2.2.106 Problems 10501 to 10600

Table 2.213: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

10501

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

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

11.561

10502

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

[‘y=_G(x,y’)‘]

85.826

10503

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

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

8.359

10504

\[ {}\left (\operatorname {b2} y+\operatorname {a2} x +\operatorname {c2} \right )^{2} {y^{\prime }}^{2}+\left (\operatorname {a1} x +\operatorname {b1} y+\operatorname {c1} \right ) y^{\prime }+\operatorname {b0} y+\operatorname {a0} +\operatorname {c0} = 0 \]

[_rational]

485.372

10505

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

[_rational]

18.070

10506

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

[_separable]

1.613

10507

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

[‘y=_G(x,y’)‘]

52.608

10508

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

[‘y=_G(x,y’)‘]

36.595

10509

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

[‘y=_G(x,y’)‘]

11.749

10510

\[ {}9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2} = 0 \]

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

16.229

10511

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

[‘y=_G(x,y’)‘]

152.391

10512

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

[[_1st_order, _with_linear_symmetries]]

163.385

10513

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

[[_1st_order, _with_linear_symmetries]]

56.882

10514

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

[‘y=_G(x,y’)‘]

151.717

10515

\[ {}{y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d = 0 \]

[_quadrature]

9.953

10516

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

[[_1st_order, _with_linear_symmetries]]

12.780

10517

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

[[_homogeneous, ‘class A‘]]

97.579

10518

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

[[_homogeneous, ‘class A‘]]

71.417

10519

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

[_quadrature]

6.014

10520

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

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

1.687

10521

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

[_quadrature]

0.441

10522

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.417

10523

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.432

10524

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

[_quadrature]

0.760

10525

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

[_quadrature]

70.127

10526

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

[_separable]

0.719

10527

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

[_quadrature]

1.145

10528

\[ {}{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5} = 0 \]

[[_1st_order, _with_linear_symmetries]]

13.336

10529

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

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

0.831

10530

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

[_dAlembert]

2.913

10531

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

[_quadrature]

68.504

10532

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

[‘y=_G(x,y’)‘]

44.756

10533

\[ {}a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d = 0 \]

[_quadrature]

0.646

10534

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.742

10535

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.754

10536

\[ {}8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y = 0 \]

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.773

10537

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

[_quadrature]

0.967

10538

\[ {}x^{3} {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+\left (3 x y^{2}+x^{6}\right ) y^{\prime }-y^{3}-2 x^{5} y = 0 \]

[[_1st_order, _with_linear_symmetries]]

12.964

10539

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

[[_homogeneous, ‘class G‘]]

15.265

10540

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

[_quadrature]

0.918

10541

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

[_quadrature]

1.822

10542

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

[[_1st_order, _with_linear_symmetries]]

108.116

10543

\[ {}16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y = 0 \]

[[_1st_order, _with_linear_symmetries]]

108.296

10544

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

[‘y=_G(x,y’)‘]

259.335

10545

\[ {}x^{7} y^{2} {y^{\prime }}^{3}-\left (3 x^{6} y^{3}-1\right ) {y^{\prime }}^{2}+3 x^{5} y^{4} y^{\prime }-x^{4} y^{5} = 0 \]

[[_homogeneous, ‘class G‘]]

119.799

10546

\[ {}{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2} = 0 \]

[_quadrature]

2.009

10547

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

[_dAlembert]

36.070

10548

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

[[_homogeneous, ‘class G‘]]

0.721

10549

\[ {}{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3} = 0 \]

[_quadrature]

4.776

10550

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

[_quadrature]

3.888

10551

\[ {}{y^{\prime }}^{r}-a y^{s}-b \,x^{\frac {r s}{r -s}} = 0 \]

[[_homogeneous, ‘class G‘]]

2.789

10552

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

[_separable]

23.102

10553

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

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

1.836

10554

\[ {}a {y^{\prime }}^{m}+b {y^{\prime }}^{n}-y = 0 \]

[_quadrature]

1.697

10555

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

[‘y=_G(x,y’)‘]

2.136

10556

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

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

1.806

10557

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

[_dAlembert]

40.149

10558

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

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

8.684

10559

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

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

39.829

10560

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

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

11.673

10561

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

[‘y=_G(x,y’)‘]

39.333

10562

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

[[_1st_order, _with_linear_symmetries]]

14.404

10563

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

[_dAlembert]

56.176

10564

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

4.574

10565

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

1.756

10566

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

[_separable]

2.856

10567

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

[_quadrature]

0.584

10568

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

[_quadrature]

0.472

10569

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

[_quadrature]

1.725

10570

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

[_Clairaut]

9.402

10571

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

[_quadrature]

1.204

10572

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

[‘y=_G(x,y’)‘]

1.304

10573

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

[‘y=_G(x,y’)‘]

0.458

10574

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

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

0.884

10575

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

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

0.895

10576

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

[[_1st_order, _with_linear_symmetries]]

0.757

10577

\[ {}y^{\prime } = -\frac {a x}{2}+F \left (y+\frac {a \,x^{2}}{4}+\frac {b x}{2}\right ) \]

[[_1st_order, _with_linear_symmetries]]

0.951

10578

\[ {}y^{\prime } = F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x} \]

[[_1st_order, _with_linear_symmetries]]

0.921

10579

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

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

2.606

10580

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

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

2.619

10581

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

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

2.656

10582

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

[[_1st_order, _with_linear_symmetries]]

1.087

10583

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

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

3.753

10584

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

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

4.006

10585

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

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

3.037

10586

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

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

2.209

10587

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

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

1.545

10588

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

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

2.635

10589

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

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

2.591

10590

\[ {}y^{\prime } = \frac {6 x^{3}+5 \sqrt {x}+5 F \left (y-\frac {2 x^{3}}{5}-2 \sqrt {x}\right )}{5 x} \]

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

3.584

10591

\[ {}y^{\prime } = \frac {F \left (y^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{x}}{\sqrt {y}} \]

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

3.152

10592

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

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

2.383

10593

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

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

2.580

10594

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

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

1.500

10595

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

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

2.591

10596

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

[[_homogeneous, ‘class D‘]]

1.834

10597

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

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

2.578

10598

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

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

1.641

10599

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

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

2.312

10600

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

[NONE]

2.911