[next] [prev] [prev-tail] [tail] [up]

2.10 Table of system of ODEs solved using Laplace method

Table 2.1213: System of differential equations using Laplace method

#

ODE

Solved

Maple

Mma

Sympy

time(sec)

2773

\begin{align*} x_{1}^{\prime }&=x_{1}-3 x_{2} \\ x_{2}^{\prime }&=-2 x_{1}+2 x_{2} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 5 \\ \end{align*}

✓

✓

✓

✓

0.599

2774

\begin{align*} x_{1}^{\prime }&=x_{1}-x_{2} \\ x_{2}^{\prime }&=5 x_{1}-3 x_{2} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 2 \\ \end{align*}

✓

✓

✓

✓

0.533

2775

\begin{align*} x_{1}^{\prime }&=3 x_{1}-2 x_{2}+t \\ x_{2}^{\prime }&=2 x_{1}-2 x_{2}+3 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 2 \\ x_{2} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.517

2776

\begin{align*} x_{1}^{\prime }&=x_{1}+x_{2}+2 \,{\mathrm e}^{t} \\ x_{2}^{\prime }&=4 x_{1}+x_{2}-{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.501

2777

\begin{align*} x_{1}^{\prime }&=3 x_{1}-4 x_{2}+{\mathrm e}^{t} \\ x_{2}^{\prime }&=x_{1}-x_{2}+{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.498

2778

\begin{align*} x_{1}^{\prime }&=2 x_{1}-5 x_{2}+\sin \left (t \right ) \\ x_{2}^{\prime }&=x_{1}-2 x_{2}+\tan \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= -1 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

530.147

2779

\begin{align*} x_{1}^{\prime }&=4 x_{1}+5 x_{2}+4 \,{\mathrm e}^{t} \cos \left (t \right ) \\ x_{2}^{\prime }&=-2 x_{1}-2 x_{2} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.700

2780

\begin{align*} x_{1}^{\prime }&=x_{2}+f_{1} \left (t \right ) \\ x_{2}^{\prime }&=-x_{1}+f_{2} \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

4.296

2781

\begin{align*} x_{1}^{\prime }&=2 x_{1}-2 x_{2} \\ x_{2}^{\prime }&=4 x_{1}-2 x_{2}+\delta \left (t -\pi \right ) \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.993

2782

\begin{align*} x_{1}^{\prime }&=3 x_{1}-2 x_{2}+1-\operatorname {Heaviside}\left (t -\pi \right ) \\ x_{2}^{\prime }&=2 x_{1}-2 x_{2} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.930

2783

\begin{align*} x_{1}^{\prime }&=x_{1}+2 x_{2}-3 x_{3} \\ x_{2}^{\prime }&=x_{1}+x_{2}+2 x_{3} \\ x_{3}^{\prime }&=x_{1}-x_{2}+4 x_{3} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 0 \\ x_{3} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.822

2784

\begin{align*} x_{1}^{\prime }&=2 x_{1}+x_{3}+{\mathrm e}^{2 t} \\ x_{2}^{\prime }&=2 x_{2} \\ x_{3}^{\prime }&=3 x_{3}+{\mathrm e}^{2 t} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 1 \\ x_{3} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.803

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*}

✓

✓

✓

✗

0.975

2786

\begin{align*} x_{1}^{\prime }&=x_{1} \\ x_{2}^{\prime }&=2 x_{1}+x_{2}-2 x_{3} \\ x_{3}^{\prime }&=3 x_{1}+2 x_{2}+x_{3}+{\mathrm e}^{t} \cos \left (2 t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 0 \\ x_{2} \left (0\right ) &= 1 \\ x_{3} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.782

2787

\begin{align*} x_{1}^{\prime }&=3 x_{1} \\ x_{2}^{\prime }&=x_{1}+3 x_{2} \\ x_{3}^{\prime }&=3 x_{3} \\ x_{4}^{\prime }&=2 x_{3}+3 x_{4} \\ \end{align*}
With initial conditions
\begin{align*} x_{1} \left (0\right ) &= 1 \\ x_{2} \left (0\right ) &= 1 \\ x_{3} \left (0\right ) &= 1 \\ x_{4} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

1.039

4551

\begin{align*} x^{\prime }-x-2 y&=16 \,{\mathrm e}^{t} t \\ 2 x-y^{\prime }-2 y&=0 \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 4 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.479

4552

\begin{align*} x^{\prime }-2 x+y&=5 \,{\mathrm e}^{t} \cos \left (t \right ) \\ x+y^{\prime }-2 y&=10 \,{\mathrm e}^{t} \sin \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.485

4553

\begin{align*} x^{\prime }-4 x+3 y&=\sin \left (t \right ) \\ 2 x+y^{\prime }-y&=2 \cos \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= x_{0} \\ y \left (0\right ) &= y_{0} \\ \end{align*}

✓

✓

✓

✓

0.488

4554

\begin{align*} x^{\prime }-2 x-y&=2 \,{\mathrm e}^{t} \\ x-y^{\prime }+2 y&=3 \,{\mathrm e}^{4 t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= x_{0} \\ y \left (0\right ) &= y_{0} \\ \end{align*}

✓

✓

✓

✓

0.454

4555

\begin{align*} x^{\prime \prime }+x^{\prime }+y^{\prime }-2 y&=40 \,{\mathrm e}^{3 t} \\ x^{\prime }+x-y^{\prime }&=36 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 1 \\ y \left (0\right ) &= 3 \\ x^{\prime }\left (0\right ) &= 1 \\ \end{align*}

✗

✓

✓

✓

0.033

4556

\begin{align*} x^{\prime }-2 x-y&=2 \,{\mathrm e}^{t} \\ y^{\prime }-2 y-4 z&=4 \,{\mathrm e}^{2 t} \\ x-z^{\prime }-z&=0 \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 9 \\ y \left (0\right ) &= 3 \\ z \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.616

4557

\begin{align*} x^{\prime \prime }+2 x-2 y^{\prime }&=0 \\ 3 x^{\prime }+y^{\prime \prime }-8 y&=240 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

✗

✓

✓

✓

0.029

4558

\begin{align*} x^{\prime }-x-2 y&=0 \\ x-y^{\prime }&=15 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= x_{0} \\ y \left (0\right ) &= y_{0} \\ \end{align*}

✓

✓

✓

✓

0.651

4559

\begin{align*} x^{\prime }-x+y&=2 \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -\pi \right )\right ) \\ 2 x-y^{\prime }-y&=0 \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.582

4560

\begin{align*} 2 x^{\prime }+x-5 y^{\prime }-4 y&=28 \,{\mathrm e}^{t} \operatorname {Heaviside}\left (-2+t \right ) \\ 3 x^{\prime }-2 x-4 y^{\prime }+y&=0 \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 2 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.589

15324

\begin{align*} x^{\prime }+y^{\prime }+x&=0 \\ x^{\prime }-x+2 y^{\prime }&={\mathrm e}^{-t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.161

18910

\begin{align*} y_{1}^{\prime }&=-5 y_{1}+y_{2} \\ y_{2}^{\prime }&=-9 y_{1}+5 y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.303

18911

\begin{align*} y_{1}^{\prime }&=5 y_{1}-2 y_{2} \\ y_{2}^{\prime }&=6 y_{1}-2 y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.298

18912

\begin{align*} y_{1}^{\prime }&=4 y_{1}-4 y_{2} \\ y_{2}^{\prime }&=5 y_{1}-4 y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.319

18913

\begin{align*} y_{1}^{\prime }&=6 y_{2} \\ y_{2}^{\prime }&=-6 y_{1} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 5 \\ y_{2} \left (0\right ) &= 4 \\ \end{align*}

✓

✓

✓

✓

0.322

18914

\begin{align*} y_{1}^{\prime }&=-4 y_{1}-y_{2} \\ y_{2}^{\prime }&=y_{1}-2 y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.293

18915

\begin{align*} y_{1}^{\prime }&=2 y_{1}-64 y_{2} \\ y_{2}^{\prime }&=y_{1}-14 y_{2} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 0 \\ y_{2} \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.300

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*}

✓

✓

✓

✓

0.353

18917

\begin{align*} y_{1}^{\prime }&=5 y_{1}-y_{2}+{\mathrm e}^{-t} \\ y_{2}^{\prime }&=y_{1}+3 y_{2}+2 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= -3 \\ y_{2} \left (0\right ) &= 2 \\ \end{align*}

✓

✓

✓

✓

0.315

18918

\begin{align*} y_{1}^{\prime }&=-y_{1}-5 y_{2}+3 \\ y_{2}^{\prime }&=y_{1}+3 y_{2}+5 \cos \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= -1 \\ \end{align*}

✓

✓

✓

✓

0.327

18919

\begin{align*} y_{1}^{\prime }&=-2 y_{1}+y_{2} \\ y_{2}^{\prime }&=y_{1}-2 y_{2}+\sin \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 0 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.346

18920

\begin{align*} y_{1}^{\prime }&=y_{2}-y_{3} \\ y_{2}^{\prime }&=y_{1}+y_{3}-{\mathrm e}^{-t} \\ y_{3}^{\prime }&=y_{1}+y_{2}+{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 2 \\ y_{3} \left (0\right ) &= 3 \\ \end{align*}

✓

✓

✓

✓

0.404

19035

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

✓

✓

✓

✓

0.316

19036

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

✓

✓

✓

✓

0.275

19037

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

✓

✓

✓

✓

0.285

19038

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

✓

✓

✓

✓

0.405

20921

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

✓

✓

✓

✓

0.202

20922

\begin{align*} x^{\prime }&=-4 x-y+{\mathrm e}^{-t} \\ y^{\prime }&=x-2 y+2 \,{\mathrm e}^{-3 t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 2 \\ y \left (0\right ) &= -1 \\ \end{align*}

✓

✓

✓

✓

0.161

20923

\begin{align*} x^{\prime }&=x-y+2 \cos \left (t \right ) \\ y^{\prime }&=x+y+3 \sin \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 3 \\ y \left (0\right ) &= 2 \\ \end{align*}

✓

✓

✓

✓

0.181

21290

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

✓

✓

✓

✗

0.162

21291

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

✓

✓

✓

✗

0.116

21292

\begin{align*} x^{\prime }&=x+y \\ y^{\prime }&=-y+\delta \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✗

0.097

21723

\begin{align*} x^{\prime }-6 x+3 y&=8 \,{\mathrm e}^{t} \\ y^{\prime }-2 x-y&=4 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 1 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.405

21724

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

✓

✓

✓

✓

0.335

21725

\begin{align*} w^{\prime }+y&=\sin \left (t \right ) \\ y^{\prime }-z&={\mathrm e}^{t} \\ w+y+z^{\prime }&=1 \\ \end{align*}
With initial conditions
\begin{align*} w \left (0\right ) &= 0 \\ z \left (0\right ) &= 1 \\ y \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✗

0.426

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*}

✓

✓

✓

✓

0.388

22256

\begin{align*} w^{\prime }+y&=\sin \left (t \right ) \\ y^{\prime }-z&={\mathrm e}^{t} \\ w+y+z^{\prime }&=1 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ z \left (0\right ) &= 1 \\ w \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✗

0.495

22257

\begin{align*} y^{\prime \prime }+z+y&=0 \\ y^{\prime }+z^{\prime }&=0 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 0 \\ z \left (0\right ) &= 1 \\ \end{align*}

✗

✓

✓

✓

0.036

22258

\begin{align*} z^{\prime \prime }+y^{\prime }&=\cos \left (t \right ) \\ y^{\prime \prime }-z&=\sin \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 0 \\ z \left (0\right ) &= -1 \\ z^{\prime }\left (0\right ) &= -1 \\ \end{align*}

✗

✓

✓

✓

0.031

22259

\begin{align*} w^{\prime \prime }-y+2 z&=3 \,{\mathrm e}^{-t} \\ -2 w^{\prime }+2 y^{\prime }+z&=0 \\ 2 w^{\prime }-2 y+z^{\prime }+2 z^{\prime \prime }&=0 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 2 \\ z \left (0\right ) &= 2 \\ z^{\prime }\left (0\right ) &= -2 \\ w \left (0\right ) &= 1 \\ w^{\prime }\left (0\right ) &= 1 \\ \end{align*}

✗

✓

✓

✗

0.056

22260

\begin{align*} y^{\prime }+z&=t \\ z^{\prime }-y&=0 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ z \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.342

22261

\begin{align*} y^{\prime }-z&=0 \\ y-z^{\prime }&=0 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ z \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.357

22262

\begin{align*} w^{\prime }-w-2 y&=1 \\ y^{\prime }-4 w-3 y&=-1 \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 2 \\ w \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.397

22263

\begin{align*} w^{\prime }-y&=0 \\ w+y^{\prime }+z&=1 \\ w-y+z^{\prime }&=2 \sin \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} y \left (0\right ) &= 1 \\ w \left (0\right ) &= 1 \\ z \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✗

0.506

22264

\begin{align*} u^{\prime \prime }-2 v&=2 \\ u+v^{\prime }&=5 \,{\mathrm e}^{2 t}+1 \\ \end{align*}
With initial conditions
\begin{align*} u \left (0\right ) &= 2 \\ u^{\prime }\left (0\right ) &= 2 \\ v \left (0\right ) &= 1 \\ \end{align*}

✗

✓

✓

✓

0.026

22265

\begin{align*} w^{\prime \prime }-2 z&=0 \\ w^{\prime }+y^{\prime }-z&=2 t \\ w^{\prime }-2 y+z^{\prime \prime }&=0 \\ \end{align*}
With initial conditions
\begin{align*} w \left (0\right ) &= 0 \\ w^{\prime }\left (0\right ) &= 0 \\ z \left (0\right ) &= 1 \\ z^{\prime }\left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

✗

✓

✓

✓

0.041

22266

\begin{align*} w^{\prime \prime }+y+z&=-1 \\ w+y^{\prime \prime }-z&=0 \\ -w-y^{\prime }+z^{\prime \prime }&=0 \\ \end{align*}
With initial conditions
\begin{align*} w \left (0\right ) &= 0 \\ w^{\prime }\left (0\right ) &= 1 \\ z \left (0\right ) &= -1 \\ z^{\prime }\left (0\right ) &= 1 \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

✗

✓

✗

✗

0.040

22902

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

✓

✓

✓

✓

1.042

22903

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

✓

✓

✓

✓

0.939

22904

\begin{align*} x^{\prime }+3 y^{\prime }+y&={\mathrm e}^{t} \\ -x+y^{\prime }&=y \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 1 \\ \end{align*}

✓

✓

✓

✓

0.904

22905

\begin{align*} x^{\prime }-3 x-6 y&=27 t^{2} \\ x^{\prime }+y^{\prime }-3 y&=5 \,{\mathrm e}^{t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 5 \\ y \left (0\right ) &= -1 \\ \end{align*}

✓

✓

✓

✓

0.917

22906

\begin{align*} x^{\prime \prime }&=-2 y \\ y^{\prime }&=y-x^{\prime } \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 10 \\ y \left (0\right ) &= 5 \\ \end{align*}

✗

✓

✓

✓

0.029

22907

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

✗

✓

✓

✓

0.027

22908

\begin{align*} x^{\prime }+y^{\prime }&=\cos \left (t \right ) \\ x+y^{\prime \prime }&=2 \\ \end{align*}
With initial conditions
\begin{align*} x \left (\pi \right ) &= 2 \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= {\frac {1}{2}} \\ \end{align*}

✗

✓

✓

✓

0.028

22909

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

✓

✓

✓

✓

1.294

22910

\begin{align*} x^{\prime }&=x+y+z \\ y^{\prime }&=2 x+5 y+3 z \\ z^{\prime }&=3 x+9 y+5 z \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= -2 \\ y \left (0\right ) &= -1 \\ z \left (0\right ) &= 3 \\ \end{align*}

✓

✓

✓

✓

1.258

22911

\begin{align*} x^{\prime \prime }&=y+4 \,{\mathrm e}^{-2 t} \\ y^{\prime \prime }&=x-{\mathrm e}^{-2 t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

✗

✓

✓

✓

0.031

23092

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

✓

✓

✓

✓

1.126

23093

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

✗

✓

✓

✓

0.038

23094

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

✓

✓

✓

✓

0.981

23095

\begin{align*} x^{\prime }+2 x+y^{\prime }+y&={\mathrm e}^{-3 t} \\ 5 x+y^{\prime }+3 y&=5 \,{\mathrm e}^{-t} \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= -1 \\ y \left (0\right ) &= 4 \\ \end{align*}

✓

✓

✓

✓

1.004

23096

\begin{align*} 4 x^{\prime }+2 y^{\prime }+3 x&=E \sin \left (t \right ) \\ 4 x+2 x^{\prime }+3 y&=0 \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

✓

✓

✓

✓

0.997

25174

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

✓

✓

✓

✓

0.434

25175

\begin{align*} y_{1}^{\prime }-y_{1}&=-2 y_{2} \\ y_{2}^{\prime }-y_{2}&=2 y_{1} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 2 \\ y_{2} \left (0\right ) &= -2 \\ \end{align*}

✓

✓

✓

✓

0.453

25176

\begin{align*} y_{1}^{\prime }-2 y_{1}&=-y_{2} \\ y_{2}^{\prime \prime }-y_{2}^{\prime }+y_{2}&=y_{1} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 0 \\ y_{2} \left (0\right ) &= -1 \\ y_{2}^{\prime }\left (0\right ) &= 2 \\ \end{align*}

✗

✓

✓

✓

0.027

25177

\begin{align*} y_{1}^{\prime }+2 y_{1}&=5 y_{2} \\ y_{2}^{\prime \prime }-2 y_{2}^{\prime }+5 y_{2}&=2 y_{1} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 0 \\ y_{2}^{\prime }\left (0\right ) &= 3 \\ \end{align*}

✗

✓

✓

✓

0.027

25178

\begin{align*} y_{1}^{\prime \prime }+2 y_{1}&=-3 y_{2} \\ y_{2}^{\prime \prime }+2 y_{2}^{\prime }-9 y_{2}&=6 y_{1} \\ \end{align*}
With initial conditions
\begin{align*} y_{1} \left (0\right ) &= 10 \\ y_{1}^{\prime }\left (0\right ) &= 0 \\ y_{2} \left (0\right ) &= 10 \\ y_{2}^{\prime }\left (0\right ) &= 0 \\ \end{align*}

✗

✓

✓

✗

0.031

26003

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

✓

✓

✓

✗

0.107

26004

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

✗

✓

✓

✓

0.035

[next] [prev] [prev-tail] [front] [up]