2.2.56 Problems 5501 to 5600

Table 2.113: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

5501

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

[_separable]

0.917

5502

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

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

0.855

5503

\[ {}4 x^{2} {y^{\prime }}^{2}-4 x y y^{\prime } = 8 x^{3}-y^{2} \]

[_linear]

0.531

5504

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

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

3.658

5505

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

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

71.651

5506

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

[_quadrature]

0.335

5507

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

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

1.229

5508

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

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

3.262

5509

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

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

12.957

5510

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

[_quadrature]

1.055

5511

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

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

1.912

5512

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

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

1.254

5513

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

[[_homogeneous, ‘class G‘]]

2.961

5514

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

[_quadrature]

0.681

5515

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

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

1.028

5516

\[ {}4 x^{5} {y^{\prime }}^{2}+12 x^{4} y y^{\prime }+9 = 0 \]

[[_homogeneous, ‘class G‘]]

4.053

5517

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

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

1.846

5518

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

[[_homogeneous, ‘class G‘]]

1.959

5519

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

[_quadrature]

0.389

5520

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

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

1.536

5521

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

[[_1st_order, _with_linear_symmetries]]

1.169

5522

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

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

1.865

5523

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

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

1.943

5524

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

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

1.574

5525

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

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

1.461

5526

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

[[_1st_order, _with_linear_symmetries]]

2.905

5527

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

[_quadrature]

0.549

5528

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

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

1.900

5529

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

[_quadrature]

0.436

5530

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

[_quadrature]

0.630

5531

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

[_quadrature]

0.689

5532

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

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

1.790

5533

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

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

0.804

5534

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

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

0.764

5535

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

[[_1st_order, _with_linear_symmetries]]

3.119

5536

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

[_quadrature]

0.586

5537

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

[_quadrature]

0.834

5538

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

[_quadrature]

0.494

5539

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

[_separable]

1.467

5540

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

[_separable]

1.208

5541

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

[_separable]

1.342

5542

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

[_rational]

141.127

5543

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

[_rational]

101.730

5544

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

[_separable]

1.424

5545

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

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

150.813

5546

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

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

72.950

5547

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

[_quadrature]

0.651

5548

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

[_quadrature]

0.546

5549

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

[[_1st_order, _with_linear_symmetries], _rational]

3.476

5550

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

[[_1st_order, _with_linear_symmetries]]

2.993

5551

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

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

72.300

5552

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

[_quadrature]

0.616

5553

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

[_separable]

0.720

5554

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

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

31.248

5555

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

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

13.391

5556

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

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

80.392

5557

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

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

10.843

5558

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

[_quadrature]

30.265

5559

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

[_quadrature]

0.681

5560

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

[‘y=_G(x,y’)‘]

84.885

5561

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

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

305.651

5562

\[ {}\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+x^{2}+\left (-4 a^{2}+1\right ) y^{2} = 0 \]

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

340.019

5563

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

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

191.463

5564

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

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

2.783

5565

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

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

2.844

5566

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

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

7.540

5567

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

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

143.572

5568

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

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

3.462

5569

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

[_separable]

0.887

5570

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

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

10.924

5571

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

[[_1st_order, _with_linear_symmetries], _rational]

3.488

5572

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

[_quadrature]

0.398

5573

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

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

327.426

5574

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

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

11.929

5575

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

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

8.647

5576

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

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

2.966

5577

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

[_rational]

18.773

5578

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

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

3.247

5579

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

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

150.332

5580

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

[[_1st_order, _with_linear_symmetries], _rational]

3.559

5581

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

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

3.509

5582

\[ {}9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }-a = 0 \]

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

3.444

5583

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

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

16.871

5584

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

[_quadrature]

0.406

5585

\[ {}{y^{\prime }}^{3} = a \,x^{n} \]

[_quadrature]

0.529

5586

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

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

0.473

5587

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

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

1.311

5588

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

[_quadrature]

6.161

5589

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

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

1.569

5590

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

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

3.332

5591

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

[_quadrature]

0.809

5592

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

[_quadrature]

0.453

5593

\[ {}{y^{\prime }}^{3}+y^{\prime } = {\mathrm e}^{y} \]

[_quadrature]

0.687

5594

\[ {}{y^{\prime }}^{3}-7 y^{\prime }+6 = 0 \]

[_quadrature]

0.705

5595

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.685

5596

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.469

5597

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

0.441

5598

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

[_quadrature]

0.779

5599

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.503

5600

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

[[_1st_order, _with_linear_symmetries], _Clairaut]

0.511