Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}2 x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
9.542 |
|
|
\[ {}\left (x +\sqrt {y^{2}-x y}\right ) y^{\prime }-y = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.41 |
|
|
\[ {}x +y-\left (x -y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.359 |
|
|
\[ {}x y^{\prime }-y-x \sin \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.28 |
|
|
\[ {}2 x^{2} y+y^{3}+\left (x y^{2}-2 x^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.447 |
|
|
\[ {}y^{2}+\left (x \sqrt {-x^{2}+y^{2}}-x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
5.064 |
|
|
\[ {}\frac {y \cos \left (\frac {y}{x}\right )}{x}-\left (\frac {x \sin \left (\frac {y}{x}\right )}{y}+\cos \left (\frac {y}{x}\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.806 |
|
|
\[ {}y+x \ln \left (\frac {y}{x}\right ) y^{\prime }-2 x y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.474 |
|
|
\[ {}2 y \,{\mathrm e}^{\frac {x}{y}}+\left (y-2 x \,{\mathrm e}^{\frac {x}{y}}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.985 |
|
|
\[ {}x \,{\mathrm e}^{\frac {y}{x}}-y \sin \left (\frac {y}{x}\right )+x \sin \left (\frac {y}{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.615 |
|
|
\[ {}x^{2}+y^{2} = 2 x y y^{\prime } \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.574 |
|
|
\[ {}x \,{\mathrm e}^{\frac {y}{x}}+y = x y^{\prime } \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.81 |
|
|
\[ {}y^{\prime }-\frac {y}{x}+\csc \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
2 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.86 |
|
|
\[ {}x y-y^{2}-x^{2} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.472 |
|
|
\[ {}x +2 y-4-\left (2 x -4 y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.886 |
|
|
\[ {}3 x +2 y+1-\left (3 x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.258 |
|
|
\[ {}x +y+1+\left (2 x +2 y+2\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.13 |
|
|
\[ {}x +y-1+\left (2 x +2 y-3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.178 |
|
|
\[ {}x +y-1-\left (x -y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.878 |
|
|
\[ {}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.141 |
|
|
\[ {}7 y-3+\left (2 x +1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeMapleC, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.137 |
|
|
\[ {}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.178 |
|
|
\[ {}x +2 y+\left (y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.835 |
|
|
\[ {}3 x -2 y+4-\left (2 x +7 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.572 |
|
|
\[ {}x +y+\left (3 x +3 y-4\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.444 |
|
|
\[ {}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.75 |
|
|
\[ {}y+7+\left (2 x +y+3\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.445 |
|
|
\[ {}x +y+2-\left (x -y-4\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.932 |
|
|
\[ {}3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
0.43 |
|
|
\[ {}\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.494 |
|
|
\[ {}2 x y+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.262 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.45 |
|
|
\[ {}\cos \left (y\right )-\left (x \sin \left (y\right )-y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.364 |
|
|
\[ {}x -2 x y+{\mathrm e}^{y}+\left (x \,{\mathrm e}^{y}+y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
1.805 |
|
|
\[ {}x^{2}-x +y^{2}-\left ({\mathrm e}^{y}-2 x y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
1.616 |
|
|
\[ {}2 x +y \cos \left (x \right )+\left (2 y+\sin \left (x \right )-\sin \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.403 |
|
|
\[ {}x \sqrt {x^{2}+y^{2}}-\frac {x^{2} y y^{\prime }}{y-\sqrt {x^{2}+y^{2}}} = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class A‘], _exact, _dAlembert] |
✓ |
✓ |
0.998 |
|
|
\[ {}4 x^{3}-\sin \left (x \right )+y^{3}-\left (y^{2}+1-3 x y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact] |
✓ |
✓ |
0.685 |
|
|
\[ {}{\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact, _Bernoulli] |
✓ |
✓ |
1.176 |
|
|
\[ {}\sin \left (x \right ) \cos \left (y\right )+\cos \left (x \right ) \sin \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
31.728 |
|
|
\[ {}y^{2} {\mathrm e}^{x y^{2}}+4 x^{3}+\left (2 x y \,{\mathrm e}^{x y^{2}}-3 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
3.813 |
|
|
\[ {}y^{2}+y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
1.007 |
|
|
\[ {}y \sec \left (x \right )+\sin \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.883 |
|
|
\[ {}{\mathrm e}^{x}-\sin \left (y\right )+\cos \left (y\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
0.29 |
|
|
\[ {}x y+\left (x^{2}+1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.407 |
|
|
\[ {}y^{3}+x y^{2}+y+\left (x^{3}+x^{2} y+x \right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.377 |
|
|
\[ {}3 y-x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.804 |
|
|
\[ {}y-3 x y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_separable] |
✓ |
✓ |
0.34 |
|
|
\[ {}y \left (2 y^{3} x^{2}+3\right )+x \left (y^{3} x^{2}-1\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.148 |
|
|
\[ {}2 x y+x^{2}+\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
0.328 |
|
|
\[ {}x^{2}+y \cos \left (x \right )+\left (y^{3}+\sin \left (x \right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.353 |
|
|
\[ {}x^{2}+y^{2}+x +x y y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[_rational, _Bernoulli] |
✓ |
✓ |
0.24 |
|
|
\[ {}x -2 x y+{\mathrm e}^{y}+\left (x \,{\mathrm e}^{y}+y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.333 |
|
|
\[ {}{\mathrm e}^{x} \sin \left (y\right )+{\mathrm e}^{-y}-\left (x \,{\mathrm e}^{-y}-{\mathrm e}^{x} \cos \left (y\right )\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.341 |
|
|
\[ {}x^{2}-y^{2}-y-\left (x^{2}-y^{2}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.471 |
|
|
\[ {}x^{4} y^{2}-y+\left (y^{4} x^{2}-x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_rational] |
✓ |
✓ |
0.397 |
|
|
\[ {}y \left (2 x +y^{3}\right )-x \left (2 x -y^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.32 |
|
|
\[ {}\arctan \left (x y\right )+\frac {x y-2 x y^{2}}{1+x^{2} y^{2}}+\frac {\left (x^{2}-2 x^{2} y\right ) y^{\prime }}{1+x^{2} y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
0.739 |
|
|
\[ {}{\mathrm e}^{x} \left (1+x \right )+\left (y \,{\mathrm e}^{y}-x \,{\mathrm e}^{x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
0.317 |
|
|
\[ {}\frac {1+x y}{y}+\frac {\left (2 y-x \right ) y^{\prime }}{y^{2}} = 0 \] |
1 |
1 |
1 |
exact |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
0.508 |
|
|
\[ {}y^{2}-3 x y-2 x^{2}+\left (x y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.293 |
|
|
\[ {}y \left (y+2 x +1\right )-x \left (x +2 y-1\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.45 |
|
|
\[ {}y \left (2 x -y-1\right )+x \left (2 y-x -1\right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.445 |
|
|
\[ {}y^{2}+12 x^{2} y+\left (2 x y+4 x^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.308 |
|
|
\[ {}3 \left (x +y\right )^{2}+x \left (2 x +3 y\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.264 |
|
|
\[ {}y-\left (x +x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.223 |
|
|
\[ {}2 x y+\left (x^{2}+y^{2}+a \right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.378 |
|
|
\[ {}2 x y+x^{2}+b +\left (x^{2}+y^{2}+a \right ) y^{\prime } = 0 \] |
1 |
1 |
3 |
exact |
[_exact, _rational] |
✓ |
✓ |
0.435 |
|
|
\[ {}x y^{\prime }+y = x^{3} \] |
1 |
1 |
1 |
exact, linear, differentialType, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.864 |
|
|
\[ {}y^{\prime }+a y = b \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.48 |
|
|
\[ {}x y^{\prime }+y = y^{2} \ln \left (x \right ) \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.133 |
|
|
\[ {}x^{\prime }+2 x y = {\mathrm e}^{-y^{2}} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.774 |
|
|
\[ {}r^{\prime } = \left (r+{\mathrm e}^{-\theta }\right ) \tan \left (\theta \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.231 |
|
|
\[ {}y^{\prime }-\frac {2 x y}{x^{2}+1} = 1 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.866 |
|
|
\[ {}y^{\prime }+y = x y^{3} \] |
1 |
2 |
2 |
bernoulli |
[_Bernoulli] |
✓ |
✓ |
0.512 |
|
|
\[ {}\left (-x^{3}+1\right ) y^{\prime }-2 \left (1+x \right ) y = y^{\frac {5}{2}} \] |
1 |
3 |
1 |
bernoulli |
[_rational, _Bernoulli] |
✓ |
✓ |
19.07 |
|
|
\[ {}\tan \left (\theta \right ) r^{\prime }-r = \tan \left (\theta \right )^{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.508 |
|
|
\[ {}y^{\prime }+2 y = 3 \,{\mathrm e}^{-2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.737 |
|
|
\[ {}y^{\prime }+2 y = \frac {3 \,{\mathrm e}^{-2 x}}{4} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime }+2 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.939 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = {\mathrm e}^{2 x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.483 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = \frac {\sin \left (2 x \right )}{2} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.425 |
|
|
\[ {}x y^{\prime }+y = x \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
0.918 |
|
|
\[ {}-y+x y^{\prime } = x^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.083 |
|
|
\[ {}x y^{\prime }+x y^{2}-y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
0.94 |
|
|
\[ {}x y^{\prime }-y \left (2 y \ln \left (x \right )-1\right ) = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.199 |
|
|
\[ {}x^{2} \left (-1+x \right ) y^{\prime }-y^{2}-x \left (-2+x \right ) y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.212 |
|
|
\[ {}y^{\prime }-y = {\mathrm e}^{x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
0.975 |
|
|
\[ {}y^{\prime }+\frac {y}{x} = \frac {y^{2}}{x} \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.67 |
|
|
\[ {}2 \cos \left (x \right ) y^{\prime } = y \sin \left (x \right )-y^{3} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
7.613 |
|
|
\[ {}\left (x -\cos \left (y\right )\right ) y^{\prime }+\tan \left (y\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
12.745 |
|
|
\[ {}y^{\prime } = x^{3}+\frac {2 y}{x}-\frac {y^{2}}{x} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
1.318 |
|
|
\[ {}y^{\prime } = 2 \tan \left (x \right ) \sec \left (x \right )-y^{2} \sin \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.563 |
|
|
\[ {}y^{\prime } = \frac {1}{x^{2}}-\frac {y}{x}-y^{2} \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
2.769 |
|
|
\[ {}y^{\prime } = 1+\frac {y}{x}-\frac {y^{2}}{x^{2}} \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
3.303 |
|
|
\[ {}2 x y y^{\prime }+\left (1+x \right ) y^{2} = {\mathrm e}^{x} \] |
1 |
1 |
2 |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.655 |
|
|
\[ {}\cos \left (y\right ) y^{\prime }+\sin \left (y\right ) = x^{2} \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.981 |
|
|
\[ {}\left (1+x \right ) y^{\prime }-y-1 = \left (1+x \right ) \sqrt {y+1} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
3.615 |
|
|
\[ {}{\mathrm e}^{y} \left (1+y^{\prime }\right ) = {\mathrm e}^{x} \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.761 |
|
|
\[ {}y^{\prime } \sin \left (y\right )+\sin \left (x \right ) \cos \left (y\right ) = \sin \left (x \right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
32.585 |
|
|
\[ {}\left (x -y\right )^{2} y^{\prime } = 4 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.873 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}+y^{2}} \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.292 |
|
|
\[ {}\left (3 x +2 y+1\right ) y^{\prime }+4 x +3 y+2 = 0 \] |
1 |
1 |
1 |
exact, differentialType, homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.098 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.274 |
|
|
\[ {}y+\left (1+y^{2} {\mathrm e}^{2 x}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.566 |
|
|
\[ {}x^{2} y+y^{2}+x^{3} y^{\prime } = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.845 |
|
|
\[ {}y^{2} {\mathrm e}^{x y^{2}}+4 x^{3}+\left (2 x y \,{\mathrm e}^{x y^{2}}-3 y^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact |
[_exact] |
✓ |
✓ |
4.291 |
|
|
\[ {}y^{\prime } = \left (x^{2}+2 y-1\right )^{\frac {2}{3}}-x \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
2.453 |
|
|
\[ {}x y^{\prime }+y = x^{2} \left (1+{\mathrm e}^{x}\right ) y^{2} \] |
1 |
1 |
1 |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.166 |
|
|
\[ {}2 y-x y \ln \left (x \right )-2 x \ln \left (x \right ) y^{\prime } = 0 \] |
1 |
1 |
1 |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.997 |
|
|
\[ {}y^{\prime }+a y = k \,{\mathrm e}^{b x} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.028 |
|
|
\[ {}y^{\prime } = \left (x +y\right )^{2} \] |
1 |
1 |
1 |
riccati, homogeneousTypeC, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class C‘], _Riccati] |
✓ |
✓ |
0.852 |
|
|
\[ {}y^{\prime }+8 x^{3} y^{3}+2 x y = 0 \] |
1 |
2 |
2 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.96 |
|
|
\[ {}\left (x y \sqrt {x^{2}-y^{2}}+x \right ) y^{\prime } = y-x^{2} \sqrt {x^{2}-y^{2}} \] |
1 |
0 |
1 |
unknown |
[NONE] |
✗ |
N/A |
2.7 |
|
|
\[ {}y^{\prime }+a y = b \sin \left (k x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.237 |
|
|
\[ {}x y^{\prime }-y^{2}+1 = 0 \] |
1 |
1 |
1 |
exact, riccati, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.028 |
|
|
\[ {}\left (y^{2}+a \sin \left (x \right )\right ) y^{\prime } = \cos \left (x \right ) \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.471 |
|
|
\[ {}x y^{\prime } = x \,{\mathrm e}^{\frac {y}{x}}+x +y \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.658 |
|
|
\[ {}y^{\prime }+y \cos \left (x \right ) = {\mathrm e}^{-\sin \left (x \right )} \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.011 |
|
|
\[ {}x y^{\prime }-y \left (\ln \left (x y\right )-1\right ) = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.353 |
|
|
\[ {}x^{3} y^{\prime }-y^{2}-x^{2} y = 0 \] |
1 |
1 |
1 |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.043 |
|
|
\[ {}x y^{\prime }+a y+b \,x^{n} = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.145 |
|
|
\[ {}x y^{\prime }-y-x \sin \left (\frac {y}{x}\right ) = 0 \] |
1 |
1 |
1 |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.015 |
|
|
\[ {}y^{2}-3 x y-2 x^{2}+\left (x y-x^{2}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.047 |
|
|
\[ {}\left (6 x y+x^{2}+3\right ) y^{\prime }+3 y^{2}+2 x y+2 x = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.602 |
|
|
\[ {}x^{2} y^{\prime }+y^{2}+x y+x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
✓ |
1.434 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+2 x y-\cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.146 |
|
|
\[ {}\left (x^{2} y-1\right ) y^{\prime }+x y^{2}-1 = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.329 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+x y-3 x y^{2} = 0 \] |
1 |
1 |
1 |
exact, riccati, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
6.486 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }-2 x y \ln \left (y\right ) = 0 \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.984 |
|
|
\[ {}\left (1+x^{2}+y^{2}\right ) y^{\prime }+2 x y+x^{2}+3 = 0 \] |
1 |
1 |
3 |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
14.271 |
|
|
\[ {}\cos \left (x \right ) y^{\prime }+y+\left (\sin \left (x \right )+1\right ) \cos \left (x \right ) = 0 \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
3.275 |
|
|
\[ {}y^{2}+12 x^{2} y+\left (2 x y+4 x^{3}\right ) y^{\prime } = 0 \] |
1 |
1 |
2 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.901 |
|
|
\[ {}\left (x^{2}-y\right ) y^{\prime }+x = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
1.16 |
|
|
\[ {}\left (x^{2}-y\right ) y^{\prime }-4 x y = 0 \] |
1 |
1 |
2 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.718 |
|
|
\[ {}x y y^{\prime }+x^{2}+y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.011 |
|
|
\[ {}2 x y y^{\prime }+3 x^{2}-y^{2} = 0 \] |
1 |
1 |
2 |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.442 |
|
|
\[ {}\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4} = 0 \] |
1 |
1 |
3 |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
10.768 |
|
|
\[ {}\left (x y-1\right )^{2} x y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \] |
1 |
1 |
1 |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.541 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right ) = 0 \] |
1 |
1 |
3 |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
12.457 |
|
|
\[ {}3 x y^{2} y^{\prime }+y^{3}-2 x = 0 \] |
1 |
1 |
3 |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.956 |
|
|
\[ {}2 y^{3} y^{\prime }+x y^{2}-x^{3} = 0 \] |
1 |
1 |
6 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
6.738 |
|
|
\[ {}\left (2 x y^{3}+x y+x^{2}\right ) y^{\prime }-x y+y^{2} = 0 \] |
1 |
1 |
1 |
exactByInspection |
[_rational] |
✓ |
✓ |
1.994 |
|
|
\[ {}\left (2 y^{3}+y\right ) y^{\prime }-2 x^{3}-x = 0 \] |
1 |
1 |
4 |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
4.438 |
|
|
\[ {}y^{\prime }-{\mathrm e}^{x -y}+{\mathrm e}^{x} = 0 \] |
1 |
1 |
1 |
exact, separable, first order special form ID 1, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.698 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.281 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.269 |
|
|
\[ {}y^{\prime \prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.059 |
|
|
\[ {}6 y^{\prime \prime }-11 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.289 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }-y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.334 |
|
|
\[ {}y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }-6 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.351 |
|
|
\[ {}y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-4 y^{\prime \prime }+4 y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.291 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+y^{\prime \prime }-4 y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.482 |
|
|
\[ {}y^{\prime \prime \prime \prime }-a^{2} y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.123 |
|
|
\[ {}y^{\prime \prime }-2 k y^{\prime }-2 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.387 |
|
|
\[ {}y^{\prime \prime }+4 k y^{\prime }-12 k^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.437 |
|
|
\[ {}y^{\prime \prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _quadrature]] |
✓ |
✓ |
0.188 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.344 |
|
|
\[ {}3 y^{\prime \prime \prime }+5 y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.312 |
|
|
\[ {}y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.105 |
|
|
\[ {}y^{\prime \prime }-2 a y^{\prime }+a^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.34 |
|
|
\[ {}y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.228 |
|
|
\[ {}y^{\prime \prime \prime \prime }-2 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.304 |
|
|
\[ {}y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-11 y^{\prime \prime }-12 y^{\prime }+36 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.394 |
|
|
\[ {}36 y^{\prime \prime \prime \prime }-37 y^{\prime \prime }+4 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.322 |
|
|
\[ {}y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+36 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
1.421 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.697 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.815 |
|
|
\[ {}y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+6 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.702 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime }+4 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.132 |
|
|
\[ {}y^{\prime \prime \prime }+8 y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.583 |
|
|
\[ {}y^{\prime \prime \prime \prime }+4 y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.363 |
|
|
\[ {}y^{\left (5\right )}+2 y^{\prime \prime \prime }+y^{\prime } = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_high_order, _missing_x]] |
✓ |
✓ |
0.151 |
|
|
\[ {}y^{\prime \prime } = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_quadrature, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _quadrature]] |
✓ |
✓ |
1.694 |
|
|
\[ {}y^{\prime \prime }+4 y^{\prime }+4 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.812 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.736 |
|
|
\[ {}y^{\prime \prime }-4 y^{\prime }+20 y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.279 |
|
|
\[ {}3 y^{\prime \prime \prime }+5 y^{\prime \prime }+y^{\prime }-y = 0 \] |
1 |
1 |
1 |
higher_order_linear_constant_coefficients_ODE |
[[_3rd_order, _missing_x]] |
✓ |
✓ |
0.64 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 4 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.433 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.566 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{i x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.761 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.567 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.55 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 8+6 \,{\mathrm e}^{x}+2 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }+y = x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.218 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }-8 y = 9 x \,{\mathrm e}^{x}+10 \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.627 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime } = 2 \,{\mathrm e}^{2 x} \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
3.536 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x^{2}+2 x \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.964 |
|
|
\[ {}y^{\prime \prime }+y^{\prime } = x +\sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_linear_constant_coeff |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
4.503 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.981 |
|
|
\[ {}y^{\prime \prime }+4 y = x \sin \left (2 x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.232 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.862 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = {\mathrm e}^{-2 x}+x^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.629 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = x \,{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.501 |
|
|
\[ {}y^{\prime \prime }+y^{\prime }-6 y = x +{\mathrm e}^{2 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.683 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )+{\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.292 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.139 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (2 x \right ) \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.018 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }-6 y = {\mathrm e}^{3 x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }-2 y = 5 \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.852 |
|
|
\[ {}y^{\prime \prime }+9 y = 8 \cos \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
1.328 |
|
|
\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = {\mathrm e}^{x} \left (2 x -3\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.731 |
|
|
\[ {}y^{\prime \prime }+y = \cot \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.792 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.793 |
|
|
\[ {}y^{\prime \prime }-y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.808 |
|
|
\[ {}y^{\prime \prime }+y = \sin \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.592 |
|
|
\[ {}y^{\prime \prime }+3 y^{\prime }+2 y = 12 \,{\mathrm e}^{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.459 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.574 |
|
|
\[ {}y^{\prime \prime }+y = 4 x \sin \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.622 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = {\mathrm e}^{-x} \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.709 |
|
|
\[ {}y^{\prime \prime }+y = \csc \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.672 |
|
|
\[ {}y^{\prime \prime }+y = \tan \left (x \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.97 |
|
|
\[ {}y^{\prime \prime }+2 y^{\prime }+y = \frac {{\mathrm e}^{-x}}{x} \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.62 |
|
|
\[ {}y^{\prime \prime }+y = \sec \left (x \right ) \csc \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.972 |
|
|
\[ {}y^{\prime \prime }-2 y^{\prime }+y = {\mathrm e}^{x} \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.708 |
|
|
\[ {}y^{\prime \prime }-3 y^{\prime }+2 y = \cos \left ({\mathrm e}^{-x}\right ) \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
0.796 |
|
|
\[ {}x^{2} y^{\prime \prime }-x y^{\prime }+y = x \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.695 |
|
|
\[ {}y^{\prime \prime }-\frac {2 y^{\prime }}{x}+\frac {2 y}{x^{2}} = x \ln \left (x \right ) \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2, linear_second_order_ode_solved_by_an_integrating_factor, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.991 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-4 y = x^{3} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.967 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-y = x^{2} {\mathrm e}^{-x} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
4.137 |
|
|
\[ {}2 x^{2} y^{\prime \prime }+3 x y^{\prime }-y = \frac {1}{x} \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, exact linear second order ode, second_order_integrable_as_is, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
2.844 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
1 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.987 |
|
|
\[ {}y^{3} y^{\prime \prime } = k \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
2.782 |
|
|
\[ {}y y^{\prime \prime } = {y^{\prime }}^{2}-1 \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
8.385 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.115 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.628 |
|
|
\[ {}\left (y+1\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2} \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.61 |
|
|
\[ {}r^{\prime \prime } = -\frac {k}{r^{2}} \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
4.896 |
|
|
\[ {}y^{\prime \prime } = \frac {3 k y^{2}}{2} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
14.666 |
|
|
\[ {}y^{\prime \prime } = 2 k y^{3} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
7.368 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{2}-y^{\prime } = 0 \] |
1 |
1 |
2 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
4.184 |
|
|
\[ {}r^{\prime \prime } = \frac {h^{2}}{r^{3}}-\frac {k}{r^{2}} \] |
1 |
2 |
2 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
8.885 |
|
|
\[ {}y y^{\prime \prime }+{y^{\prime }}^{3}-{y^{\prime }}^{2} = 0 \] |
1 |
1 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
2.019 |
|
|
\[ {}y y^{\prime \prime }-3 {y^{\prime }}^{2} = 0 \] |
1 |
2 |
3 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
2.004 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1 = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]] |
✓ |
✓ |
1.483 |
|
|
\[ {}\left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (1+y^{\prime }\right ) = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
2.169 |
|
|
\[ {}\left (y+1\right ) y^{\prime \prime } = 3 {y^{\prime }}^{2} \] |
1 |
1 |
1 |
second_order_ode_missing_x |
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.104 |
|
|
\[ {}y^{\prime \prime } = y^{\prime } {\mathrm e}^{y} \] |
1 |
0 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
3.116 |
|
|
\[ {}y^{\prime \prime } = 2 y y^{\prime } \] |
1 |
0 |
1 |
second_order_integrable_as_is, second_order_ode_missing_x, exact nonlinear second order ode |
[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
1.908 |
|
|
\[ {}2 y^{\prime \prime } = {\mathrm e}^{y} \] |
1 |
2 |
1 |
second_order_ode_missing_x, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]] |
✓ |
✓ |
63.523 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime } = 1 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.986 |
|
|
\[ {}x y^{\prime \prime }-y^{\prime } = x^{2} \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is, second_order_ode_missing_y, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
1.478 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+y y^{\prime } = 0 \] |
1 |
1 |
2 |
second_order_nonlinear_solved_by_mainardi_lioville_method |
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.139 |
|
|
\[ {}x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime } = 0 \] |
1 |
1 |
3 |
second_order_integrable_as_is, second_order_nonlinear_solved_by_mainardi_lioville_method |
[[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✓ |
✓ |
0.439 |
|
|
\[ {}x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime } = 0 \] |
1 |
0 |
2 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]] |
✗ |
N/A |
0.03 |
|
|
|
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