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

1808

\begin{align*} y-2 y^{\prime }+y^{\prime \prime }&=14 x^{{3}/{2}} {\mathrm e}^{x} \\ \end{align*}

0.678

5402

1915

\begin{align*} \left (6+4 x \right ) y^{\prime \prime }+\left (2 x +1\right ) y&=0 \\ y \left (-1\right ) &= -1 \\ y^{\prime }\left (-1\right ) &= 2 \\ \end{align*}

Series expansion around \(x=-1\).

0.678

5403

12804

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

0.678

5404

15203

\begin{align*} 4 y^{\prime \prime }+40 y^{\prime }+101 y&=0 \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= -5 \\ \end{align*}

Using Laplace transform method.

0.678

5405

16989

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

0.678

5406

17835

\begin{align*} x^{\prime \prime }+16 x&=t \sin \left (t \right ) \\ \end{align*}

0.678

5407

17855

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

0.678

5408

18822

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

0.678

5409

3160

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

0.679

5410

8634

\begin{align*} y^{\prime }-6 y&=0 \\ y \left (-1\right ) &= 4 \\ \end{align*}

Using Laplace transform method.

0.679

5411

10521

\begin{align*} \left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y&=0 \\ \end{align*}

0.679

5412

12591

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

0.679

5413

15525

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

0.679

5414

17504

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&={\mathrm e}^{-3 t} \ln \left (t \right ) \\ \end{align*}

0.679

5415

19460

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

0.679

5416

2187

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

0.680

5417

8163

\begin{align*} y^{\prime \prime }-6 y^{\prime }+13 y&=0 \\ \end{align*}

0.680

5418

15193

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

Using Laplace transform method.

0.680

5419

18670

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

With initial conditions

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

0.680

5420

19627

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

Using Laplace transform method.

0.680

5421

20057

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

0.680

5422

20512

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

0.680

5423

463

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

Series expansion around \(x=0\).

0.681

5424

1933

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

Series expansion around \(x=0\).

0.681

5425

3506

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

Series expansion around \(z=0\).

0.681

5426

16035

\begin{align*} y^{\prime \prime }-6 y^{\prime }-7 y&=0 \\ \end{align*}

0.681

5427

18916

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

With initial conditions

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

0.681

5428

19028

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

0.681

5429

19626

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

Using Laplace transform method.

0.681

5430

20396

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

0.681

5431

20936

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

0.681

5432

27632

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

0.681

5433

8028

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

0.682

5434

14387

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

0.682

5435

15189

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

Using Laplace transform method.

0.682

5436

16488

\begin{align*} y^{\prime \prime }-7 y^{\prime }+10 y&=0 \\ \end{align*}

0.682

5437

16930

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

0.682

5438

875

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

0.683

5439

3994

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

Series expansion around \(x=0\).

0.683

5440

4878

\begin{align*} x^{2} y^{\prime }&=a +b \,x^{n}+y^{2} x^{2} \\ \end{align*}

0.683

5441

10471

\begin{align*} \left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y&=0 \\ \end{align*}

0.683

5442

10515

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

0.683

5443

14874

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

0.683

5444

18675

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

0.683

5445

593

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

0.684

5446

1021

\begin{align*} x_{1}^{\prime }&=x_{3} \\ x_{2}^{\prime }&=-5 x_{1}-x_{2}-5 x_{3} \\ x_{3}^{\prime }&=4 x_{1}+x_{2}-2 x_{3} \\ \end{align*}

0.684

5447

1518

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

Using Laplace transform method.

0.684

5448

2195

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y&=5 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}-4 \cos \left (x \right )+4 \sin \left (x \right ) \\ \end{align*}

0.684

5449

2487

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

0.684

5450

2838

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

0.684

5451

3509

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

Series expansion around \(z=0\).

0.684

5452

5393

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

0.684

5453

8623

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

Using Laplace transform method.

0.684

5454

13207

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

0.684

5455

17432

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

0.684

5456

17446

\begin{align*} y^{\prime \prime }-9 y&=54 t \sin \left (2 t \right ) \\ \end{align*}

0.684

5457

17704

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

Series expansion around \(x=0\).

0.684

5458

18650

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

0.684

5459

18700

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

With initial conditions

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

0.684

5460

22479

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

0.684

5461

1639

\begin{align*} y^{\prime }-4 y&=\frac {48 x}{y^{2}} \\ y \left (0\right ) &= 1 \\ \end{align*}

0.685

5462

1923

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

Series expansion around \(x=0\).

0.685

5463

1926

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

Series expansion around \(x=0\).

0.685

5464

3990

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

Series expansion around \(x=0\).

0.685

5465

4500

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

0.685

5466

5736

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

0.685

5467

14716

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

0.685

5468

14816

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

Using Laplace transform method.

0.685

5469

17808

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

0.685

5470

18699

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

With initial conditions

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

0.685

5471

19630

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

Using Laplace transform method.

0.685

5472

21490

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

0.685

5473

26808

\begin{align*} 6 y^{\prime \prime \prime \prime }+29 y^{\prime \prime \prime }+45 y^{\prime \prime }+24 y^{\prime }+20 y&=0 \\ \end{align*}

0.685

5474

868

\begin{align*} x^{\prime \prime }+10 x^{\prime }+125 x&=0 \\ x \left (0\right ) &= 6 \\ x^{\prime }\left (0\right ) &= 50 \\ \end{align*}

0.686

5475

3737

\begin{align*} y^{\prime \prime }-y^{\prime }-2 y&=40 \sin \left (x \right )^{2} \\ \end{align*}

0.686

5476

6416

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

0.686

5477

12538

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

0.686

5478

13164

\(\left [\begin {array}{cc} 0 & 1 \\ -1 & 0 \end {array}\right ]\)

N/A

N/A

N/A

0.686

5479

18212

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

0.686

5480

1389

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

Series expansion around \(x=0\).

0.687

5481

2397

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

0.687

5482

3585

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

0.687

5483

4026

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

Series expansion around \(x=0\).

0.687

5484

9221

\begin{align*} y^{\prime \prime }-6 y^{\prime }+25 y&=0 \\ \end{align*}

0.687

5485

9728

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

0.687

5486

12667

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

0.687

5487

17498

\begin{align*} y^{\prime \prime }-25 y&=\frac {1}{1-{\mathrm e}^{5 t}} \\ \end{align*}

0.687

5488

17791

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

Series expansion around \(x=0\).

0.687

5489

18456

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

Using Laplace transform method.

0.687

5490

19208

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

0.687

5491

20202

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

0.687

5492

20847

\begin{align*} 2 y-3 y^{\prime }+y^{\prime \prime }&=10 \\ \end{align*}

0.687

5493

24502

\begin{align*} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+9 y^{\prime \prime }&=0 \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ y^{\prime }\left (\infty \right ) &= 1 \\ y^{\prime \prime }\left (0\right ) &= 6 \\ \end{align*}

0.687

5494

292

\begin{align*} 9 y^{\prime \prime }+6 y^{\prime }+4 y&=0 \\ y \left (0\right ) &= 3 \\ y^{\prime }\left (0\right ) &= 4 \\ \end{align*}

0.688

5495

654

\begin{align*} y^{\prime }&=\frac {1}{x^{2}} \\ y \left (1\right ) &= 5 \\ \end{align*}

0.688

5496

5386

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

0.688

5497

6648

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

0.688

5498

7883

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

0.688

5499

18636

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

0.688

5500

19019

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

0.688