2.2.190 Problems 18901 to 19000

Table 2.381: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

18901

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.714

18902

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

[_Gegenbauer]

0.607

18903

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

[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_poly_yn]]

152.296

18904

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

[[_2nd_order, _missing_y]]

1.638

18905

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

[[_3rd_order, _exact, _linear, _homogeneous]]

4.026

18906

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

[[_3rd_order, _exact, _linear, _nonhomogeneous]]

0.457

18907

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

[[_high_order, _missing_x]]

0.058

18908

\[ {}{y^{\prime }}^{2}-y y^{\prime \prime } = n \sqrt {{y^{\prime }}^{2}+a^{2} {y^{\prime \prime }}^{2}} \]

[[_2nd_order, _missing_x]]

18.904

18909

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

[[_3rd_order, _fully, _exact, _linear]]

0.408

18910

\[ {}y^{\prime \prime }+y^{\prime }+{y^{\prime }}^{3} = 0 \]

[[_2nd_order, _missing_x]]

3.225

18911

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

[[_2nd_order, _missing_y]]

1.299

18912

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

[[_2nd_order, _exact, _linear, _homogeneous]]

1.649

18913

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

[[_3rd_order, _missing_y]]

0.335

18914

\[ {}y^{\prime \prime } = \frac {a}{x} \]

[[_2nd_order, _quadrature]]

1.847

18915

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

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

0.538

18916

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

[[_3rd_order, _quadrature]]

0.151

18917

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

[[_2nd_order, _missing_y]]

1.849

18918

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

[[_3rd_order, _fully, _exact, _linear]]

0.697

18919

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

[[_2nd_order, _with_linear_symmetries]]

0.862

18920

\[ {}a y^{\prime \prime } = y^{\prime } \]

[[_2nd_order, _missing_x]]

1.202

18921

\[ {}y^{3} y^{\prime \prime } = a \]

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

2.227

18922

\[ {}y^{\prime \prime \prime } = f \left (x \right ) \]

[[_3rd_order, _quadrature]]

0.166

18923

\[ {}y^{\prime \prime } = a^{2}+k^{2} {y^{\prime }}^{2} \]

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

1.646

18924

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

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

1.066

18925

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

[[_2nd_order, _with_linear_symmetries]]

1.674

18926

\[ {}\left (3-x \right ) y^{\prime \prime }-\left (9-4 x \right ) y^{\prime }+\left (6-3 x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.168

18927

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

[[_2nd_order, _exact, _linear, _homogeneous]]

1.239

18928

\[ {}3 x^{2} y^{\prime \prime }+\left (-6 x^{2}+2\right ) y^{\prime }-4 y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.228

18929

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

[[_2nd_order, _missing_x]]

11.664

18930

\[ {}y^{\prime \prime }+\frac {y^{\prime }}{x^{{1}/{3}}}+\left (\frac {1}{4 x^{{2}/{3}}}-\frac {1}{6 x^{{1}/{3}}}-\frac {6}{x^{2}}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.765

18931

\[ {}4 x^{2} y^{\prime \prime }+4 x^{5} y^{\prime }+\left (x^{8}+6 x^{4}+4\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.408

18932

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

[[_2nd_order, _with_linear_symmetries]]

4.881

18933

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

[[_2nd_order, _with_linear_symmetries]]

0.635

18934

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

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

1.125

18935

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

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

1.759

18936

\[ {}x y^{\prime \prime }-y^{\prime }+4 x^{3} y = x^{5} \]

[[_2nd_order, _linear, _nonhomogeneous]]

3.679

18937

\[ {}x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+a^{2} y = \frac {1}{x^{2}} \]

[[_2nd_order, _linear, _nonhomogeneous]]

4.364

18938

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

[[_2nd_order, _with_linear_symmetries]]

1.195

18939

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

[[_2nd_order, _with_linear_symmetries]]

1.954

18940

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

[[_2nd_order, _with_linear_symmetries]]

2.472

18941

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

[[_2nd_order, _exact, _linear, _homogeneous]]

1.458

18942

\[ {}\left (x -3\right ) y^{\prime \prime }-\left (4 x -9\right ) y^{\prime }+3 \left (-2+x \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.115

18943

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

[[_2nd_order, _with_linear_symmetries]]

0.543

18944

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

[[_2nd_order, _with_linear_symmetries]]

0.828

18945

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

[[_2nd_order, _with_linear_symmetries]]

0.099

18946

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

[[_2nd_order, _with_linear_symmetries]]

2.309

18947

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

[[_2nd_order, _with_linear_symmetries]]

0.159

18948

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

[[_3rd_order, _with_linear_symmetries]]

0.038

18949

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

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

0.183

18950

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

[[_2nd_order, _with_linear_symmetries]]

1.534

18951

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

[[_2nd_order, _with_linear_symmetries]]

1.483

18952

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

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

3.088

18953

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

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

2.406

18954

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

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

0.131

18955

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

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

2.309

18956

\[ {}x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y = 0 \]

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

1.133

18957

\[ {}\left [\begin {array}{c} x^{\prime }+2 x+y^{\prime }+y=0 \\ 5 x+y^{\prime }+3 y=0 \end {array}\right ] \]

system_of_ODEs

0.532

18958

\[ {}\left [\begin {array}{c} x^{\prime }-7 x+y=0 \\ y^{\prime }-2 x-5 y=0 \end {array}\right ] \]

system_of_ODEs

0.544

18959

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

system_of_ODEs

0.519

18960

\[ {}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+44 x+49 y=t \\ 3 x^{\prime }+7 y^{\prime }+34 x+38 y={\mathrm e}^{t} \end {array}\right ] \]

system_of_ODEs

0.643

18961

\[ {}\left [\begin {array}{c} x^{\prime \prime }-3 x-4 y=0 \\ y^{\prime \prime }+x+y=0 \end {array}\right ] \]

system_of_ODEs

0.025

18962

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

system_of_ODEs

0.608

18963

\[ {}\left [\begin {array}{c} 4 x^{\prime }+9 y^{\prime }+2 x+31 y={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+x+24 y=3 \end {array}\right ] \]

system_of_ODEs

0.888

18964

\[ {}\left [\begin {array}{c} x^{\prime }+4 x+3 y=t \\ y^{\prime }+2 x+5 y={\mathrm e}^{t} \end {array}\right ] \]

system_of_ODEs

0.634

18965

\[ {}\left [\begin {array}{c} x^{\prime }=n y-m z \\ y^{\prime }=L z-m x \\ z^{\prime }=m x-L y \end {array}\right ] \]

system_of_ODEs

108.300

18966

\[ {}x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y = \frac {1}{x} \]

[[_2nd_order, _with_linear_symmetries]]

1.665

18967

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

[[_2nd_order, _missing_y]]

0.473

18968

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

[_linear]

2.796

18969

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

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

2.466

18970

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

[_Bernoulli]

2.660

18971

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

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

4.395

18972

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

[[_1st_order, _with_linear_symmetries], _exact, _rational]

1.912

18973

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

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

27.130

18974

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

[_Bernoulli]

2.441

18975

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

[_linear]

1.874

18976

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

[_linear]

1.583

18977

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

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

4.369

18978

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

[_separable]

4.627

18979

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

[_separable]

1.569

18980

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

[_separable]

1.545

18981

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

[_separable]

2.066

18982

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

[_separable]

2.776

18983

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

[_separable]

1.552

18984

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

[_separable]

9.441

18985

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

[_separable]

4.383

18986

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

[_separable]

2.066

18987

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

[_separable]

6.791

18988

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

[_separable]

36.338

18989

\[ {}\cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right ) = \cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime } \]

[_separable]

27.982

18990

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

[_separable]

1.700

18991

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

[_separable]

35.467

18992

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

[_separable]

3.306

18993

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

[_separable]

1.605

18994

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

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

1.825

18995

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

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

2.000

18996

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

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

1.923

18997

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

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

5.841

18998

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

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

7.737

18999

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

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

8.918

19000

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

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

3.996