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]]

0.905

12

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

[[_2nd_order, _quadrature]]

0.903

13

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

[[_2nd_order, _quadrature]]

1.895

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]]

2.032

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]]

1.999

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]]

1.310

17

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

[[_2nd_order, _quadrature]]

1.340

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]]

1.270

3088

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

[[_2nd_order, _quadrature]]

0.550

3243

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

[[_2nd_order, _quadrature]]

0.743

3248

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

[[_2nd_order, _quadrature]]

0.737

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]]

1.836

3583

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

[[_2nd_order, _quadrature]]

0.938

3584

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

[[_2nd_order, _quadrature]]

0.997

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]]

1.218

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]]

1.214

4126

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

[[_2nd_order, _quadrature]]

0.978

5710

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

[[_2nd_order, _quadrature]]

1.964

5711

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

[[_2nd_order, _quadrature]]

2.159

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]]

2.667

5713

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

[[_2nd_order, _quadrature]]

1.833

5714

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

[[_2nd_order, _quadrature]]

2.455

5953

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

[[_2nd_order, _quadrature]]

1.664

6187

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

[[_2nd_order, _quadrature]]

1.251

6297

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

[[_2nd_order, _quadrature]]

1.586

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]]

33.559

7069

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

[[_2nd_order, _quadrature]]

0.737

7789

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

[[_2nd_order, _quadrature]]

0.931

8202

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

[[_2nd_order, _quadrature]]

0.966

8856

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

[[_2nd_order, _quadrature]]

1.198

8864

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

[[_2nd_order, _quadrature]]

1.152

8890

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

[[_2nd_order, _quadrature]]

0.870

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]]

7.053

10040

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

[[_2nd_order, _quadrature]]

0.500

10041

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

[[_2nd_order, _quadrature]]

0.906

10042

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

[[_2nd_order, _quadrature]]

0.692

10043

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

[[_2nd_order, _quadrature]]

0.675

10046

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

[[_2nd_order, _quadrature]]

0.751

10360

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

[[_2nd_order, _quadrature]]

0.595

10363

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

[[_2nd_order, _quadrature]]

0.644

10366

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

[[_2nd_order, _quadrature]]

0.753

10368

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

[[_2nd_order, _quadrature]]

0.783

12281

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

[[_2nd_order, _quadrature]]

0.792

14159

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

[[_2nd_order, _quadrature]]

1.546

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]]

2.302

16157

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

[[_2nd_order, _quadrature]]

1.133

16158

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

[[_2nd_order, _quadrature]]

0.480

16171

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

[[_2nd_order, _quadrature]]

0.878

16172

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

[[_2nd_order, _quadrature]]

0.815

16180

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

[[_2nd_order, _quadrature]]

1.151

16623

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

[[_2nd_order, _quadrature]]

1.201

17379

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

[[_2nd_order, _quadrature]]

0.872

17434

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

[[_2nd_order, _quadrature]]

1.273

17456

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

[[_2nd_order, _quadrature]]

1.888

18082

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

[[_2nd_order, _quadrature]]

1.307

18090

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

[[_2nd_order, _quadrature]]

1.024

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]]

1.678

18092

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

[[_2nd_order, _quadrature]]

2.007

19065

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

[[_2nd_order, _quadrature]]

0.625

19426

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

[[_2nd_order, _quadrature]]

0.836

19847

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

[[_2nd_order, _quadrature]]

1.145

19848

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

[[_2nd_order, _quadrature]]

1.159

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]]

1.210

19850

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

[[_2nd_order, _quadrature]]

1.112

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]]

1.266

19867

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

[[_2nd_order, _quadrature]]

0.920

19868

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

[[_2nd_order, _quadrature]]

0.700

20125

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

[[_2nd_order, _quadrature]]

1.253

20162

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

[[_2nd_order, _quadrature]]

1.240

20535

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

[[_2nd_order, _quadrature]]

1.011

20536

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

[[_2nd_order, _quadrature]]

0.934

20537

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

[[_2nd_order, _quadrature]]

0.831

20539

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

[[_2nd_order, _quadrature]]

1.080

20541

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

[[_2nd_order, _quadrature]]

1.220

20542

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

[[_2nd_order, _quadrature]]

0.635

20771

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

[[_2nd_order, _quadrature]]

1.116

20772

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

[[_2nd_order, _quadrature]]

1.158

21484

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

[[_2nd_order, _quadrature]]

0.977

21515

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

[[_2nd_order, _quadrature]]

1.342

21567

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

[[_2nd_order, _quadrature]]

1.310

22100

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

[[_2nd_order, _quadrature]]

0.971

22133

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

[[_2nd_order, _quadrature]]

1.342

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]]

1.806

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]]

1.293

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]]

1.869

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]]

2.388

22477

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

[[_2nd_order, _quadrature]]

2.126

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]]

2.313

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]]

1.333

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]]

0.771

23108

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

[[_2nd_order, _quadrature]]

0.861

23261

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

[[_2nd_order, _quadrature]]

0.802

23262

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

[[_2nd_order, _quadrature]]

0.806

23501

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

[[_2nd_order, _quadrature]]

0.808

23763

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

[[_2nd_order, _quadrature]]

1.869

23764

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

[[_2nd_order, _quadrature]]

0.858

23921

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

[[_2nd_order, _quadrature]]

0.832

24926

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

[[_2nd_order, _quadrature]]

1.075

24927

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

[[_2nd_order, _quadrature]]

1.091

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]]

1.434

25094

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

[[_2nd_order, _quadrature]]

1.282

25529

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

[[_2nd_order, _quadrature]]

1.306

25530

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

[[_2nd_order, _quadrature]]

1.237

25543

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

[[_2nd_order, _quadrature]]

1.438

25544

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

[[_2nd_order, _quadrature]]

2.253

25585

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

[[_2nd_order, _quadrature]]

1.161

25622

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

[[_2nd_order, _quadrature]]

1.196

25623

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

[[_2nd_order, _quadrature]]

1.207

27557

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

[[_2nd_order, _quadrature]]

0.650