2.2.59 Problems 5801 to 5900

Table 2.119: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

5801

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

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

0.302

5802

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

[_exact]

0.406

5803

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

[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

0.329

5804

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

[_exact]

1.925

5805

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

[_exact]

1.717

5806

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

[_exact]

0.372

5807

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

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

0.458

5808

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

[_exact]

0.510

5809

\[ {}{\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime } = 0 \]
i.c.

[_exact, _Bernoulli]

0.908

5810

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

[_separable]

1.178

5811

\[ {}y^{2} {\mathrm e}^{x y^{2}}+4 x^{3}+\left (2 x y \,{\mathrm e}^{x y^{2}}-3 y^{2}\right ) y^{\prime } = 0 \]
i.c.

[_exact]

0.527

5812

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

[_separable]

0.297

5813

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

[_separable]

0.334

5814

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

[‘y=_G(x,y’)‘]

0.375

5815

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

[_separable]

0.316

5816

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

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

0.851

5817

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

[_separable]

0.264

5818

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

[_separable]

0.308

5819

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

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

3.151

5820

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

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

0.303

5821

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

[_exact]

0.336

5822

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

[_rational, _Bernoulli]

0.526

5823

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

[_exact]

0.367

5824

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

[_exact]

0.463

5825

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

[[_1st_order, _with_linear_symmetries], _rational]

1.847

5826

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

[_rational]

0.454

5827

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

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

0.375

5828

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

[_exact]

0.540

5829

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

[‘y=_G(x,y’)‘]

0.401

5830

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

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

0.495

5831

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

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

0.620

5832

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

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

0.491

5833

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

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

0.508

5834

\[ {}y^{2}+12 x^{2} y+\left (2 x y+4 x^{3}\right ) y^{\prime } = 0 \]

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

0.677

5835

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

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

0.642

5836

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

[_rational]

1.308

5837

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

[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

0.158

5838

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

[_exact, _rational]

0.169

5839

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

[_linear]

1.556

5840

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

[_quadrature]

0.857

5841

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

[_Bernoulli]

2.449

5842

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

[_linear]

1.709

5843

\[ {}r^{\prime } = \left (r+{\mathrm e}^{-\theta }\right ) \tan \left (\theta \right ) \]

[_linear]

2.125

5844

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

[_linear]

1.859

5845

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

[_Bernoulli]

0.640

5846

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

[_rational, _Bernoulli]

8.898

5847

\[ {}\tan \left (\theta \right ) r^{\prime }-r = \tan \left (\theta \right )^{2} \]

[_linear]

1.836

5848

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

[[_linear, ‘class A‘]]

1.340

5849

\[ {}y^{\prime }+2 y = \frac {3 \,{\mathrm e}^{-2 x}}{4} \]

[[_linear, ‘class A‘]]

1.376

5850

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

[[_linear, ‘class A‘]]

1.483

5851

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

[_linear]

1.910

5852

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

[_linear]

2.507

5853

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

[_linear]

1.445

5854

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

[_linear]

1.731

5855

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

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

2.552

5856

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

[_Bernoulli]

2.596

5857

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

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

2.539

5858

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

[[_linear, ‘class A‘]]

1.499

5859

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

[_separable]

3.326

5860

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

[_Bernoulli]

22.290

5861

\[ {}\left (x -\cos \left (y\right )\right ) y^{\prime }+\tan \left (y\right ) = 0 \]
i.c.

[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

6.175

5862

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

[_rational, _Riccati]

1.256

5863

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

[_Riccati]

4.220

5864

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

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

2.217

5865

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

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

2.991

5866

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

[_Bernoulli]

2.049

5867

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

[‘y=_G(x,y’)‘]

2.162

5868

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

[[_1st_order, _with_linear_symmetries]]

4.486

5869

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

[[_homogeneous, ‘class C‘], _dAlembert]

1.717

5870

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

[_separable]

41.869

5871

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

[[_homogeneous, ‘class C‘], _dAlembert]

3.021

5872

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

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

8.457

5873

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

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

3.506

5874

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

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

5.820

5875

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

[[_1st_order, _with_linear_symmetries]]

1.473

5876

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

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

2.240

5877

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

[_exact]

36.058

5878

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

[[_1st_order, _with_linear_symmetries]]

1.566

5879

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

[_Bernoulli]

3.132

5880

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

[_separable]

2.172

5881

\[ {}y^{\prime }+a y = k \,{\mathrm e}^{b x} \]

[[_linear, ‘class A‘]]

1.040

5882

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

[[_homogeneous, ‘class C‘], _Riccati]

1.548

5883

\[ {}y^{\prime }+8 x^{3} y^{3}+2 x y = 0 \]

[_Bernoulli]

1.522

5884

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

[NONE]

48.997

5885

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

[[_linear, ‘class A‘]]

1.477

5886

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

[_separable]

1.877

5887

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

[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

2.124

5888

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

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

11.479

5889

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

[_linear]

1.888

5890

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

[[_homogeneous, ‘class G‘]]

2.086

5891

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

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

2.333

5892

\[ {}x y^{\prime }+a y+b \,x^{n} = 0 \]

[_linear]

1.228

5893

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

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

6.591

5894

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

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

77.861

5895

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

[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1.611

5896

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

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

2.226

5897

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

[_linear]

2.871

5898

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

[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

1.439

5899

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

[_separable]

3.033

5900

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

[_separable]

2.510