221

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

[_Riccati]

55.134

12222

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

[_Riccati]

415.276

12223

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

[_Riccati]

416.554

12224

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

[_Riccati]

0.974

12225

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

[_Riccati]

1.633

12226

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

[_Riccati]

1.659

12227

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

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

1.624

12228

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

[_Riccati]

103.430

12229

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

[_Riccati]

104.859

12230

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

[_Riccati]

2.062

12231

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

[_Riccati]

2.030

12232

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

[_Riccati]

0.959

12233

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

[_Riccati]

1.610

12234

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

[_Riccati]

1.773

12235

\[ {}y^{\prime } = y^{2}+\frac {f \left (\frac {a x +b}{c x +d}\right )}{\left (c x +d \right )^{4}} \]

[_Riccati]

4.892

12236

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

[_Riccati]

2.466

12237

\[ {}x^{2} y^{\prime } = x^{4} y^{2}+x^{2 n} f \left (a \,x^{n}+b \right )-\frac {n^{2}}{4}+\frac {1}{4} \]

[_Riccati]

6.967

12238

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

[_Riccati]

2.427

12239

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

[_Riccati]

3.916

12240

\[ {}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {a \,{\mathrm e}^{\lambda x}+b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}} \]

[_Riccati]

50.589

12241

\[ {}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\coth \left (\lambda x \right )\right )}{\sinh \left (\lambda x \right )^{4}} \]

[_Riccati]

59.915

12242

\[ {}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\tanh \left (\lambda x \right )\right )}{\cosh \left (\lambda x \right )^{4}} \]

[_Riccati]

26.648

12243

\[ {}x^{2} y^{\prime } = y^{2} x^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4} \]

[_Riccati]

3.332

12244

\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\cot \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{4}} \]

[_Riccati]

113.087

12245

\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\tan \left (\lambda x \right )\right )}{\cos \left (\lambda x \right )^{4}} \]

[_Riccati]

31.570

12246

\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\frac {\sin \left (\lambda x +a \right )}{\sin \left (\lambda x +b \right )}\right )}{\sin \left (\lambda x +b \right )^{4}} \]

[_Riccati]

464.509

12247

\[ {}y y^{\prime }-y = A \]

[_quadrature]

1.036

12248

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

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

10.385

12249

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \]

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

7.437

12250

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

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

7.925

12251

\[ {}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \]

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

1.576

12252

\[ {}y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1} \]

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

2.739

12253

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

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

1.353

12254

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

[[_Abel, ‘2nd type‘, ‘class A‘]]

1.655

12255

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

[[_Abel, ‘2nd type‘, ‘class A‘]]

1.608

12256

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

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

2.382

12257

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \]

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

8.740

12258

\[ {}y y^{\prime }-y = \frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (m +3\right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (m +3\right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}} \]

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

43.362

12259

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

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

2.118

12260

\[ {}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \]

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

1.469

12261

\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {5 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}} \]

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

4.875

12262

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

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

0.768

12263

\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \]

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

11.903

12264

\[ {}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

3.049

12265

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

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

1.023

12266

\[ {}y y^{\prime }-y = -\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25} \]

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

31.229

12267

\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {a^{2}+x^{2}}}{8}-\frac {a^{2}}{16 \sqrt {a^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

6.065

12268

\[ {}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}} \]

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

7.965

12269

\[ {}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{{5}/{3}}} \]

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

42.415

12270

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49} \]

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

8.556

12271

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98} \]

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

8.088

12272

\[ {}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}} \]

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

5.993

12273

\[ {}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{{7}/{5}}} \]

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

51.063

12274

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {6 A \left (-3 \sqrt {x}+23 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \]

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

7.896

12275

\[ {}y y^{\prime }-y = -\frac {30 x}{121}+\frac {3 A \left (21 \sqrt {x}+35 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{242} \]

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

8.796

12276

\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{{5}/{3}}} \]

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

40.982

12277

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {4 A \left (-10 \sqrt {x}+27 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{49} \]

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

7.130

12278

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

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

2.149

12279

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

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

0.784

12280

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

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

17.682

12281

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

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

18.391

12282

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

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

13.435

12283

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

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

2.505

12284

\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {6 A \left (\sqrt {x}+8 A +\frac {5 A^{2}}{\sqrt {x}}\right )}{49} \]

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

8.714

12285

\[ {}y y^{\prime }-y = -\frac {6 x}{25}+\frac {6 A \left (2 \sqrt {x}+7 A +\frac {4 A^{2}}{\sqrt {x}}\right )}{25} \]

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

7.941

12286

\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{{1}/{3}}}-\frac {12 A^{2}}{x^{{5}/{3}}} \]

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

6.027

12287

\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {b^{2}+x^{2}}}{8}+\frac {3 b^{2}}{16 \sqrt {b^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

6.071

12288

\[ {}y y^{\prime }-y = \frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

7.381

12289

\[ {}y y^{\prime }-y = -\frac {3 x}{32}-\frac {3 \sqrt {a^{2}+x^{2}}}{32}+\frac {15 a^{2}}{64 \sqrt {a^{2}+x^{2}}} \]

[[_Abel, ‘2nd type‘, ‘class B‘]]

6.360

12290

\[ {}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A} \]

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

1.035

12291

\[ {}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2} \]

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

0.931

12292

\[ {}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2} \]

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

0.924

12293

\[ {}y y^{\prime }-y = 12 x +\frac {A}{x^{{5}/{2}}} \]

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

6.492

12294

\[ {}y y^{\prime }-y = \frac {63 x}{4}+\frac {A}{x^{{5}/{3}}} \]

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

47.651

12295

\[ {}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \]

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

8.313

12296

\[ {}y y^{\prime }-y = 2 x +2 A \left (-10 \sqrt {x}+19 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \]

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

7.401

12297

\[ {}y y^{\prime }-y = -\frac {28 x}{121}+\frac {2 A \left (5 \sqrt {x}+106 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{121} \]

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

9.160

12298

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49} \]

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

8.234

12299

\[ {}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x} \]

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

5.826

12300

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

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

0.983