2.3.131 Problems 13001 to 13100

Table 2.793: Main lookup table. Sorted by time used to solve.

#

ID

ODE

Solved?

Maple

Mma

Sympy

time(sec)

13001

8586

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

0.784

13002

12548

\begin{align*} \left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y-3 x -1&=0 \\ \end{align*}

0.784

13003

13023

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

0.784

13004

19226

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

0.784

13005

20525

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

0.784

13006

1715

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

0.785

13007

14761

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

0.785

13008

17794

\begin{align*} x \left (x +1\right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }-10 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.785

13009

25342

\begin{align*} t y^{\prime \prime }+\left (1-t \right ) y^{\prime }+\lambda y&=0 \\ \end{align*}
Series expansion around \(t=0\).

0.785

13010

997

\begin{align*} x_{1}^{\prime }&=147 x_{1}+23 x_{2}-202 x_{3} \\ x_{2}^{\prime }&=-90 x_{1}-9 x_{2}+129 x_{3} \\ x_{3}^{\prime }&=90 x_{1}+15 x_{2}-123 x_{3} \\ \end{align*}

0.786

13011

2181

\begin{align*} y^{\prime \prime \prime }-7 y^{\prime \prime }+20 y^{\prime }-24 y&=-{\mathrm e}^{2 x} \left (\left (13-8 x \right ) \cos \left (2 x \right )-\left (8-4 x \right ) \sin \left (2 x \right )\right ) \\ \end{align*}

0.786

13012

2406

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=t^{{5}/{2}} {\mathrm e}^{-2 t} \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}

0.786

13013

2683

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} 2 & 0\le t \le 3 \\ 3 t -7 & 3<t \end {array}\right . \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}
Using Laplace transform method.

0.786

13014

9276

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

0.786

13015

9722

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

0.786

13016

13736

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

0.786

13017

14652

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

0.786

13018

15502

\begin{align*} x^{2} y^{\prime \prime }-5 y^{\prime } x +9 y&=0 \\ \end{align*}

0.786

13019

17486

\begin{align*} y^{\prime \prime }+\frac {y}{4}&=\sec \left (\frac {t}{2}\right )+\csc \left (\frac {t}{2}\right ) \\ \end{align*}

0.786

13020

24897

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

0.786

13021

2581

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

0.787

13022

3357

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

0.787

13023

4524

\begin{align*} y^{\prime \prime }+4 y&=8 \left (t^{2}+t -1\right ) \operatorname {Heaviside}\left (t -2\right ) \\ y \left (0\right ) &= 1 \\ y^{\prime }\left (0\right ) &= 2 \\ \end{align*}
Using Laplace transform method.

0.787

13024

9494

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

0.787

13025

10098

\begin{align*} y^{\prime \prime }-y^{\prime }-y x -x^{3}+1&=0 \\ \end{align*}

0.787

13026

17379

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

0.787

13027

19392

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

0.787

13028

21143

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

0.787

13029

21752

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

0.787

13030

22661

\begin{align*} u^{\prime \prime }+16 u&=0 \\ u \left (0\right ) &= 0 \\ u^{\prime }\left (0\right ) &= 4 \\ \end{align*}

0.787

13031

25285

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 4 & 0\le t <2 \\ 8 t & 2\le t <\infty \end {array}\right . \\ \end{align*}
Using Laplace transform method.

0.787

13032

2004

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

0.788

13033

2248

\begin{align*} y_{1}^{\prime }&=y_{1}-y_{2}+2 y_{3} \\ y_{2}^{\prime }&=12 y_{1}-4 y_{2}+10 y_{3} \\ y_{3}^{\prime }&=-6 y_{1}+y_{2}-7 y_{3} \\ \end{align*}

0.788

13034

5934

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

0.788

13035

14977

\begin{align*} 2 y^{\prime \prime } x +y^{\prime }-2 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.788

13036

21122

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

0.788

13037

602

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

0.789

13038

749

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

0.789

13039

2564

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

0.789

13040

18518

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

0.789

13041

21302

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

0.789

13042

23626

\begin{align*} x^{\prime }&=7 x+4 y-4 z \\ y^{\prime }&=4 x-8 y-z \\ z^{\prime }&=-4 x-y-8 z \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 3 \\ y \left (0\right ) &= 5 \\ z \left (0\right ) &= -1 \\ \end{align*}

0.789

13043

2445

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

0.790

13044

3332

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

0.790

13045

6085

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

0.790

13046

8273

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

0.790

13047

10393

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

0.790

13048

20897

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+\frac {3 y^{\prime }}{2+x}+\frac {\left (1-x \right )^{2} y}{x +3}&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.790

13049

10427

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

0.791

13050

18331

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

0.791

13051

18629

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

0.791

13052

860

\begin{align*} x^{2} y^{\prime \prime }+y^{\prime } x +9 y&=0 \\ \end{align*}

0.792

13053

2180

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+2 y&={\mathrm e}^{x} \left (\left (20+4 x \right ) \cos \left (x \right )-\left (12+12 x \right ) \sin \left (x \right )\right ) \\ \end{align*}

0.792

13054

4458

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

0.792

13055

18810

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

0.792

13056

23131

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

0.792

13057

25700

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

0.792

13058

1301

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

0.793

13059

2027

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

0.793

13060

3248

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

0.793

13061

3901

\begin{align*} x_{1}^{\prime }&=-4 x_{1} \\ x_{2}^{\prime }&=2 x_{1}+5 x_{2}-9 x_{3} \\ x_{3}^{\prime }&=5 x_{2}-x_{3} \\ \end{align*}

0.793

13062

5650

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

0.793

13063

9892

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

0.793

13064

24535

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

0.793

13065

7307

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

0.794

13066

21271

\begin{align*} 4 t^{2} x^{\prime \prime }+4 t x^{\prime }-x&=0 \\ \end{align*}
Series expansion around \(t=0\).

0.794

13067

2026

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

0.795

13068

15066

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

0.795

13069

20516

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

0.795

13070

21053

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

0.795

13071

23736

\begin{align*} \left (1-x \right ) x y^{\prime \prime }+\left (\frac {2}{3}-3 x \right ) y^{\prime }-y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.795

13072

1346

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

0.796

13073

3088

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

0.796

13074

3365

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

0.796

13075

6052

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

0.796

13076

6527

\begin{align*} 3 x y^{2}-12 x^{2} y y^{\prime }+4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}+8 \left (-x^{3}+1\right ) y y^{\prime \prime }&=0 \\ \end{align*}

0.796

13077

15090

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

0.796

13078

15177

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

0.796

13079

24878

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

0.796

13080

5835

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

0.797

13081

9635

\begin{align*} y^{\prime \prime }+16 y&=\left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 1 \\ \end{align*}
Using Laplace transform method.

0.797

13082

10117

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

0.797

13083

15558

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

0.797

13084

2187

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

0.798

13085

2691

\begin{align*} y^{\prime \prime }+y&=\left \{\begin {array}{cc} t^{2} & 0\le t <1 \\ 0 & 1\le t \end {array}\right . \\ y \left (0\right ) &= 0 \\ y^{\prime }\left (0\right ) &= 0 \\ \end{align*}
Using Laplace transform method.

0.798

13086

3380

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

0.798

13087

13007

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

0.798

13088

15447

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

0.798

13089

19981

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

0.798

13090

21753

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

0.798

13091

25618

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

0.798

13092

1946

\begin{align*} x^{2} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+x \left (7 x^{2}+8 x +5\right ) y^{\prime }-\left (-9 x^{2}-2 x +1\right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.799

13093

6017

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

0.799

13094

15409

\begin{align*} y^{\prime \prime \prime } y^{\prime }-3 {y^{\prime \prime }}^{2}&=0 \\ \end{align*}

0.799

13095

17793

\begin{align*} \left (1-x \right ) x y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }+10 y&=0 \\ \end{align*}
Series expansion around \(x=0\).

0.799

13096

18876

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

0.799

13097

19643

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

0.799

13098

23836

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

0.799

13099

2708

\begin{align*} x^{\prime }&=y+\textit {f\_1} \left (t \right ) \\ y^{\prime }&=-x+f_{2} \left (t \right ) \\ \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) &= 0 \\ y \left (0\right ) &= 0 \\ \end{align*}

0.800

13100

6212

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

0.800