2.8.4 Second order ode Frobenius with complex roots

Table 2.1263: Second order ode Frobenius with complex roots [45]

#

ODE

solved

Maple

Mma

Sympy

time(sec)

422

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

Series expansion around \(x=0\).

0.869

495

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

Series expansion around \(x=0\).

0.659

1064

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

Series expansion around \(x=0\).

0.783

1944

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

Series expansion around \(x=0\).

2.084

2442

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

Series expansion around \(t=0\).

0.905

2445

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

Series expansion around \(t=0\).

1.226

2466

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

Series expansion around \(t=0\).

0.768

2639

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

Series expansion around \(t=0\).

0.851

2642

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

Series expansion around \(t=0\).

0.661

2664

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

Series expansion around \(t=0\).

0.789

3392

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

Series expansion around \(x=0\).

1.536

3396

\begin{align*} 9 x^{2} y^{\prime \prime }+10 x y^{\prime }+y&=x -1 \\ \end{align*}

Series expansion around \(x=0\).

0.746

4006

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

Series expansion around \(x=0\).

0.866

4012

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

Series expansion around \(x=0\).

1.087

4021

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

Series expansion around \(x=0\).

0.697

4030

\begin{align*} x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y&=0 \\ \end{align*}

Series expansion around \(x=0\).

0.924

4177

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

Series expansion around \(x=0\).

0.710

8112

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

Series expansion around \(x=0\).

0.645

8113

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

Series expansion around \(x=0\).

0.638

8114

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {25}{4}\right ) y&=0 \\ \end{align*}

Series expansion around \(x=0\).

0.613

8133

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

Series expansion around \(x=0\).

0.691

8989

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

Series expansion around \(x=0\).

1.079

8992

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

Series expansion around \(x=0\).

1.387

8995

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

Series expansion around \(x=0\).

1.183

9391

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

Series expansion around \(x=0\).

1.966

9427

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

Series expansion around \(x=0\).

1.378

9428

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

Series expansion around \(x=0\).

1.131

9430

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

Series expansion around \(x=0\).

1.148

9433

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

Series expansion around \(x=0\).

1.110

10176

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

Series expansion around \(x=0\).

1.283

10197

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

Series expansion around \(x=0\).

0.417

10222

\begin{align*} 2 x^{2} y^{\prime \prime }+5 x y^{\prime }+4 y&=0 \\ \end{align*}

Series expansion around \(x=0\).

0.458

14746

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

Series expansion around \(x=0\).

4.716

16888

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

Series expansion around \(x=0\).

2.989

16892

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

Series expansion around \(x=4\).

2.017

17705

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

Series expansion around \(x=0\).

0.741

19592

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

Series expansion around \(x=0\).

0.891

19598

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

Series expansion around \(x=0\).

0.920

21669

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

Series expansion around \(x=1\).

3.175

22179

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

Series expansion around \(x=0\).

0.786

23725

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+\frac {y}{16}&=0 \\ \end{align*}

Series expansion around \(x=0\).

0.665

25341

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

Series expansion around \(t=0\).

1.476

25353

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

Series expansion around \(t=0\).

0.929

25355

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

Series expansion around \(t=0\).

0.898

27780

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

Series expansion around \(x=0\).

0.384