2.8.1 First order Taylor and ordinary point series

Table 2.1209: 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 \(x=0\).

0.260

401

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

0.248

402

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

0.248

403

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

0.274

404

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

0.235

405

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

0.265

406

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

0.284

407

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

0.268

408

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

0.272

409

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

0.272

424

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

0.128

1042

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

0.216

1043

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

0.238

1044

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

0.251

1045

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

0.283

1046

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

0.277

1047

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

0.274

1048

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

0.279

1049

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

0.308

1050

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

0.279

1051

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

0.290

1396

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

0.253

1397

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

0.281

1398

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

0.295

3334

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

0.216

3335

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

0.505

3336

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

0.306

3337

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

0.487

3338

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

0.469

3339

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

0.291

3340

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

0.365

3341

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

0.291

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 \(x=\frac {\pi }{2}\).

0.809

7362

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

0.626

7631

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

0.787

7632

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

0.578

7633

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

0.736

7645

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

0.797

7646

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

0.891

7651

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

0.830

7652

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

0.797

7656

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

0.817

7657

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

0.856

8067

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

0.388

8068

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

0.423

8070

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

1.395

8071

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

0.386

8116

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

0.450

8125

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

0.364

8566

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

0.356

8567

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

0.411

8575

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

0.442

8578

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

0.527

9347

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

0.559

9349

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

0.499

9351

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

0.497

9353

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

0.462

9355

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

0.450

9357

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

0.553

9366

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

0.587

9367

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

0.490

9368

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

0.568

9487

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

0.361

9489

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

0.674

9491

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

0.985

10346

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

0.548

10347

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

0.529

16820

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

0.402

16821

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

0.394

16822

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

0.418

16823

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

0.362

16824

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

0.376

16825

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

0.398

16826

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

0.483

16827

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

0.475

16828

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

0.489

16829

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

0.446

16830

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

0.455

16831

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

0.469

16853

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

0.339

16854

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

0.523

16855

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

0.613

16864

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

0.507

16865

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

0.578

16866

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

0.543

16867

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

0.643

16874

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

0.826

16875

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

1.154

16876

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

1.243

16877

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

1.073

18371

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

0.570

18372

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

0.408

18373

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

0.776

18379

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

0.641

18385

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

0.468

18387

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

0.575

19578

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

0.469

19579

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

0.419

19582

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

0.278

19583

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

0.494

20881

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

0.434

20882

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

0.537

20883

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

0.473

20884

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

1.477

20886

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

0.442

20887

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

0.456

20986

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

1.177

20987

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

0.295

20988

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

0.269

21637

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

0.516

21644

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

0.544

21645

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

0.489

21665

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

0.375

21906

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

0.608

22821

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

0.359

22822

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

0.415

22823

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

0.404

22838

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

0.366

22839

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

0.462

22840

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

0.396

22841

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

0.430

23450

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

0.437

23666

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

0.417

24080

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

0.521

24115

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

0.467

24116

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

0.464

24117

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

0.442

24118

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

0.484

24119

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

0.537