2.8.1 First order Taylor and ordinary point series

Table 2.1181: First order Taylor and ordinary point [129]

#

ODE

solved

Maple

Mma

Sympy

time(sec)

400

\begin{align*} y^{\prime }&=y \\ \end{align*}
Series expansion around \(x=0\).

0.210

401

\begin{align*} y^{\prime }&=4 y \\ \end{align*}
Series expansion around \(x=0\).

0.237

402

\begin{align*} 2 y^{\prime }+3 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.246

403

\begin{align*} y^{\prime }+2 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.260

404

\begin{align*} y^{\prime }&=y x \\ \end{align*}
Series expansion around \(x=0\).

0.238

405

\begin{align*} \left (x -2\right ) y^{\prime }+y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.263

406

\begin{align*} \left (2 x -1\right ) y^{\prime }+2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.276

407

\begin{align*} 2 \left (x +1\right ) y^{\prime }+y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.264

408

\begin{align*} \left (-1+x \right ) y^{\prime }+2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.264

409

\begin{align*} 2 \left (-1+x \right ) y^{\prime }&=3 y \\ \end{align*}
Series expansion around \(x=0\).

0.268

424

\begin{align*} y^{\prime }&=1+y^{2} \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

1.139

1042

\begin{align*} y^{\prime }&=y \\ \end{align*}
Series expansion around \(x=0\).

0.225

1043

\begin{align*} y^{\prime }&=4 y \\ \end{align*}
Series expansion around \(x=0\).

0.237

1044

\begin{align*} 2 y^{\prime }+3 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.245

1045

\begin{align*} y^{\prime }+2 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.273

1046

\begin{align*} y^{\prime }&=x^{2} y \\ \end{align*}
Series expansion around \(x=0\).

0.278

1047

\begin{align*} \left (x -2\right ) y^{\prime }+y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.262

1048

\begin{align*} \left (2 x -1\right ) y^{\prime }+2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.280

1049

\begin{align*} 2 \left (x +1\right ) y^{\prime }&=y \\ \end{align*}
Series expansion around \(x=0\).

0.302

1050

\begin{align*} \left (-1+x \right ) y^{\prime }+2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.269

1051

\begin{align*} 2 \left (-1+x \right ) y^{\prime }&=3 y \\ \end{align*}
Series expansion around \(x=0\).

0.274

1396

\begin{align*} y^{\prime }-y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.245

1397

\begin{align*} y^{\prime }-y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.274

1398

\begin{align*} \left (1-x \right ) y^{\prime }&=y \\ \end{align*}
Series expansion around \(x=0\).

0.289

3334

\begin{align*} y^{\prime }&=\sqrt {1-y} \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.126

3335

\begin{align*} y^{\prime }&=y x -x^{2} \\ y \left (0\right ) &= 2 \\ \end{align*}
Series expansion around \(x=0\).

0.314

3336

\begin{align*} y^{\prime }&=y^{2} x^{2} \\ y \left (1\right ) &= 0 \\ \end{align*}
Series expansion around \(x=1\).

0.188

3337

\begin{align*} y^{\prime }&=3 x +\frac {y}{x} \\ y \left (1\right ) &= 3 \\ \end{align*}
Series expansion around \(x=1\).

0.308

3338

\begin{align*} y^{\prime }&=\ln \left (y x \right ) \\ y \left (1\right ) &= 1 \\ \end{align*}
Series expansion around \(x=1\).

0.270

3339

\begin{align*} y^{\prime }&=1+y^{2} \\ y \left (1\right ) &= -1 \\ \end{align*}
Series expansion around \(x=1\).

0.177

3340

\begin{align*} y^{\prime }&=x^{2}+y^{2} \\ y \left (2\right ) &= 0 \\ \end{align*}
Series expansion around \(x=2\).

0.235

3341

\begin{align*} y^{\prime }&=\sqrt {y x +1} \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.164

3342

\begin{align*} y^{\prime }&=\cos \left (x \right )+\sin \left (y\right ) \\ y \left (\frac {\pi }{2}\right ) &= \frac {\pi }{2} \\ \end{align*}
Series expansion around \(x=\frac {\pi }{2}\).

0.575

7362

\begin{align*} y^{\prime }&=3 x^{2} y \\ \end{align*}
Series expansion around \(x=0\).

0.571

7631

\begin{align*} y^{\prime }+\left (2+x \right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.635

7632

\begin{align*} y^{\prime }-y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.459

7633

\begin{align*} z^{\prime }-x^{2} z&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.609

7645

\begin{align*} y^{\prime }+2 \left (-1+x \right ) y&=0 \\ \end{align*}
Series expansion around \(x=1\).

0.649

7646

\begin{align*} y^{\prime }-2 y x&=0 \\ \end{align*}
Series expansion around \(x=1\).

0.750

7651

\begin{align*} x^{\prime }+x \sin \left (t \right )&=0 \\ x \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(t=0\).

0.669

7652

\begin{align*} y^{\prime }-{\mathrm e}^{x} y&=0 \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.631

7656

\begin{align*} y^{\prime }-y x&=\sin \left (x \right ) \\ \end{align*}
Series expansion around \(x=0\).

0.646

7657

\begin{align*} w^{\prime }+w x&={\mathrm e}^{x} \\ \end{align*}
Series expansion around \(x=0\).

0.698

8067

\begin{align*} \left (1-x \right ) y^{\prime }&=x^{2}-y \\ \end{align*}
Series expansion around \(x=0\).

0.300

8068

\begin{align*} y^{\prime } x&=1-x +2 y \\ \end{align*}
Series expansion around \(x=1\).

0.281

8070

\begin{align*} y^{\prime }&=2 x^{2}+3 y \\ \end{align*}
Series expansion around \(x=0\).

0.293

8071

\begin{align*} \left (x +1\right ) y^{\prime }&=x^{2}-2 x +y \\ \end{align*}
Series expansion around \(x=0\).

0.253

8116

\begin{align*} y^{\prime }+y x&=\cos \left (x \right ) \\ \end{align*}
Series expansion around \(x=0\).

0.306

8125

\begin{align*} y^{\prime }-y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.230

8566

\begin{align*} \left (x +1\right ) y^{\prime }&=y \\ \end{align*}
Series expansion around \(x=0\).

0.240

8567

\begin{align*} y^{\prime }&=-2 y x \\ \end{align*}
Series expansion around \(x=0\).

0.297

8575

\begin{align*} y^{\prime }+4 y&=1 \\ y \left (0\right ) &= {\frac {5}{4}} \\ \end{align*}
Series expansion around \(x=0\).

0.303

8578

\begin{align*} \left (x -2\right ) y^{\prime }&=y x \\ y \left (0\right ) &= 4 \\ \end{align*}
Series expansion around \(x=0\).

0.364

9347

\begin{align*} y^{\prime }&=2 y x \\ \end{align*}
Series expansion around \(x=0\).

0.420

9349

\begin{align*} y^{\prime }+y&=1 \\ \end{align*}
Series expansion around \(x=0\).

0.365

9351

\begin{align*} y^{\prime }-y&=2 \\ \end{align*}
Series expansion around \(x=0\).

0.355

9353

\begin{align*} y^{\prime }+y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.312

9355

\begin{align*} y^{\prime }-y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.316

9357

\begin{align*} y^{\prime }-y&=x^{2} \\ \end{align*}
Series expansion around \(x=0\).

0.392

9366

\begin{align*} y^{\prime }&=\frac {1}{\sqrt {-x^{2}+1}} \\ \end{align*}
Series expansion around \(x=0\).

0.386

9367

\begin{align*} y^{\prime }&=1+y \\ \end{align*}
Series expansion around \(x=0\).

0.352

9368

\begin{align*} y^{\prime }&=x -y \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.401

9487

\begin{align*} y^{\prime }&=y^{2}-x \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.246

9489

\begin{align*} y^{\prime }-2 y&=x^{2} \\ y \left (1\right ) &= 1 \\ \end{align*}
Series expansion around \(x=1\).

0.493

9491

\begin{align*} y^{\prime }&=y+x \,{\mathrm e}^{y} \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.756

10346

\begin{align*} y^{\prime }+2 y x&=x \\ \end{align*}
Series expansion around \(x=0\).

0.516

10347

\begin{align*} y^{\prime }+y&=\sin \left (x \right ) \\ \end{align*}
Series expansion around \(x=0\).

0.472

16820

\begin{align*} y^{\prime }-2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.232

16821

\begin{align*} y^{\prime }-2 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.215

16822

\begin{align*} y^{\prime }+\frac {2 y}{2 x -1}&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.218

16823

\begin{align*} \left (x -3\right ) y^{\prime }-2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.187

16824

\begin{align*} \left (x^{2}+1\right ) y^{\prime }-2 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.191

16825

\begin{align*} y^{\prime }+\frac {y}{-1+x}&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.206

16826

\begin{align*} y^{\prime }+\frac {y}{-1+x}&=0 \\ \end{align*}
Series expansion around \(x=3\).

0.240

16827

\begin{align*} \left (1-x \right ) y^{\prime }-2 y&=0 \\ \end{align*}
Series expansion around \(x=5\).

0.241

16828

\begin{align*} \left (-x^{3}+2\right ) y^{\prime }-3 x^{2} y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.269

16829

\begin{align*} \left (-x^{3}+2\right ) y^{\prime }+3 x^{2} y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.227

16830

\begin{align*} \left (x +1\right ) y^{\prime }-y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.239

16831

\begin{align*} \left (x +1\right ) y^{\prime }+\left (1-x \right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.250

16853

\begin{align*} y^{\prime }+\cos \left (y\right )&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.166

16854

\begin{align*} y^{\prime }-{\mathrm e}^{x} y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.282

16855

\begin{align*} y^{\prime }-\tan \left (x \right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.382

16864

\begin{align*} y^{\prime }-{\mathrm e}^{x} y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.262

16865

\begin{align*} y^{\prime }+y \,{\mathrm e}^{2 x}&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.305

16866

\begin{align*} y^{\prime }+\cos \left (x \right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.301

16867

\begin{align*} y^{\prime }+y \ln \left (x \right )&=0 \\ \end{align*}
Series expansion around \(x=1\).

0.334

16874

\begin{align*} y^{\prime }-{\mathrm e}^{x} y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.490

16875

\begin{align*} y^{\prime }+y \sqrt {x^{2}+1}&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.547

16876

\begin{align*} \cos \left (x \right ) y^{\prime }+y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.923

16877

\begin{align*} y^{\prime }+\sqrt {2 x^{2}+1}\, y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.557

18371

\begin{align*} y^{\prime }&=-y x +1 \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.378

18372

\begin{align*} y^{\prime }&=\frac {-x +y}{x +y} \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.246

18373

\begin{align*} y^{\prime }&=\sin \left (x \right ) y \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.519

18379

\begin{align*} y^{\prime }-2 y x&=0 \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.392

18385

\begin{align*} y^{\prime }&={\mathrm e}^{y}+y x \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.306

18387

\begin{align*} \left (x +1\right ) y^{\prime }-n y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.416

19578

\begin{align*} y^{\prime }&=2 y x \\ \end{align*}
Series expansion around \(x=0\).

0.488

19579

\begin{align*} y^{\prime }+y&=1 \\ \end{align*}
Series expansion around \(x=0\).

0.416

19582

\begin{align*} y^{\prime }&=1+y^{2} \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.222

19583

\begin{align*} y^{\prime }&=x -y \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.455

20881

\begin{align*} y^{\prime }&=y \\ y \left (0\right ) &= 2 \\ \end{align*}
Series expansion around \(x=0\).

0.280

20882

\begin{align*} y^{\prime }&=2 y x -x^{3} \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.377

20883

\begin{align*} \left (x +1\right ) y^{\prime }&=y p \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.312

20884

\begin{align*} y^{\prime }&=\sqrt {x^{2}+y^{2}} \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.287

20886

\begin{align*} y^{\prime }&=-x +y \\ y \left (0\right ) &= 2 \\ \end{align*}
Series expansion around \(x=0\).

0.302

20887

\begin{align*} \left (x +1\right ) y^{\prime }&=y p \\ \end{align*}
Series expansion around \(x=0\).

0.303

20986

\begin{align*} y^{\prime }&={\mathrm e}^{x}+x \cos \left (y\right ) \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

1.108

20987

\begin{align*} y^{\prime }&=x^{3}+y^{3} \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.218

20988

\begin{align*} u^{\prime }&=u^{3} \\ u \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.166

21637

\begin{align*} y^{\prime }-y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.332

21644

\begin{align*} y^{\prime }&=x^{2} \\ y \left (2\right ) &= 1 \\ \end{align*}
Series expansion around \(x=2\).

0.386

21645

\begin{align*} y^{\prime }&=y \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.341

21665

\begin{align*} x^{\prime }+x^{2}&=0 \\ x \left (-\frac {1}{2}\right ) &= 0 \\ \end{align*}
Series expansion around \(t=-{\frac {1}{2}}\).

0.232

21906

\begin{align*} y^{\prime }-y^{2}-x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.342

22821

\begin{align*} y^{\prime }+y&=0 \\ y \left (0\right ) &= 4 \\ \end{align*}
Series expansion around \(x=0\).

0.590

22822

\begin{align*} y^{\prime }&=y x \\ y \left (0\right ) &= 5 \\ \end{align*}
Series expansion around \(x=0\).

0.659

22823

\begin{align*} y^{\prime }&=2 x -y \\ \end{align*}
Series expansion around \(x=0\).

0.712

22838

\begin{align*} y^{\prime }+3 y&=0 \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.585

22839

\begin{align*} y^{\prime }+y&=x^{2} \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.682

22840

\begin{align*} y^{\prime }&={\mathrm e}^{x}+y \\ y \left (0\right ) &= 0 \\ \end{align*}
Series expansion around \(x=0\).

0.648

22841

\begin{align*} 2 y^{\prime }+y x -y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.725

23450

\begin{align*} y^{\prime }+3 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.339

23666

\begin{align*} y^{\prime }&=3 y \\ \end{align*}
Series expansion around \(x=0\).

0.317

24080

\begin{align*} \left (-x^{2}+1\right ) y^{\prime }+3 y x&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.349

24115

\begin{align*} y^{\prime }&=2 y \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.284

24116

\begin{align*} y^{\prime }&=x +2 y \\ y \left (0\right ) &= 2 \\ \end{align*}
Series expansion around \(x=0\).

0.303

24117

\begin{align*} y^{\prime }-y&=x^{2}+1 \\ y \left (0\right ) &= 1 \\ \end{align*}
Series expansion around \(x=0\).

0.288

24118

\begin{align*} y^{\prime }-y&=x^{2}+1 \\ y \left (0\right ) &= -3 \\ \end{align*}
Series expansion around \(x=0\).

0.311

24119

\begin{align*} y^{\prime }-7 y&=-x^{4}+2 \\ y \left (0\right ) &= a \\ \end{align*}
Series expansion around \(x=0\).

0.351