2.2.50 Problems 4901 to 5000

Table 2.101: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

4901

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

[_rational, _Abel]

1.286

4902

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

[‘y=_G(x,y’)‘]

16.555

4903

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

[_linear]

1.998

4904

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

[_rational, _Bernoulli]

1.433

4905

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

[_linear]

1.179

4906

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

[_separable]

3.706

4907

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

[_rational, _Bernoulli]

2.369

4908

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

[_rational, _Riccati]

160.245

4909

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

[_separable]

2.671

4910

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

[_linear]

1.098

4911

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

[_linear]

1.107

4912

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

[_linear]

1.098

4913

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

[_separable]

1.332

4914

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

[_linear]

1.270

4915

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

[_linear]

1.417

4916

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

[_linear]

1.168

4917

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

[_linear]

1.362

4918

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

[_linear]

1.447

4919

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

[_separable]

2.566

4920

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

[_separable]

1.186

4921

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

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

42.557

4922

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

[_separable]

2.079

4923

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

[_linear]

1.554

4924

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

[_separable]

1.789

4925

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

[_separable]

3.460

4926

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

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

2.917

4927

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

[_separable]

1.349

4928

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

[_linear]

1.985

4929

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

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

1.632

4930

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

[[_homogeneous, ‘class D‘], _Riccati]

49.634

4931

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

[_linear]

2.403

4932

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

[_linear]

1.138

4933

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

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

2.392

4934

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

[_linear]

1.267

4935

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

[_rational, _Riccati]

2.185

4936

\[ {}2 \left (x^{2}+x +1\right ) y^{\prime } = 1+8 x^{2}-\left (2 x +1\right ) y \]

[_linear]

3.825

4937

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

[_linear]

1.172

4938

\[ {}a \,x^{2} y^{\prime } = x^{2}+a x y+b^{2} y^{2} \]

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

23.751

4939

\[ {}\left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2} \]

[_separable]

2.505

4940

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

[_separable]

1.937

4941

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

[_separable]

0.990

4942

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

[_rational, _Abel]

1.974

4943

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

[_linear]

1.161

4944

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

[_linear]

1.018

4945

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

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

1.158

4946

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

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

1.727

4947

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

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

2.105

4948

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

[_separable]

1.322

4949

\[ {}x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right ) = 0 \]

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

1.899

4950

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

[_rational, _Riccati]

1.753

4951

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

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

92.586

4952

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

[‘y=_G(x,y’)‘]

45.219

4953

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

[_linear]

1.105

4954

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

[_linear]

1.168

4955

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

[_linear]

1.044

4956

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

[_linear]

1.114

4957

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

[_separable]

1.382

4958

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

[_separable]

1.521

4959

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

[_linear]

1.142

4960

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

[_linear]

2.648

4961

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

[_linear]

1.174

4962

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

[_linear]

1.217

4963

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

[_rational, _Riccati]

108.021

4964

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

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

2.034

4965

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

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

89.667

4966

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

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

14.138

4967

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

[_rational, _Bernoulli]

2.119

4968

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

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

2.731

4969

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

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

1.747

4970

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

[_rational, [_Riccati, _special]]

2.958

4971

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

[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

12.951

4972

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

[_separable]

2.381

4973

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

[_linear]

1.214

4974

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

[_rational, _Riccati]

1.690

4975

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

[_rational, _Bernoulli]

1.373

4976

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

[_separable]

1.495

4977

\[ {}\left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0 \]

[_rational, _Riccati]

3.735

4978

\[ {}x^{5} y^{\prime } = 1-3 x^{4} y \]

[_linear]

1.220

4979

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

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

2.195

4980

\[ {}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0 \]

[_rational, _Abel]

1.246

4981

\[ {}x^{n} y^{\prime } = a +b \,x^{n -1} y \]

[_linear]

1.163

4982

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

[_Riccati]

1.883

4983

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

[_Riccati]

9.745

4984

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

[[_homogeneous, ‘class G‘], _Riccati]

4.224

4985

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

[_rational, _Riccati]

2.495

4986

\[ {}x^{k} y^{\prime } = a \,x^{m}+b y^{n} \]

[_Chini]

1.643

4987

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

[_linear]

1.558

4988

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

[_separable]

2.983

4989

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

[_separable]

2.543

4990

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

[_linear]

1.520

4991

\[ {}y^{\prime } \sqrt {b^{2}+x^{2}} = \sqrt {y^{2}+a^{2}} \]

[_separable]

13.111

4992

\[ {}y^{\prime } \sqrt {b^{2}-x^{2}} = \sqrt {a^{2}-y^{2}} \]

[_separable]

13.316

4993

\[ {}x y^{\prime } \sqrt {a^{2}+x^{2}} = y \sqrt {b^{2}+y^{2}} \]

[_separable]

2.165

4994

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

[_separable]

21.060

4995

\[ {}y^{\prime } \sqrt {X}+\sqrt {Y} = 0 \]

[_quadrature]

0.406

4996

\[ {}y^{\prime } \sqrt {X} = \sqrt {Y} \]

[_quadrature]

0.408

4997

\[ {}x^{{3}/{2}} y^{\prime } = a +b \,x^{{3}/{2}} y^{2} \]

[_rational, [_Riccati, _special]]

1.779

4998

\[ {}y^{\prime } \sqrt {x^{3}+1} = \sqrt {y^{3}+1} \]

[_separable]

2.131

4999

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

[_separable]

3.080

5000

\[ {}y^{\prime } \sqrt {-x^{4}+1} = \sqrt {1-y^{4}} \]

[_separable]

2.706