2.2.91 Problems 9001 to 9100

Table 2.199: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

Maple

Mma

Sympy

time(sec)

9001

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

Series expansion around \(x=0\).

[[_2nd_order, _with_linear_symmetries]]

1.227

9002

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

Series expansion around \(x=0\).

[[_2nd_order, _with_linear_symmetries]]

1.320

9003

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

Series expansion around \(x=0\).

[_Bessel]

5.208

9004

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

Series expansion around \(x=0\).

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

5.746

9005

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

Series expansion around \(x=0\).

[_Gegenbauer]

0.669

9006

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

[_separable]

5.063

9007

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

[_separable]

13.756

9008

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

[_separable]

3.704

9009

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

[_separable]

2.989

9010

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

[_separable]

8.346

9011

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

[_quadrature]

7.497

9012

\begin{align*} y^{\prime }&=2 \sqrt {y} \\ y \left (x_{0} \right ) &= y_{0} \\ \end{align*}

[_quadrature]

4.609

9013

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

[_quadrature]

4.164

9014

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

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

16.635

9015

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

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

27.178

9016

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

[[_homogeneous, ‘class A‘], _rational, _Riccati]

8.809

9017

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

[[_homogeneous, ‘class A‘], _dAlembert]

11.801

9018

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

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

21.145

9019

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

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

45.260

9020

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

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

13.236

9021

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

[[_homogeneous, ‘class C‘], _rational, _Riccati]

8.361

9022

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

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

0.281

9023

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

[_quadrature]

0.250

9024

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

[_separable]

0.575

9025

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

[_separable]

1.724

9026

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

[_separable]

0.376

9027

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

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

1.191

9028

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

[_exact]

0.536

9029

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

[_linear]

0.337

9030

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

[_separable]

0.395

9031

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

[_separable]

0.614

9032

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

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

0.581

9033

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

[_quadrature]

0.414

9034

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

[[_2nd_order, _missing_x]]

1.665

9035

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

[[_2nd_order, _missing_y]]

0.428

9036

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

0.546

9037

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

[[_2nd_order, _missing_x]]

3.222

9038

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

[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

1.744

9039

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

[[_2nd_order, _missing_y]]

1.376

9040

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

6.766

9041

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

1.495

9042

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

10.668

9043

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

[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

5.722

9044

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) &= 1 \\ y_{2} \left (0\right ) &= 2 \\ \end{align*}

system_of_ODEs

0.493

9045

\begin{align*} y_{1}^{\prime }&=y_{2} \\ y_{2}^{\prime }&=6 y_{1}+y_{2} \\ \end{align*}

With initial conditions

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

system_of_ODEs

0.665

9046

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) &= 0 \\ y_{2} \left (0\right ) &= 0 \\ \end{align*}

system_of_ODEs

0.767

9047

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

system_of_ODEs

0.063

9048

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

[_quadrature]

1.580

9049

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

[_separable]

7.090

9050

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

[_separable]

4.972

9051

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

[_quadrature]

2.041

9052

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

[[_2nd_order, _missing_x]]

1.853

9053

\begin{align*} y^{\prime \prime }-4 y&=0 \\ \end{align*}

[[_2nd_order, _missing_x]]

2.251

9054

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

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

34.799

9055

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

[[_homogeneous, ‘class D‘], _rational, _Riccati]

3.490

9056

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

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

13.060

9057

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

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

18.533

9058

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

[[_homogeneous, ‘class G‘], _rational]

2.974

9059

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

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

27.710

9060

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

[[_1st_order, _with_linear_symmetries]]

4.939

9061

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

[_quadrature]

0.675

9062

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

[_quadrature]

0.454

9063

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

[_quadrature]

0.444

9064

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

[_quadrature]

0.588

9065

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

[_quadrature]

0.523

9066

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

[_quadrature]

0.478

9067

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

[_quadrature]

0.736

9068

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

[_quadrature]

0.408

9069

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

[_quadrature]

0.944

9070

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

[_quadrature]

0.800

9071

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

[_quadrature]

0.665

9072

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

[_quadrature]

0.542

9073

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

[_quadrature]

0.593

9074

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

[_quadrature]

0.725

9075

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

[_quadrature]

0.711

9076

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

[_quadrature]

0.925

9077

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

[_quadrature]

0.842

9078

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

[_linear]

2.241

9079

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

[[_2nd_order, _missing_x]]

0.335

9080

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

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

34.647

9081

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

[[_3rd_order, _missing_x]]

0.089

9082

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

[_separable]

14.411

9083

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

[_separable]

5.006

9084

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

[_separable]

4.736

9085

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

[_separable]

5.616

9086

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

[_separable]

8.771

9087

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

[_separable]

3.961

9088

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

[_separable]

3.373

9089

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

[_separable]

4.868

9090

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

[_separable]

9.548

9091

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

[_separable]

4.806

9092

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

[_separable]

6.135

9093

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

[_separable]

13.283

9094

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

[_separable]

3.820

9095

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

[_separable]

7.277

9096

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

[_separable]

12.135

9097

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

[_separable]

6.381

9098

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

[[_2nd_order, _missing_y]]

2.118

9099

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

[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

8.279

9100

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

[_separable]

0.199