2.3.55 Problems 5401 to 5500

Table 2.683: Main lookup table. Sorted by time used to solve.

#

ID

ODE

Solved?

Maple

Mma

Sympy

time(sec)

5401

2785

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 0 \\ x_{3} \left (0\right ) &= 0 \\ \end{align*}

0.361

5402

2833

\begin{align*} x_{1}^{\prime }&=4 x_{2} \\ x_{2}^{\prime }&=-9 x_{1} \\ \end{align*}

0.361

5403

6930

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

0.361

5404

9727

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

0.361

5405

10523

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x&=0 \\ \end{align*}

0.361

5406

10844

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

0.361

5407

14927

\begin{align*} 4 x^{\prime \prime }-20 x^{\prime }+21 x&=0 \\ x \left (0\right ) &= -4 \\ x^{\prime }\left (0\right ) &= -12 \\ \end{align*}

0.361

5408

14975

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

Series expansion around \(x=0\).

0.361

5409

19123

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

0.361

5410

21460

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

0.361

5411

21591

\begin{align*} y^{\prime \prime }+y^{\prime }-12 y&={\mathrm e}^{x} x^{2} \\ \end{align*}

0.361

5412

22812

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=12 t \\ y \left (0\right ) &= 4 \\ y^{\prime }\left (0\right ) &= 1 \\ \end{align*}

Using Laplace transform method.

0.361

5413

22813

\begin{align*} y^{\prime \prime }+8 y^{\prime }+25 y&=100 \\ y \left (0\right ) &= 2 \\ y^{\prime }\left (0\right ) &= 20 \\ \end{align*}

Using Laplace transform method.

0.361

5414

23417

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

0.361

5415

23634

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\ y \left (0\right ) &= 3 \\ y^{\prime }\left (0\right ) &= -3 \\ \end{align*}

Using Laplace transform method.

0.361

5416

24598

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

0.361

5417

24614

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

0.361

5418

25159

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

0.361

5419

26622

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

0.361

5420

1013

\begin{align*} x_{1}^{\prime }&=x_{1}-4 x_{2} \\ x_{2}^{\prime }&=4 x_{1}+9 x_{2} \\ \end{align*}

0.362

5421

1067

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

Series expansion around \(x=0\).

0.362

5422

1417

\begin{align*} x_{1}^{\prime }&=4 x_{1}-2 x_{2} \\ x_{2}^{\prime }&=8 x_{1}-4 x_{2} \\ \end{align*}

0.362

5423

2620

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

Series expansion around \(t=0\).

0.362

5424

3115

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

0.362

5425

4750

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

0.362

5426

6238

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

0.362

5427

6511

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

0.362

5428

9732

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

0.362

5429

10037

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

0.362

5430

10538

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

0.362

5431

10943

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x&=0 \\ \end{align*}

0.362

5432

14377

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

0.362

5433

14428

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

0.362

5434

16196

\begin{align*} y^{\prime }&=\left \{\begin {array}{cc} 0 & x <1 \\ 1 & 1\le x <2 \\ 0 & 2\le x \end {array}\right . \\ y \left (0\right ) &= 0 \\ \end{align*}

0.362

5435

16262

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

0.362

5436

19361

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

0.362

5437

19634

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

Using Laplace transform method.

0.362

5438

19689

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

0.362

5439

19970

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

0.362

5440

20487

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

0.362

5441

21222

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

0.362

5442

22762

\begin{align*} 3 x^{2} y^{\prime \prime }+x^{3} y^{\prime \prime \prime }&=x +1 \\ \end{align*}

0.362

5443

22913

\begin{align*} x^{\prime }-x+2 y^{\prime }+7 y&=0 \\ 2 x^{\prime }+y^{\prime }+x+5 y&=0 \\ \end{align*}

0.362

5444

23029

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

0.362

5445

23765

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

0.362

5446

24605

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

0.362

5447

24711

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

0.362

5448

27366

\begin{align*} {y^{\prime }}^{2}-2 y^{\prime } x&=8 x^{2} \\ \end{align*}

0.362

5449

540

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

With initial conditions

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

0.363

5450

569

\begin{align*} x^{\prime \prime }+9 x&=\delta \left (t -3 \pi \right )+\cos \left (3 t \right ) \\ x \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 0 \\ \end{align*}

Using Laplace transform method.

0.363

5451

1450

\begin{align*} x_{1}^{\prime }&=3 x_{1}-4 x_{2} \\ x_{2}^{\prime }&=x_{1}-x_{2} \\ \end{align*}

0.363

5452

5492

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

0.363

5453

8329

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

0.363

5454

9714

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

0.363

5455

12745

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

0.363

5456

15702

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

Using Laplace transform method.

0.363

5457

17568

\begin{align*} y^{\prime \prime \prime \prime }-16 y&=0 \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 2 \\ y^{\prime \prime }\left (0\right ) &= 4 \\ y^{\prime \prime \prime }\left (0\right ) &= -24 \\ \end{align*}

0.363

5458

18658

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

0.363

5459

22183

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

Series expansion around \(x=0\).

0.363

5460

25616

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&={\mathrm e}^{t} \\ \end{align*}

0.363

5461

1383

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

Series expansion around \(x=0\).

0.364

5462

9331

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

0.364

5463

9605

\begin{align*} y^{\prime \prime }+y&=\sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) \\ y \left (0\right ) &= 10 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

Using Laplace transform method.

0.364

5464

10771

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

0.364

5465

10863

\begin{align*} 5 y^{\prime \prime }-2 y^{\prime } x +10 y&=0 \\ \end{align*}

0.364

5466

18193

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

0.364

5467

18410

\begin{align*} x^{\prime }&=-9 y \\ y^{\prime }&=x \\ \end{align*}

0.364

5468

18428

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

0.364

5469

19422

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

0.364

5470

20593

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

0.364

5471

24034

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

0.364

5472

24671

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&={\mathrm e}^{x} \cos \left (2 x \right ) \\ \end{align*}

0.364

5473

578

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

With initial conditions

\begin{align*} x \left (0\right ) &= 1 \\ y \left (0\right ) &= 0 \\ \end{align*}

0.365

5474

1267

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

0.365

5475

1370

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

Series expansion around \(x=0\).

0.365

5476

1756

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

0.365

5477

2030

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

Series expansion around \(x=0\).

0.365

5478

2398

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

0.365

5479

2602

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&=2 \cos \left (t \right )^{2} {\mathrm e}^{t} \\ \end{align*}

0.365

5480

3942

\begin{align*} y^{\prime \prime }+4 y&=10 \,{\mathrm e}^{-t} \\ y \left (0\right ) &= 4 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

Using Laplace transform method.

0.365

5481

5693

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

0.365

5482

9028

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

0.365

5483

9451

\begin{align*} y^{\prime \prime }-y^{\prime }+y&=3 \,{\mathrm e}^{-t} \\ y \left (0\right ) &= 3 \\ y^{\prime }\left (0\right ) &= 2 \\ \end{align*}

Using Laplace transform method.

0.365

5484

10388

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

0.365

5485

10469

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

0.365

5486

11293

\begin{align*} y^{\prime \prime }&=\left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y \\ \end{align*}

0.365

5487

11312

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

0.365

5488

15497

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

0.365

5489

18000

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

0.365

5490

18375

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

Series expansion around \(x=0\).

0.365

5491

19583

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

Series expansion around \(x=0\).

0.365

5492

21211

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

With initial conditions

\begin{align*} x \left (0\right ) &= a \\ y \left (0\right ) &= 0 \\ \end{align*}

0.365

5493

21959

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

0.365

5494

22278

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

0.365

5495

22291

\begin{align*} y^{\prime \prime }-4 y^{\prime }-5 y&={\mathrm e}^{3 x} \\ \end{align*}

0.365

5496

23638

\begin{align*} y^{\prime \prime }+10 y^{\prime }+25 y&=2 \,{\mathrm e}^{-5 t} \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= -1 \\ \end{align*}

Using Laplace transform method.

0.365

5497

24431

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

0.365

5498

24587

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

0.365

5499

25132

\begin{align*} y^{\prime \prime }+2 y^{\prime }-15 y&=16 \,{\mathrm e}^{t} \\ \end{align*}

0.365

5500

25167

\begin{align*} y_{1}^{\prime }-3 y_{1}&=-4 y_{2} \\ y_{2}^{\prime }+y_{2}&=y_{1} \\ \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 1 \\ \end{align*}

0.365