2.2.130 Problems 12901 to 13000

Table 2.261: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

12901

\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (-x^{2}+2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.070

12902

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.033

12903

\[ {}x y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+\left (-1+x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.744

12904

\[ {}x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (x^{2}+6\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.338

12905

\[ {}\left (2 x^{3}-1\right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.563

12906

\[ {}x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+2 \left (x +1\right ) y = x^{3} \]

[[_2nd_order, _with_linear_symmetries]]

1.430

12907

\[ {}x^{2} y^{\prime \prime }-2 n x \left (x +1\right ) y^{\prime }+\left (a^{2} x^{2}+n^{2}+n \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.217

12908

\[ {}x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+n^{2} y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.897

12909

\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+1+{y^{\prime }}^{2} = 0 \]

[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

0.752

12910

\[ {}\left (x y^{\prime \prime \prime }-y^{\prime \prime }\right )^{2} = {y^{\prime \prime \prime }}^{2}+1 \]

[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]

1.102

12911

\[ {}y^{\prime \prime }+x y^{\prime } = x \]

[[_2nd_order, _missing_y]]

1.190

12912

\[ {}y^{\prime \prime } = x \,{\mathrm e}^{x} \]

[[_2nd_order, _quadrature]]

1.820

12913

\[ {}\left (y^{\prime }-x y^{\prime \prime }\right )^{2} = 1+{y^{\prime \prime }}^{2} \]

[[_2nd_order, _missing_y]]

0.876

12914

\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}-y^{2} y^{\prime } = 0 \]

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

0.563

12915

\[ {}y^{\prime \prime } y-{y^{\prime }}^{2}+1 = 0 \]

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

1.143

12916

\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \]

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

1.614

12917

\[ {}y^{\prime \prime } y+2 y^{\prime }-{y^{\prime }}^{2} = 0 \]

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

0.349

12918

\[ {}\left (x^{2}-2 x +2\right ) y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

0.039

12919

\[ {}x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y = -x^{2}+1 \]

[[_3rd_order, _with_linear_symmetries]]

0.041

12920

\[ {}\left (x +2\right )^{2} y^{\prime \prime \prime }+\left (x +2\right ) y^{\prime \prime }+y^{\prime } = 1 \]

[[_3rd_order, _missing_y]]

0.546

12921

\[ {}x^{2} y^{\prime \prime }+3 x y^{\prime }+y = x \]

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

1.538

12922

\[ {}\left (-1+x \right )^{2} y^{\prime \prime }+4 \left (-1+x \right ) y^{\prime }+2 y = \cos \left (x \right ) \]

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

1.855

12923

\[ {}\left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y = 0 \]

[[_3rd_order, _fully, _exact, _linear]]

0.316

12924

\[ {}2 x^{3} y y^{\prime \prime \prime }+6 x^{3} y^{\prime } y^{\prime \prime }+18 x^{2} y y^{\prime \prime }+18 x^{2} {y^{\prime }}^{2}+36 x y y^{\prime }+6 y^{2} = 0 \]

[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

0.047

12925

\[ {}x^{5} y^{\prime \prime }+\left (2 x^{4}-x \right ) y^{\prime }-\left (2 x^{3}-1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.203

12926

\[ {}x^{2} \left (-x^{3}+1\right ) y^{\prime \prime }-x^{3} y^{\prime }-2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

139.868

12927

\[ {}x^{2} y^{\prime \prime \prime }-5 x y^{\prime \prime }+\left (4 x^{4}+5\right ) y^{\prime }-8 x^{3} y = 0 \]

[[_3rd_order, _with_linear_symmetries]]

0.039

12928

\[ {}y^{\prime \prime }+2 \cot \left (x \right ) y^{\prime }+2 \tan \left (x \right ) {y^{\prime }}^{2} = 0 \]

[[_2nd_order, _missing_y]]

1.233

12929

\[ {}x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

0.125

12930

\[ {}x^{3} y^{\prime \prime }-\left (-y+x y^{\prime }\right )^{2} = 0 \]

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

0.134

12931

\[ {}y^{\prime \prime } y-{y^{\prime }}^{2} = y^{2} \ln \left (y\right )-y^{2} x^{2} \]

[[_2nd_order, _reducible, _mu_xy]]

0.133

12932

\[ {}\sin \left (x \right )^{2} y^{\prime \prime }-2 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.849

12933

\[ {}y^{\prime \prime } = 1+{y^{\prime }}^{2} \]

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

1.442

12934

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime } = 2 \]

[[_2nd_order, _missing_y]]

1.328

12935

\[ {}y^{\prime \prime }+y y^{\prime } = 0 \]

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

1.014

12936

\[ {}\left (x^{3}+1\right ) y^{\prime \prime \prime }+9 x^{2} y^{\prime \prime }+18 x y^{\prime }+6 y = 0 \]

[[_3rd_order, _fully, _exact, _linear]]

0.175

12937

\[ {}\left (x^{2}-x \right ) y^{\prime \prime }+\left (4 x +2\right ) y^{\prime }+2 y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

1.411

12938

\[ {}y \left (1-\ln \left (y\right )\right ) y^{\prime \prime }+\left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2} = 0 \]

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

0.518

12939

\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x} = 0 \]

[[_2nd_order, _missing_y]]

0.564

12940

\[ {}x \left (x +2 y\right ) y^{\prime \prime }+2 x {y^{\prime }}^{2}+4 \left (x +y\right ) y^{\prime }+2 y+x^{2} = 0 \]

[[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

1.253

12941

\[ {}y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \]

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

1.714

12942

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-\frac {y^{\prime }}{x}+x^{2} = 0 \]

[[_2nd_order, _missing_y]]

1.497

12943

\[ {}4 x^{2} y^{\prime \prime \prime }+8 x y^{\prime \prime }+y^{\prime } = 0 \]

[[_3rd_order, _missing_y]]

0.217

12944

\[ {}\sin \left (x \right ) y^{\prime \prime }-\cos \left (x \right ) y^{\prime }+2 \sin \left (x \right ) y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

1.625

12945

\[ {}\left [\begin {array}{c} 3 x^{\prime }+3 x+2 y={\mathrm e}^{t} \\ 4 x-3 y^{\prime }+3 y=3 t \end {array}\right ] \]

system_of_ODEs

0.515

12946

\[ {}x^{\prime } = \frac {2 x}{t} \]

[_separable]

2.158

12947

\[ {}x^{\prime } = -\frac {t}{x} \]

[_separable]

4.022

12948

\[ {}x^{\prime } = -x^{2} \]

[_quadrature]

1.942

12949

\[ {}x^{\prime \prime }+2 x^{\prime }+2 x = 0 \]

[[_2nd_order, _missing_x]]

1.973

12950

\[ {}x^{\prime } = {\mathrm e}^{-x} \]

[_quadrature]

1.588

12951

\[ {}x^{\prime }+2 x = t^{2}+4 t +7 \]

[[_linear, ‘class A‘]]

1.354

12952

\[ {}2 t x^{\prime } = x \]

[_separable]

2.228

12953

\[ {}t^{2} x^{\prime \prime }-6 x = 0 \]

[[_Emden, _Fowler]]

0.432

12954

\[ {}2 x^{\prime \prime }-5 x^{\prime }-3 x = 0 \]

[[_2nd_order, _missing_x]]

0.860

12955

\[ {}x^{\prime } = x \left (1-\frac {x}{4}\right ) \]

[_quadrature]

2.294

12956

\[ {}x^{\prime } = x^{2}+t^{2} \]

[[_Riccati, _special]]

1.129

12957

\[ {}x^{\prime } = t \cos \left (t^{2}\right ) \]
i.c.

[_quadrature]

0.904

12958

\[ {}x^{\prime } = \frac {t +1}{\sqrt {t}} \]
i.c.

[_quadrature]

0.730

12959

\[ {}x^{\prime \prime } = -3 \sqrt {t} \]
i.c.

[[_2nd_order, _quadrature]]

1.741

12960

\[ {}x^{\prime } = t \,{\mathrm e}^{-2 t} \]

[_quadrature]

0.552

12961

\[ {}x^{\prime } = \frac {1}{t \ln \left (t \right )} \]

[_quadrature]

0.466

12962

\[ {}\sqrt {t}\, x^{\prime } = \cos \left (\sqrt {t}\right ) \]

[_quadrature]

0.621

12963

\[ {}x^{\prime } = \frac {{\mathrm e}^{-t}}{\sqrt {t}} \]
i.c.

[_quadrature]

0.623

12964

\[ {}x^{\prime }+t x^{\prime \prime } = 1 \]
i.c.

[[_2nd_order, _missing_y]]

1.364

12965

\[ {}x^{\prime } = \sqrt {x} \]
i.c.

[_quadrature]

1.595

12966

\[ {}x^{\prime } = {\mathrm e}^{-2 x} \]
i.c.

[_quadrature]

3.519

12967

\[ {}y^{\prime } = 1+y^{2} \]

[_quadrature]

3.349

12968

\[ {}u^{\prime } = \frac {1}{5-2 u} \]

[_quadrature]

1.869

12969

\[ {}x^{\prime } = a x+b \]

[_quadrature]

0.896

12970

\[ {}Q^{\prime } = \frac {Q}{4+Q^{2}} \]

[_quadrature]

1.496

12971

\[ {}x^{\prime } = {\mathrm e}^{x^{2}} \]

[_quadrature]

2.079

12972

\[ {}y^{\prime } = r \left (a -y\right ) \]

[_quadrature]

0.832

12973

\[ {}x^{\prime } = \frac {2 x}{t +1} \]

[_separable]

1.956

12974

\[ {}\theta ^{\prime } = t \sqrt {t^{2}+1}\, \sec \left (\theta \right ) \]

[_separable]

2.038

12975

\[ {}\left (2 u+1\right ) u^{\prime }-t -1 = 0 \]

[_separable]

2.720

12976

\[ {}R^{\prime } = \left (t +1\right ) \left (1+R^{2}\right ) \]

[_separable]

2.274

12977

\[ {}y^{\prime }+y+\frac {1}{y} = 0 \]

[_quadrature]

8.546

12978

\[ {}\left (t +1\right ) x^{\prime }+x^{2} = 0 \]

[_separable]

1.516

12979

\[ {}y^{\prime } = \frac {1}{2 y+1} \]
i.c.

[_quadrature]

2.181

12980

\[ {}x^{\prime } = \left (4 t -x\right )^{2} \]
i.c.

[[_homogeneous, ‘class C‘], _Riccati]

2.411

12981

\[ {}x^{\prime } = 2 t x^{2} \]
i.c.

[_separable]

2.328

12982

\[ {}x^{\prime } = t^{2} {\mathrm e}^{-x} \]
i.c.

[_separable]

2.400

12983

\[ {}x^{\prime } = x \left (4+x\right ) \]
i.c.

[_quadrature]

2.745

12984

\[ {}x^{\prime } = {\mathrm e}^{t +x} \]
i.c.

[_separable]

4.296

12985

\[ {}T^{\prime } = 2 a t \left (T^{2}-a^{2}\right ) \]
i.c.

[_separable]

2.028

12986

\[ {}y^{\prime } = t^{2} \tan \left (y\right ) \]
i.c.

[_separable]

1.996

12987

\[ {}x^{\prime } = \frac {\left (4+2 t \right ) x}{\ln \left (x\right )} \]
i.c.

[_separable]

3.212

12988

\[ {}y^{\prime } = \frac {2 t y^{2}}{t^{2}+1} \]
i.c.

[_separable]

2.402

12989

\[ {}x^{\prime } = \frac {t^{2}}{1-x^{2}} \]
i.c.

[_separable]

3.952

12990

\[ {}x^{\prime } = 6 t \left (x-1\right )^{{2}/{3}} \]

[_separable]

3.656

12991

\[ {}x^{\prime } = \frac {4 t^{2}+3 x^{2}}{2 t x} \]

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

7.102

12992

\[ {}x^{\prime } {\mathrm e}^{2 t}+2 x \,{\mathrm e}^{2 t} = {\mathrm e}^{-t} \]
i.c.

[[_linear, ‘class A‘]]

1.866

12993

\[ {}\frac {x^{\prime }+t x^{\prime \prime }}{t} = -2 \]

[[_2nd_order, _missing_y]]

0.845

12994

\[ {}y^{\prime } = \frac {y^{2}+2 t y}{t^{2}} \]

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

2.972

12995

\[ {}y^{\prime } = -y^{2} {\mathrm e}^{-t^{2}} \]
i.c.

[_separable]

2.312

12996

\[ {}x^{\prime } = 2 t^{3} x-6 \]

[_linear]

1.504

12997

\[ {}\cos \left (t \right ) x^{\prime }-2 x \sin \left (x\right ) = 0 \]

[_separable]

2.726

12998

\[ {}x^{\prime } = t -x^{2} \]

[[_Riccati, _special]]

1.047

12999

\[ {}7 t^{2} x^{\prime } = 3 x-2 t \]

[_linear]

1.315

13000

\[ {}x x^{\prime } = 1-t x \]

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

0.757