2.4.10 second order ode can be made integrable

Table 2.393: second order ode can be made integrable

#

ODE

CAS classification

Solved?

11

\[ {}x^{\prime \prime } = 50 \]
i.c.

[[_2nd_order, _quadrature]]

12

\[ {}x^{\prime \prime } = -20 \]
i.c.

[[_2nd_order, _quadrature]]

149

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

[[_2nd_order, _missing_x]]

215

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

[[_2nd_order, _missing_x]]

216

\[ {}y^{\prime \prime }-9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

218

\[ {}y^{\prime \prime }+25 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

258

\[ {}y^{\prime \prime }-4 y = 12 \]
i.c.

[[_2nd_order, _missing_x]]

271

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

[[_2nd_order, _missing_x]]

311

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

[[_2nd_order, _missing_x]]

807

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

[[_2nd_order, _missing_x]]

808

\[ {}y^{\prime \prime }-9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

809

\[ {}y^{\prime \prime }+4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

810

\[ {}y^{\prime \prime }+25 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

839

\[ {}y^{\prime \prime }-4 y = 12 \]
i.c.

[[_2nd_order, _missing_x]]

842

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

[[_2nd_order, _missing_x]]

845

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

[[_2nd_order, _missing_x]]

859

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

[[_2nd_order, _missing_x]]

1254

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

[[_2nd_order, _missing_x]]

1264

\[ {}4 y^{\prime \prime }-y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1265

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

[[_2nd_order, _missing_x]]

1268

\[ {}4 y^{\prime \prime }-y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1278

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

[[_2nd_order, _missing_x]]

1283

\[ {}y^{\prime \prime }+4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

1286

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

[[_2nd_order, _missing_x]]

1355

\[ {}u^{\prime \prime }+2 u = 0 \]

[[_2nd_order, _missing_x]]

2364

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

[[_2nd_order, _missing_x]]

2545

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

[[_2nd_order, _missing_x]]

2564

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

[[_2nd_order, _missing_x]]

2992

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

[[_2nd_order, _missing_x]]

3022

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

[[_2nd_order, _quadrature]]

3178

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

[[_2nd_order, _missing_x]]

3179

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

[[_2nd_order, _missing_x]]

3199

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

[[_2nd_order, _missing_x]]

3206

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

3215

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

[[_2nd_order, _missing_x]]

3491

\[ {}y^{\prime \prime }-25 y = 0 \]

[[_2nd_order, _missing_x]]

3492

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

[[_2nd_order, _missing_x]]

3497

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

[[_2nd_order, _missing_x]]

3631

\[ {}y^{\prime \prime }-36 y = 0 \]

[[_2nd_order, _missing_x]]

5478

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

[[_2nd_order, _missing_x]]

5505

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

[[_2nd_order, _quadrature]]

5562

\[ {}y^{\prime \prime } = \frac {3 k y^{2}}{2} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

5563

\[ {}y^{\prime \prime } = 2 k y^{3} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

5573

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

5700

\[ {}y^{\prime \prime }+16 y = 0 \]

[[_2nd_order, _missing_x]]

5803

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

[[_2nd_order, _missing_x]]

5805

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

[[_2nd_order, _missing_x]]

5949

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

[[_2nd_order, _missing_x]]

6136

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

[[_2nd_order, _missing_x]]

6267

\[ {}y^{\prime \prime }+25 y = 0 \]

[[_2nd_order, _missing_x]]

6821

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

[[_2nd_order, _missing_x]]

6822

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

[[_2nd_order, _missing_x]]

6823

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

[[_2nd_order, _missing_x]]

6848

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

[[_2nd_order, _missing_x]]

6849

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

[[_2nd_order, _missing_x]]

6850

\[ {}y^{\prime \prime }+16 y = 0 \]

[[_2nd_order, _missing_x]]

6851

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

[[_2nd_order, _quadrature]]

6858

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

[[_2nd_order, _missing_x]]

6859

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

[[_2nd_order, _missing_x]]

6860

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

[[_2nd_order, _missing_x]]

6864

\[ {}y^{\prime \prime }+10 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

6886

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

[[_2nd_order, _missing_x]]

6887

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

[[_2nd_order, _missing_x]]

6998

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

[[_2nd_order, _missing_x]]

7002

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

7003

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

7004

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

7013

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

[[_2nd_order, _missing_x]]

7014

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

[[_2nd_order, _missing_x]]

7143

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

[[_2nd_order, _missing_x]]

7175

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

[[_2nd_order, _missing_x]]

7181

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

[[_2nd_order, _missing_x]]

7185

\[ {}y^{\prime \prime } = 4 y \]

[[_2nd_order, _missing_x]]

7283

\[ {}y^{\prime \prime }+9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

7306

\[ {}y^{\prime \prime } = -3 y \]
i.c.

[[_2nd_order, _missing_x]]

7307

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

7455

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

[[_2nd_order, _missing_x]]

7457

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

[[_2nd_order, _missing_x]]

7543

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

[[_2nd_order, _missing_x]]

7736

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

[[_2nd_order, _missing_x]]

7742

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

7743

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

7744

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

8002

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

[[_2nd_order, _quadrature]]

8003

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

[[_2nd_order, _quadrature]]

8005

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

[[_2nd_order, _quadrature]]

8198

\[ {}y^{\prime \prime } = A y^{{2}/{3}} \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

8219

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

8294

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

[[_2nd_order, _quadrature]]

8297

\[ {}a y^{\prime \prime } = 0 \]

[[_2nd_order, _quadrature]]

8300

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

[[_2nd_order, _quadrature]]

8331

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

[[_2nd_order, _missing_x]]

8350

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

8351

\[ {}y^{\prime \prime } y^{\prime }+y^{n} = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10233

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

[[_2nd_order, _quadrature]]

10234

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

[[_2nd_order, _missing_x]]

10238

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

[[_2nd_order, _missing_x]]

10241

\[ {}y^{\prime \prime }+l y = 0 \]

[[_2nd_order, _missing_x]]

10812

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10813

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10815

\[ {}y^{\prime \prime }-6 y^{2}+4 y = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10818

\[ {}y^{\prime \prime }-a y^{3} = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10821

\[ {}y^{\prime \prime }+d +b y^{2}+c y+a y^{3} = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10823

\[ {}y^{\prime \prime }+6 a^{10} y^{11}-y = 0 \]

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10825

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10828

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

11723

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

[[_2nd_order, _missing_x]]

12217

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

12345

\[ {}x^{\prime \prime }-12 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

12489

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

[[_2nd_order, _missing_x]]

12639

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

[[_2nd_order, _missing_x]]

12640

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

[[_2nd_order, _missing_x]]

12917

\[ {}\theta ^{\prime \prime }+4 \theta = 0 \]
i.c.

[[_2nd_order, _missing_x]]

12924

\[ {}y^{\prime \prime }-4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

12926

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

[[_2nd_order, _missing_x]]

13311

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

[[_2nd_order, _missing_x]]

13312

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

[[_2nd_order, _missing_x]]

13388

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

13390

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

[[_2nd_order, _missing_x]]

13396

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

13399

\[ {}y^{\prime \prime } = 9 y \]

[[_2nd_order, _missing_x]]

13400

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

[[_2nd_order, _missing_x]]

13401

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

[[_2nd_order, _missing_x]]

13466

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

[[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

13467

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

[[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

13470

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

13499

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

[[_2nd_order, _missing_x]]

13648

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

[[_2nd_order, _missing_x]]

13649

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

[[_2nd_order, _missing_x]]

13652

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

[[_2nd_order, _missing_x]]

13655

\[ {}y^{\prime \prime }-4 y = 31 \]
i.c.

[[_2nd_order, _missing_x]]

13668

\[ {}y^{\prime \prime }+\alpha y = 0 \]

[[_2nd_order, _missing_x]]

14058

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

[[_2nd_order, _missing_x]]

14378

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

14397

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

14458

\[ {}y^{\prime \prime }+4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14459

\[ {}y^{\prime \prime }-4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14471

\[ {}y^{\prime \prime }-4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14479

\[ {}y^{\prime \prime }-25 y = 0 \]

[[_2nd_order, _missing_x]]

14481

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

[[_2nd_order, _missing_x]]

14486

\[ {}y^{\prime \prime }-9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14487

\[ {}y^{\prime \prime }-9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14488

\[ {}y^{\prime \prime }-9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14501

\[ {}y^{\prime \prime }+25 y = 0 \]

[[_2nd_order, _missing_x]]

14506

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

[[_2nd_order, _missing_x]]

14508

\[ {}y^{\prime \prime }+16 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14509

\[ {}y^{\prime \prime }+16 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

14577

\[ {}y^{\prime \prime }-9 y = 36 \]
i.c.

[[_2nd_order, _missing_x]]

14697

\[ {}y^{\prime \prime }+36 y = 0 \]

[[_2nd_order, _missing_x]]

14700

\[ {}y^{\prime \prime }-36 y = 0 \]

[[_2nd_order, _missing_x]]

14706

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

[[_2nd_order, _missing_x]]

15339

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

[[_2nd_order, _missing_x]]

15342

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

[[_2nd_order, _missing_x]]

15344

\[ {}y^{\prime \prime }+9 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15349

\[ {}y^{\prime \prime }+16 y = 0 \]

[[_2nd_order, _missing_x]]

15368

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

[[_2nd_order, _quadrature]]

15375

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

[[_2nd_order, _missing_x]]

15376

\[ {}y^{\prime \prime }+16 y = 0 \]

[[_2nd_order, _missing_x]]

15377

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

[[_2nd_order, _missing_x]]

15378

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

[[_2nd_order, _missing_x]]

15389

\[ {}y^{\prime \prime }+36 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15390

\[ {}y^{\prime \prime }+100 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15408

\[ {}y^{\prime \prime }-16 y = 0 \]

[[_2nd_order, _missing_x]]

15447

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

[[_2nd_order, _missing_x]]

15470

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

[[_2nd_order, _missing_x]]

15753

\[ {}y^{\prime \prime }+16 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15786

\[ {}9 x^{\prime \prime }+4 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15787

\[ {}x^{\prime \prime }+64 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15788

\[ {}x^{\prime \prime }+100 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15789

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

[[_2nd_order, _missing_x]]

15790

\[ {}x^{\prime \prime }+4 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15791

\[ {}x^{\prime \prime }+16 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15792

\[ {}x^{\prime \prime }+256 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15793

\[ {}x^{\prime \prime }+9 x = 0 \]
i.c.

[[_2nd_order, _missing_x]]

15794

\[ {}10 x^{\prime \prime }+\frac {x}{10} = 0 \]
i.c.

[[_2nd_order, _missing_x]]

16073

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

[[_2nd_order, _missing_x]]

16111

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

16114

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

[[_2nd_order, _missing_x]]

16151

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

[[_2nd_order, _missing_x]]

16174

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

[[_2nd_order, _missing_x]]

16271

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

[[_2nd_order, _missing_x]]

16343

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

[[_2nd_order, _missing_x]]

16344

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

[[_2nd_order, _missing_x]]

16348

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

[[_2nd_order, _missing_x]]

16354

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

[[_2nd_order, _missing_x]]