2.3.5 Problems 401 to 500

Table 2.541: Main lookup table. Sorted by time used to solve.

#

ID

ODE

Solved?

Maple

Mma

Sympy

time(sec)

401

13039

\begin{align*} \left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2}-4 x y \left (-y+y^{\prime } x \right )^{3}&=0 \\ \end{align*}

0.042

402

13112

\begin{align*} t x^{\prime }&=2 x-t \\ t^{3} y^{\prime }&=-x+t^{2} y+t \\ t^{4} z^{\prime }&=-x-t^{2} y+t^{3} z+t \\ \end{align*}

0.042

403

14094

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }&=0 \\ \end{align*}

0.042

404

14344

\begin{align*} x^{\prime \prime \prime }+x^{\prime }&=0 \\ \end{align*}

0.042

405

16468

\begin{align*} x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y&=0 \\ \end{align*}

0.042

406

21701

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }+y^{\prime } x -y&=-\ln \left (x \right ) \\ \end{align*}
Series expansion around \(x=0\).

0.042

407

22673

\begin{align*} y^{\prime \prime \prime }-y&=0 \\ \end{align*}

0.042

408

25148

\begin{align*} y^{\prime \prime \prime \prime }-y&=0 \\ \end{align*}

0.042

409

4549

\begin{align*} x^{\prime }+4 x+2 y&=\frac {2}{{\mathrm e}^{t}-1} \\ 6 x-y^{\prime }+3 y&=\frac {3}{{\mathrm e}^{t}-1} \\ \end{align*}

0.043

410

6642

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }-3 y&=0 \\ \end{align*}

0.043

411

6645

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y&=0 \\ \end{align*}

0.043

412

6674

\begin{align*} y-y^{\prime } x -y^{\prime \prime }+x y^{\prime \prime \prime }&=-x^{2}+1 \\ \end{align*}

0.043

413

12751

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

0.043

414

12780

\begin{align*} y^{\prime \prime \prime }+y^{\prime } x +n y&=0 \\ \end{align*}

0.043

415

15734

\begin{align*} y_{1}^{\prime }&=\frac {5 y_{1}}{x}+\frac {4 y_{2}}{x}-2 x \\ y_{2}^{\prime }&=-\frac {6 y_{1}}{x}-\frac {5 y_{2}}{x}+5 x \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (-1\right ) &= 3 \\ y_{2} \left (-1\right ) &= -3 \\ \end{align*}

0.043

416

15736

\begin{align*} y_{1}^{\prime }&=\sin \left (x \right ) y_{1}+\sqrt {x}\, y_{2}+\ln \left (x \right ) \\ y_{2}^{\prime }&=\tan \left (x \right ) y_{1}-{\mathrm e}^{x} y_{2}+1 \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (1\right ) &= 1 \\ y_{2} \left (1\right ) &= -1 \\ \end{align*}

0.043

417

15737

\begin{align*} y_{1}^{\prime }&=\sin \left (x \right ) y_{1}+\sqrt {x}\, y_{2}+\ln \left (x \right ) \\ y_{2}^{\prime }&=\tan \left (x \right ) y_{1}-{\mathrm e}^{x} y_{2}+1 \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (2\right ) &= 1 \\ y_{2} \left (2\right ) &= -1 \\ \end{align*}

0.043

418

17540

\begin{align*} y^{\prime \prime \prime }&=0 \\ \end{align*}

0.043

419

18140

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y&=0 \\ \end{align*}

0.043

420

18142

\begin{align*} y^{\prime \prime \prime \prime }-y&=0 \\ \end{align*}

0.043

421

18968

\begin{align*} t \left (t -1\right ) y^{\prime \prime \prime \prime }+{\mathrm e}^{t} y^{\prime \prime }+4 t^{2} y&=0 \\ \end{align*}

0.043

422

18969

\begin{align*} y^{\prime \prime \prime }+t y^{\prime \prime }+t^{2} y^{\prime }+t^{2} y&=\ln \left (t \right ) \\ \end{align*}

0.043

423

18974

\begin{align*} t \left (t -1\right ) y^{\prime \prime \prime \prime }+{\mathrm e}^{t} y^{\prime \prime }+7 t^{2} y&=0 \\ \end{align*}

0.043

424

18980

\begin{align*} y^{\prime \prime \prime }+y^{\prime }&=0 \\ \end{align*}

0.043

425

19532

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }+3 y^{\prime }+y&=0 \\ \end{align*}

0.043

426

19581

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

0.043

427

20754

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

0.043

428

21310

\begin{align*} x^{\prime \prime \prime }+5 x^{\prime \prime }+9 x^{\prime }+5 x&=0 \\ \end{align*}

0.043

429

21900

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

0.043

430

22111

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=0 \\ \end{align*}

0.043

431

22301

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }-5 y^{\prime }+6 y&=0 \\ \end{align*}

0.043

432

23440

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

0.043

433

23780

\begin{align*} x_{1}^{\prime }&=x_{2} \\ x_{2}^{\prime }&=\sin \left (x_{1}\right ) \\ \end{align*}

0.043

434

6641

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y&=0 \\ \end{align*}

0.044

435

6661

\begin{align*} a^{3} x^{3} y+3 a^{2} x^{2} y^{\prime }+3 a x y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.044

436

6666

\begin{align*} f \left (x \right ) y+y^{\prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.044

437

12726

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

0.044

438

12753

\begin{align*} x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y&=0 \\ \end{align*}

0.044

439

12772

\begin{align*} x^{3} \left (x^{2}+1\right ) y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y&=0 \\ \end{align*}

0.044

440

14171

\begin{align*} 2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 y y^{\prime } x +6 y^{2}&=0 \\ \end{align*}

0.044

441

15670

\begin{align*} -4 y+6 y^{\prime }-4 y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.044

442

15674

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime }&=0 \\ \end{align*}

0.044

443

18130

\begin{align*} y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y&=0 \\ \end{align*}

0.044

444

18967

\begin{align*} t y^{\prime \prime \prime }+\sin \left (t \right ) y^{\prime \prime }+8 y&=\cos \left (t \right ) \\ \end{align*}

0.044

445

20155

\begin{align*} y^{\prime \prime \prime \prime }-a^{2} y^{\prime \prime }&=0 \\ \end{align*}

0.044

446

25149

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }&=0 \\ \end{align*}

0.044

447

25395

\begin{align*} y_{1}^{\prime }&=3 \sec \left (t \right ) y_{1}+5 \sec \left (t \right ) y_{2} \\ y_{2}^{\prime }&=-\sec \left (t \right ) y_{1}-3 \sec \left (t \right ) y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 2 \\ y_{2} \left (0\right ) &= 1 \\ \end{align*}

0.044

448

6604

\begin{align*} y^{\prime \prime \prime }&=y \\ \end{align*}

0.045

449

6687

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

0.045

450

6688

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

0.045

451

6720

\begin{align*} -4 \left (1+3 x \right ) y+2 x \left (2+5 x \right ) y^{\prime }-2 x^{2} \left (2 x +1\right ) y^{\prime \prime }+x^{3} \left (x +1\right ) y^{\prime \prime \prime }&=0 \\ \end{align*}

0.045

452

6734

\begin{align*} y-2 y^{\prime \prime }+y^{\prime \prime \prime \prime }&=0 \\ \end{align*}

0.045

453

7054

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y&=0 \\ \end{align*}

0.045

454

14090

\begin{align*} 2 y-y^{\prime }-2 y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.045

455

14096

\begin{align*} y^{\prime }-y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.045

456

18145

\begin{align*} 2 y^{\prime \prime \prime }-3 y^{\prime \prime }+y^{\prime }&=0 \\ \end{align*}

0.045

457

18977

\begin{align*} \left (x^{2}-25\right ) y^{\left (6\right )}+x^{2} y^{\prime \prime }+5 y&=0 \\ \end{align*}

0.045

458

19225

\begin{align*} t x^{\prime }+6 x-y-3 z&=0 \\ y^{\prime } t +23 x-6 y-9 z&=0 \\ t z^{\prime }+x+y-2 z&=0 \\ \end{align*}

0.045

459

20534

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

0.045

460

21253

\begin{align*} x^{\prime }&=x \left (3-y\right ) \\ y^{\prime }&=y \left (x-5\right ) \\ \end{align*}

0.045

461

22123

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y&=0 \\ \end{align*}

0.045

462

302

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-3 y^{\prime \prime }-5 y^{\prime }-2 y&=0 \\ \end{align*}

0.046

463

937

\begin{align*} y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }&=0 \\ \end{align*}

0.046

464

950

\begin{align*} 2 y^{\prime \prime \prime }-y^{\prime \prime }-5 y^{\prime }-2 y&=0 \\ \end{align*}

0.046

465

1058

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

0.046

466

2789

\begin{align*} x^{\prime }&=-b x y+m \\ y^{\prime }&=b x y-g y \\ \end{align*}

0.046

467

2791

\begin{align*} x^{\prime }&=-x-x \,y^{2} \\ y^{\prime }&=-y-y \,x^{2} \\ z^{\prime }&=1-z+x^{2} \\ \end{align*}

0.046

468

2792

\begin{align*} x^{\prime }&=x \,y^{2}-x \\ y^{\prime }&=x \sin \left (\pi y\right ) \\ \end{align*}

0.046

469

3831

\begin{align*} x_{1}^{\prime }&=\frac {x_{1}}{t} \\ x_{2}^{\prime }&=x_{2} \\ \end{align*}

0.046

470

4145

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y&=0 \\ \end{align*}

0.046

471

4573

\begin{align*} x_{1}^{\prime }&=-4 x_{1}-2 x_{2}+\frac {2}{{\mathrm e}^{t}-1} \\ x_{2}^{\prime }&=6 x_{1}+3 x_{2}-\frac {3}{{\mathrm e}^{t}-1} \\ \end{align*}

0.046

472

6623

\begin{align*} y^{\prime }-y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.046

473

6628

\begin{align*} 4 y+4 y^{\prime }+y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.046

474

6639

\begin{align*} 2 a^{2} y-a^{2} y^{\prime }-2 y^{\prime \prime }+y^{\prime \prime \prime }&=0 \\ \end{align*}

0.046

475

6658

\begin{align*} y^{\prime \prime \prime }&=a y^{\prime \prime } \\ \end{align*}

0.046

476

8917

\begin{align*} y^{\prime \prime \prime }-5 y^{\prime \prime }+6 y^{\prime }&=0 \\ \end{align*}

0.046

477

10132

\begin{align*} y^{\prime \prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3}&=0 \\ \end{align*}

0.046

478

12714

\begin{align*} y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-a b y&=0 \\ \end{align*}

0.046

479

12750

\begin{align*} x^{2} y^{\prime \prime \prime }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y&=0 \\ \end{align*}

0.046

480

12762

\begin{align*} -2 \left (x^{2}+4\right ) y+x \left (x^{2}+8\right ) y^{\prime }-4 x^{2} y^{\prime \prime }+x^{3} y^{\prime \prime \prime }&=0 \\ \end{align*}

0.046

481

14349

\begin{align*} x^{\prime \prime \prime }-8 x&=0 \\ \end{align*}

0.046

482

14597

\begin{align*} y^{\left (5\right )}&=0 \\ \end{align*}

0.046

483

15069

\begin{align*} y^{\prime }+y^{\prime \prime \prime }-3 y^{\prime \prime }&=0 \\ \end{align*}

0.046

484

17422

\begin{align*} {y^{\prime \prime }}^{2}-2 y^{\prime \prime } y^{\prime }+y^{2}&=0 \\ \end{align*}

0.046

485

18124

\begin{align*} y^{\prime \prime \prime }&=3 y y^{\prime } \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 1 \\ y^{\prime \prime }\left (0\right ) &= {\frac {3}{2}} \\ \end{align*}

0.046

486

18141

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+2 y^{\prime }&=0 \\ \end{align*}

0.046

487

18705

\begin{align*} x^{\prime }&=-x+y+x^{2} \\ y^{\prime }&=y-2 x y \\ \end{align*}

0.046

488

18981

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }&=0 \\ \end{align*}

0.046

489

20041

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-9 y^{\prime \prime }-11 y^{\prime }-4 y&=0 \\ \end{align*}

0.046

490

20340

\begin{align*} y^{\prime \prime \prime \prime }-y&=0 \\ \end{align*}

0.046

491

21252

\begin{align*} x^{\prime }&=x-4 x y \\ y^{\prime }&=-2 y+x y \\ \end{align*}

0.046

492

21899

\begin{align*} y^{\prime }+y-x^{\prime \prime }+x&={\mathrm e}^{t} \\ y^{\prime }-x^{\prime }+x&={\mathrm e}^{-t} \\ \end{align*}

0.046

493

22125

\begin{align*} -y-2 y^{\prime }+2 y^{\prime \prime \prime }+y^{\prime \prime \prime \prime }&=0 \\ \end{align*}

0.046

494

22669

\begin{align*} y^{\prime \prime \prime }-y&=0 \\ \end{align*}

0.046

495

23552

\begin{align*} y^{\left (5\right )}-y^{\prime }-\frac {4 y}{x}&=0 \\ \end{align*}

0.046

496

25394

\begin{align*} y_{1}^{\prime }&=\frac {4 t y_{1}}{t^{2}+1}+\frac {6 y_{2} t}{t^{2}+1}-3 t \\ y_{2}^{\prime }&=-\frac {2 t y_{1}}{t^{2}+1}-\frac {4 y_{2} t}{t^{2}+1}+t \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (1\right ) &= 1 \\ y_{2} \left (1\right ) &= -1 \\ \end{align*}

0.046

497

2790

\begin{align*} x^{\prime }&=a x-b x y \\ y^{\prime }&=-c y+d x y \\ z^{\prime }&=z+x^{2}+y^{2} \\ \end{align*}

0.047

498

3097

\begin{align*} y^{\prime \prime \prime \prime }-y&=0 \\ \end{align*}

0.047

499

4146

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime }&=0 \\ \end{align*}

0.047

500

4150

\begin{align*} y^{\prime \prime \prime }-y&=0 \\ \end{align*}

0.047