2.2.91 Problems 9001 to 9100

Table 2.195: 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 y^{\prime } x +\left (-x^{3}+3\right ) y&=0 \\ \end{align*}
Series expansion around \(x=0\).

[[_2nd_order, _with_linear_symmetries]]

0.581

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]]

0.724

9003

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

[_Bessel]

1.489

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)]‘]]

2.239

9005

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

[_Gegenbauer]

0.356

9006

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

[_separable]

1.573

9007

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

[_separable]

3.650

9008

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

[_separable]

1.631

9009

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

[_separable]

1.643

9010

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

[_separable]

3.261

9011

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

[_quadrature]

1.595

9012

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

[_quadrature]

1.407

9013

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

[_quadrature]

2.207

9014

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

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

3.833

9015

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

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

7.201

9016

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

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

2.786

9017

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

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

4.777

9018

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

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

4.180

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‘]]

16.981

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‘]]

5.319

9021

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

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

3.434

9022

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

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

0.134

9023

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

[_quadrature]

0.100

9024

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

[_separable]

0.184

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]

0.375

9026

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

[_separable]

0.158

9027

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

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

0.244

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]

0.181

9029

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

[_linear]

0.148

9030

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

[_separable]

0.162

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]

0.230

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‘]]

0.244

9033

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

[_quadrature]

0.186

9034

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

[[_2nd_order, _missing_x]]

0.819

9035

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

[[_2nd_order, _missing_y]]

0.230

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]]

0.215

9037

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

[[_2nd_order, _missing_x]]

0.856

9038

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

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

0.693

9039

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

[[_2nd_order, _missing_y]]

0.760

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]]

2.697

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]]

5.797

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]]

38.250

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]]

33.349

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

0.316

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

0.444

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

0.584

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

system_of_ODEs

0.046

9048

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

[_quadrature]

0.293

9049

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

[_separable]

2.045

9050

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

[_separable]

1.606

9051

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

[_quadrature]

0.559

9052

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

[[_2nd_order, _missing_x]]

0.869

9053

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

[[_2nd_order, _missing_x]]

0.850

9054

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

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

3.710

9055

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

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

1.079

9056

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

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

3.605

9057

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

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

5.534

9058

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

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

0.502

9059

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

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

7.336

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]]

2.333

9061

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

[_quadrature]

0.260

9062

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

[_quadrature]

0.202

9063

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

[_quadrature]

0.208

9064

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

[_quadrature]

0.244

9065

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

[_quadrature]

0.238

9066

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

[_quadrature]

0.240

9067

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

[_quadrature]

0.280

9068

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

[_quadrature]

0.200

9069

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

[_quadrature]

0.284

9070

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

[_quadrature]

0.351

9071

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

[_quadrature]

0.267

9072

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

[_quadrature]

0.306

9073

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

[_quadrature]

0.359

9074

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

[_quadrature]

0.335

9075

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

[_quadrature]

0.368

9076

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

[_quadrature]

0.402

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]

0.503

9078

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

[_linear]

1.089

9079

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

[[_2nd_order, _missing_x]]

0.174

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‘]]

3.951

9081

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

[[_3rd_order, _missing_x]]

0.049

9082

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

[_separable]

4.726

9083

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

[_separable]

1.579

9084

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

[_separable]

1.554

9085

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

[_separable]

2.400

9086

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

[_separable]

2.291

9087

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

[_separable]

2.183

9088

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

[_separable]

1.615

9089

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

[_separable]

1.649

9090

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

[_separable]

2.697

9091

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

[_separable]

1.800

9092

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

[_separable]

1.916

9093

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

[_separable]

2.658

9094

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

[_separable]

2.600

9095

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

[_separable]

1.967

9096

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

[_separable]

4.164

9097

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

[_separable]

1.737

9098

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

[[_2nd_order, _missing_y]]

2.727

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]]

4.947

9100

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

[_separable]

0.064