2.5.13 second order ode quadrature

Table 2.1221: second order ode quadrature [109]

#

ODE

CAS classification

Solved

Maple

Mma

Sympy

time(sec)

11

\begin{align*} x^{\prime \prime }&=50 \\ x \left (0\right ) &= 20 \\ x^{\prime }\left (0\right ) &= 10 \\ \end{align*}

[[_2nd_order, _quadrature]]

0.880

12

\begin{align*} x^{\prime \prime }&=-20 \\ x \left (0\right ) &= 5 \\ x^{\prime }\left (0\right ) &= -15 \\ \end{align*}

[[_2nd_order, _quadrature]]

0.865

13

\begin{align*} x^{\prime \prime }&=3 t \\ x \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 5 \\ \end{align*}

[[_2nd_order, _quadrature]]

0.895

14

\begin{align*} x^{\prime \prime }&=2 t +1 \\ x \left (0\right ) &= 4 \\ x^{\prime }\left (0\right ) &= -7 \\ \end{align*}

[[_2nd_order, _quadrature]]

0.920

15

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

[[_2nd_order, _quadrature]]

1.052

16

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

[[_2nd_order, _quadrature]]

2.149

17

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

[[_2nd_order, _quadrature]]

1.085

18

\begin{align*} x^{\prime \prime }&=50 \sin \left (5 t \right ) \\ x \left (0\right ) &= 8 \\ x^{\prime }\left (0\right ) &= -10 \\ \end{align*}

[[_2nd_order, _quadrature]]

1.183

3088

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

[[_2nd_order, _quadrature]]

0.622

3243

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

[[_2nd_order, _quadrature]]

0.860

3248

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

[[_2nd_order, _quadrature]]

0.862

3271

\begin{align*} y^{\prime \prime }&=\sec \left (x \right ) \tan \left (x \right ) \\ y \left (0\right ) &= \frac {\pi }{4} \\ y^{\prime }\left (0\right ) &= 1 \\ \end{align*}

[[_2nd_order, _quadrature]]

2.131

3583

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

[[_2nd_order, _quadrature]]

0.975

3584

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

[[_2nd_order, _quadrature]]

1.032

3586

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

[[_2nd_order, _quadrature]]

1.282

3588

\begin{align*} y^{\prime \prime }&=x \,{\mathrm e}^{x} \\ y \left (0\right ) &= 3 \\ y^{\prime }\left (0\right ) &= 4 \\ \end{align*}

[[_2nd_order, _quadrature]]

1.283

4126

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

[[_2nd_order, _quadrature]]

0.708

5710

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

[[_2nd_order, _quadrature]]

0.951

5711

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

[[_2nd_order, _quadrature]]

1.522

5712

\begin{align*} y^{\prime \prime }&=\operatorname {c1} \cos \left (a x \right )+\operatorname {c2} \sin \left (b x \right ) \\ \end{align*}

[[_2nd_order, _quadrature]]

1.924

5713

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

[[_2nd_order, _quadrature]]

1.259

5714

\begin{align*} y^{\prime \prime }&=\operatorname {c1} \,{\mathrm e}^{a x}+\operatorname {c2} \,{\mathrm e}^{-b x} \\ \end{align*}

[[_2nd_order, _quadrature]]

1.743

5953

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

[[_2nd_order, _quadrature]]

0.859

6187

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

[[_2nd_order, _quadrature]]

0.652

6297

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

[[_2nd_order, _quadrature]]

0.985

6415

\begin{align*} \left (c \,x^{2}+2 b x +a \right )^{{3}/{2}} y^{\prime \prime }&=f \left (\frac {x}{\sqrt {c \,x^{2}+2 b x +a}}\right ) \\ \end{align*}

[[_2nd_order, _quadrature]]

28.047

7069

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

[[_2nd_order, _quadrature]]

1.662

7789

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

[[_2nd_order, _quadrature]]

1.644

8202

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

[[_2nd_order, _quadrature]]

2.338

8856

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

[[_2nd_order, _quadrature]]

2.237

8864

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

[[_2nd_order, _quadrature]]

2.079

8890

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

[[_2nd_order, _quadrature]]

1.500

9328

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

[[_2nd_order, _quadrature]]

4.190

10040

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

[[_2nd_order, _quadrature]]

1.614

10041

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

[[_2nd_order, _quadrature]]

2.019

10042

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

[[_2nd_order, _quadrature]]

2.821

10043

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

[[_2nd_order, _quadrature]]

2.296

10046

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

[[_2nd_order, _quadrature]]

2.612

10360

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

[[_2nd_order, _quadrature]]

1.831

10363

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

[[_2nd_order, _quadrature]]

2.064

10366

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

[[_2nd_order, _quadrature]]

2.058

10368

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

[[_2nd_order, _quadrature]]

2.113

12281

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

[[_2nd_order, _quadrature]]

0.722

14159

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

[[_2nd_order, _quadrature]]

1.574

14205

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

[[_2nd_order, _quadrature]]

2.662

16157

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

[[_2nd_order, _quadrature]]

2.501

16158

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

[[_2nd_order, _quadrature]]

1.178

16171

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

[[_2nd_order, _quadrature]]

2.236

16172

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

[[_2nd_order, _quadrature]]

2.193

16180

\begin{align*} y^{\prime \prime } x +2&=\sqrt {x} \\ y \left (1\right ) &= 8 \\ y^{\prime }\left (1\right ) &= 6 \\ \end{align*}

[[_2nd_order, _quadrature]]

3.584

16623

\begin{align*} y^{\prime \prime }&=6 \,{\mathrm e}^{x} \sin \left (x \right ) x \\ \end{align*}

[[_2nd_order, _quadrature]]

3.132

17379

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

[[_2nd_order, _quadrature]]

0.575

17434

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

[[_2nd_order, _quadrature]]

0.796

17456

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

[[_2nd_order, _quadrature]]

1.111

18082

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

[[_2nd_order, _quadrature]]

0.889

18090

\begin{align*} y^{\prime \prime } \left (2+x \right )^{5}&=1 \\ y \left (-1\right ) &= {\frac {1}{12}} \\ y^{\prime }\left (-1\right ) &= -{\frac {1}{4}} \\ \end{align*}

[[_2nd_order, _quadrature]]

0.978

18091

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

[[_2nd_order, _quadrature]]

1.826

18092

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

[[_2nd_order, _quadrature]]

1.348

19065

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

[[_2nd_order, _quadrature]]

1.235

19426

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

[[_2nd_order, _quadrature]]

1.363

19847

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

[[_2nd_order, _quadrature]]

1.934

19848

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

[[_2nd_order, _quadrature]]

1.852

19849

\begin{align*} e y^{\prime \prime }&=-\frac {\left (w L +P \right ) x}{2}-\frac {w \,x^{2}}{2} \\ \end{align*}

[[_2nd_order, _quadrature]]

1.927

19850

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

[[_2nd_order, _quadrature]]

1.689

19851

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

[[_2nd_order, _quadrature]]

1.809

19867

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

[[_2nd_order, _quadrature]]

1.429

19868

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

[[_2nd_order, _quadrature]]

0.977

20125

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

[[_2nd_order, _quadrature]]

2.187

20162

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

[[_2nd_order, _quadrature]]

1.813

20535

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

[[_2nd_order, _quadrature]]

2.024

20536

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

[[_2nd_order, _quadrature]]

1.631

20537

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

[[_2nd_order, _quadrature]]

1.075

20539

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

[[_2nd_order, _quadrature]]

2.002

20541

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

[[_2nd_order, _quadrature]]

1.945

20542

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

[[_2nd_order, _quadrature]]

1.067

20771

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

[[_2nd_order, _quadrature]]

2.346

20772

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

[[_2nd_order, _quadrature]]

1.374

21484

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

[[_2nd_order, _quadrature]]

1.515

21515

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

[[_2nd_order, _quadrature]]

1.973

21567

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

[[_2nd_order, _quadrature]]

1.885

22100

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

[[_2nd_order, _quadrature]]

1.477

22133

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

[[_2nd_order, _quadrature]]

1.974

22306

\begin{align*} x^{\prime \prime }&=t^{2}-4 t +8 \\ x \left (0\right ) &= 1 \\ x^{\prime }\left (0\right ) &= -3 \\ \end{align*}

[[_2nd_order, _quadrature]]

3.598

22308

\begin{align*} y^{\prime \prime }&=12 x \left (4-x \right ) \\ y \left (0\right ) &= 7 \\ y \left (1\right ) &= 0 \\ \end{align*}

[[_2nd_order, _quadrature]]

2.632

22310

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

[[_2nd_order, _quadrature]]

3.842

22311

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

[[_2nd_order, _quadrature]]

2.905

22477

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

[[_2nd_order, _quadrature]]

4.041

22481

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

[[_2nd_order, _quadrature]]

4.287

22482

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

[[_2nd_order, _quadrature]]

1.583

23053

\begin{align*} s^{\prime \prime }&=5 t^{2}-7 t \\ s \left (0\right ) &= 0 \\ s \left (1\right ) &= {\frac {1}{4}} \\ \end{align*}

[[_2nd_order, _quadrature]]

3.494

23108

\begin{align*} m s^{\prime \prime }&=\frac {g \,t^{2}}{2} \\ \end{align*}

[[_2nd_order, _quadrature]]

3.303

23261

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

[[_2nd_order, _quadrature]]

2.684

23262

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

[[_2nd_order, _quadrature]]

2.396

23501

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

[[_2nd_order, _quadrature]]

2.509

23763

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

[[_2nd_order, _quadrature]]

3.598

23764

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

[[_2nd_order, _quadrature]]

3.438

23921

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

[[_2nd_order, _quadrature]]

2.621

24926

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

[[_2nd_order, _quadrature]]

3.579

24927

\begin{align*} y^{\prime \prime }&=6 \sin \left (3 t \right ) \\ \end{align*}

[[_2nd_order, _quadrature]]

3.314

24934

\begin{align*} y^{\prime \prime }&=6 \sin \left (3 t \right ) \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 2 \\ \end{align*}

[[_2nd_order, _quadrature]]

5.926

25094

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

[[_2nd_order, _quadrature]]

4.682

25529

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

[[_2nd_order, _quadrature]]

5.138

25530

\begin{align*} y^{\prime \prime }&={\mathrm e}^{i \omega t} \\ \end{align*}

[[_2nd_order, _quadrature]]

3.217

25543

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

[[_2nd_order, _quadrature]]

5.876

25544

\begin{align*} y^{\prime \prime }&=\operatorname {Direct}_{t} \\ \end{align*}

[[_2nd_order, _quadrature]]

9.026

25585

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

[[_2nd_order, _quadrature]]

3.718

25622

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

[[_2nd_order, _quadrature]]

3.842

25623

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

[[_2nd_order, _quadrature]]

3.807

27557

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

[[_2nd_order, _quadrature]]

4.528