2.2.128 Problems 12701 to 12800

Table 2.257: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

12701

\[ {}y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.751

12702

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

[[_2nd_order, _with_linear_symmetries]]

1.092

12703

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

[[_2nd_order, _with_linear_symmetries]]

0.634

12704

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

[[_2nd_order, _with_linear_symmetries]]

0.983

12705

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

[[_2nd_order, _with_linear_symmetries]]

0.636

12706

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

[[_2nd_order, _with_linear_symmetries]]

0.699

12707

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

[[_2nd_order, _with_linear_symmetries]]

0.793

12708

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

[[_2nd_order, _with_linear_symmetries]]

0.716

12709

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

[[_2nd_order, _with_linear_symmetries]]

39.367

12710

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

[[_2nd_order, _with_linear_symmetries]]

0.761

12711

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

[[_2nd_order, _with_linear_symmetries]]

0.734

12712

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.691

12713

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

[[_2nd_order, _with_linear_symmetries]]

0.782

12714

\[ {}y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.880

12715

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.835

12716

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

[[_2nd_order, _with_linear_symmetries]]

0.973

12717

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right ) y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

1.383

12718

\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}+{\mathrm e}^{\lambda x} a c +b \mu \,{\mathrm e}^{\mu x}\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.904

12719

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

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

1.637

12720

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

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

43.760

12721

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

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

18.623

12722

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

[_linear]

2.728

12723

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

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

3.444

12724

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

[_separable]

7.178

12725

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

[_separable]

1.888

12726

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

[_separable]

5.588

12727

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

[_separable]

3.801

12728

\[ {}x \,{\mathrm e}^{\frac {y}{x}}+y-x y^{\prime } = 0 \]

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

16.189

12729

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

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

43.631

12730

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

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

2.414

12731

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

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

73.415

12732

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

[_separable]

5.238

12733

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

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

5.404

12734

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

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

2.743

12735

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

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

36.918

12736

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

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

2.055

12737

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

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

1.958

12738

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

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

10.855

12739

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

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

1.915

12740

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

[_linear]

1.838

12741

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

[_linear]

1.454

12742

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

[_linear]

1.619

12743

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

[_linear]

1.356

12744

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

[_linear]

1.832

12745

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

[_rational, _Bernoulli]

1.944

12746

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

[_separable]

2.131

12747

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

[_separable]

29.934

12748

\[ {}4 x y^{\prime }+3 y+{\mathrm e}^{x} x^{4} y^{5} = 0 \]

[_Bernoulli]

2.373

12749

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

[[_1st_order, _with_linear_symmetries]]

4.238

12750

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

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

4.432

12751

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

[_separable]

2.263

12752

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

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

5.063

12753

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

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

2.416

12754

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

[‘y=_G(x,y’)‘]

2.016

12755

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

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

39.520

12756

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

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

7.570

12757

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

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

2.360

12758

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

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

1.752

12759

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

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

41.014

12760

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

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

1.274

12761

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

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

2.426

12762

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

[_separable]

1.713

12763

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

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

12.822

12764

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

[_linear]

1.710

12765

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

[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

1.767

12766

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

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

38.083

12767

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

[_rational, _Riccati]

2.817

12768

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

[_separable]

3.731

12769

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

[_separable]

40.450

12770

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

[_linear]

2.175

12771

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

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

3.017

12772

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

[_Bernoulli]

3.761

12773

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

[_separable]

1.437

12774

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

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

3.166

12775

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

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

8.071

12776

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

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

88.928

12777

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

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

4.102

12778

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

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

3.908

12779

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

[_linear]

3.993

12780

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

[_separable]

2.275

12781

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

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

5.165

12782

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

[_linear]

2.039

12783

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

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

2.607

12784

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

[_linear]

2.404

12785

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

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

2.391

12786

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

[_separable]

1.351

12787

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

[_separable]

2.344

12788

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

[_rational]

3.010

12789

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

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

1.881

12790

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

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

50.793

12791

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

[_rational]

1.645

12792

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

[[_1st_order, _with_linear_symmetries]]

2.938

12793

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

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

5.369

12794

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

[_Bernoulli]

2.355

12795

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

[_rational]

3.082

12796

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

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

24.313

12797

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

[_quadrature]

0.491

12798

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

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

0.951

12799

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

[_quadrature]

0.646

12800

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

[_linear]

0.581