2.3.86 Problems 8501 to 8600

Table 2.745: Main lookup table. Sorted by time used to solve.

#

ID

ODE

Solved?

Maple

Mma

Sympy

time(sec)

8501

7663

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

Series expansion around \(x=0\).

0.625

8502

15682

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

0.625

8503

19065

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

0.625

8504

20210

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

0.625

8505

23967

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

0.625

8506

25265

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

0.625

8507

27032

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

Using Laplace transform method.

0.625

8508

27353

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

0.625

8509

887

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

0.626

8510

1937

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

Series expansion around \(x=-2\).

0.626

8511

4594

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

Series expansion around \(x=0\).

0.626

8512

5947

\begin{align*} -y-2 y^{\prime }+4 y^{\prime \prime }&=0 \\ \end{align*}

0.626

8513

6850

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

0.626

8514

6932

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

0.626

8515

7381

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

0.626

8516

9265

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

0.626

8517

9727

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

0.626

8518

14431

\begin{align*} y^{\prime \prime }-y^{\prime }-12 y&=0 \\ y \left (0\right ) &= 5 \\ y^{\prime }\left (0\right ) &= 6 \\ \end{align*}

0.626

8519

16687

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

0.626

8520

2282

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

0.627

8521

5366

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

0.627

8522

7313

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

0.627

8523

8288

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=6 x +4 \\ y \left (-1\right ) &= 0 \\ y^{\prime }\left (-1\right ) &= 1 \\ \end{align*}

0.627

8524

16798

\begin{align*} y^{\prime }&=\left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1<t <3 \\ 0 & 3<t \end {array}\right . \\ y \left (0\right ) &= 0 \\ \end{align*}

Using Laplace transform method.

0.627

8525

21137

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

0.627

8526

24709

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

0.627

8527

25588

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

0.627

8528

25592

\begin{align*} y^{\prime \prime }+9 y&={\mathrm e}^{2 t} t \\ \end{align*}

0.627

8529

592

\begin{align*} x^{\prime }&=x+9 y \\ y^{\prime }&=-2 x-5 y \\ \end{align*}

With initial conditions

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

0.628

8530

3742

\begin{align*} y^{\prime \prime }-2 y^{\prime }+10 y&=24 \,{\mathrm e}^{x} \cos \left (3 x \right ) \\ \end{align*}

0.628

8531

4413

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

0.628

8532

7209

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}&=0 \\ \end{align*}

0.628

8533

7803

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

0.628

8534

8476

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

Series expansion around \(x=0\).

0.628

8535

8908

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=3 \,{\mathrm e}^{-x}+2 x^{2} \\ \end{align*}

0.628

8536

9363

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

Series expansion around \(x=0\).

0.628

8537

9806

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

0.628

8538

10201

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

Series expansion around \(x=0\).

0.628

8539

10223

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y&=0 \\ \end{align*}

Series expansion around \(x=0\).

0.628

8540

10354

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

Series expansion around \(x=0\).

0.628

8541

10401

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

0.628

8542

10519

\begin{align*} \left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y&=0 \\ \end{align*}

0.628

8543

12567

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

0.628

8544

13151

\(\left [\begin {array}{ccc} 5 & 0 & 0 \\ 4 & -4 & -2 \\ -2 & 12 & 6 \end {array}\right ]\)

N/A

N/A

N/A

0.628

8545

15515

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

0.628

8546

17445

\begin{align*} y^{\prime \prime }-6 y^{\prime }+13 y&=25 \sin \left (2 t \right ) \\ \end{align*}

0.628

8547

17605

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

0.628

8548

17766

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

0.628

8549

18797

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

0.628

8550

18843

\begin{align*} y^{\prime \prime }-3 y^{\prime }-4 y&=2 \,{\mathrm e}^{-t} \\ \end{align*}

0.628

8551

20366

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

0.628

8552

24616

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

0.628

8553

24758

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

0.628

8554

25937

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

0.628

8555

25965

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

0.628

8556

639

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

0.629

8557

899

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

0.629

8558

3184

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

0.629

8559

9472

\begin{align*} x^{\prime }&=3 x-4 y \\ y^{\prime }&=4 x-7 y \\ \end{align*}

0.629

8560

9664

\begin{align*} x^{\prime }&=3 x-4 y \\ y^{\prime }&=4 x-7 y \\ \end{align*}

0.629

8561

10081

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

0.629

8562

12394

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

0.629

8563

13191

\(\left [\begin {array}{ccc} 2 & 0 & 0 \\ -6 & 11 & 2 \\ 6 & -15 & 0 \end {array}\right ]\)

N/A

N/A

N/A

0.629

8564

17028

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

0.629

8565

18452

\begin{align*} x^{\prime }&=6 x+y \\ y^{\prime }&=4 x+3 y \\ \end{align*}

0.629

8566

20791

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{4}/{3}}}-\frac {6}{x^{2}}\right ) y&=0 \\ \end{align*}

0.629

8567

22255

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

With initial conditions

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

0.629

8568

23922

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

0.629

8569

27178

\begin{align*} x_{1}^{\prime }&=x_{1}+x_{2}+6 \,{\mathrm e}^{2 t} \\ x_{2}^{\prime }&=x_{1}+x_{2}+2 \,{\mathrm e}^{2 t} \\ \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) &= 6 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

0.629

8570

1455

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

0.630

8571

3716

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=5 \,{\mathrm e}^{-2 x} x \\ \end{align*}

0.630

8572

3802

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

0.630

8573

6670

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

0.630

8574

11739

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

0.630

8575

12291

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

0.630

8576

13150

\(\left [\begin {array}{ccc} 2 & 0 & 0 \\ 2 & -2 & -1 \\ -2 & 6 & 3 \end {array}\right ]\)

N/A

N/A

N/A

0.630

8577

15207

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+13 y^{\prime }&=0 \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 1 \\ y^{\prime \prime }\left (0\right ) &= 6 \\ \end{align*}

Using Laplace transform method.

0.630

8578

18860

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

0.630

8579

20160

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

0.630

8580

21132

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

0.630

8581

22917

\begin{align*} x^{\prime }+4 x+3 y^{\prime }+4 y&=0 \\ x^{\prime }+2 x+2 y^{\prime }+2 y&=0 \\ \end{align*}

With initial conditions

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

0.630

8582

23500

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

0.630

8583

27028

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

Using Laplace transform method.

0.630

8584

27670

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

0.630

8585

921

\begin{align*} x^{\prime \prime }+8 x^{\prime }+25 x&=200 \cos \left (t \right )+520 \sin \left (t \right ) \\ x \left (0\right ) &= -30 \\ x^{\prime }\left (0\right ) &= -10 \\ \end{align*}

0.631

8586

1413

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

0.631

8587

5492

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

0.631

8588

9608

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

Using Laplace transform method.

0.631

8589

10135

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

0.631

8590

12675

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

0.631

8591

20207

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

0.631

8592

23530

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=\frac {{\mathrm e}^{-3 x}}{x^{3}} \\ \end{align*}

0.631

8593

23983

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

0.631

8594

24573

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

0.631

8595

26559

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

0.631

8596

1358

\begin{align*} u^{\prime \prime }+\frac {u^{\prime }}{8}+4 u&=3 \cos \left (2 t \right ) \\ u \left (0\right ) &= 2 \\ u^{\prime }\left (0\right ) &= 0 \\ \end{align*}

0.632

8597

3130

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

0.632

8598

4158

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-4 y^{\prime }+4 y&=2 x^{2}-4 x -1+2 x^{2} {\mathrm e}^{2 x}+5 x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} \\ \end{align*}

0.632

8599

5880

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

0.632

8600

14282

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

0.632