|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}x \left (x -2 y\right ) y^{\prime }+\left (2 x -y\right ) y = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.526 |
|
|
\[ {}x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y = 0 \] |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.194 |
|
|
\[ {}x \left (1-x -2 y\right ) y^{\prime }+\left (1+2 x +y\right ) y = 0 \] |
exactWithIntegrationFactor |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.174 |
|
|
\[ {}2 x \left (2 x^{2}+y\right ) y^{\prime }+\left (12 x^{2}+y\right ) y = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.415 |
|
|
\[ {}2 \left (1+x \right ) y y^{\prime }+2 x -3 x^{2}+y^{2} = 0 \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[_exact, _rational, _Bernoulli] |
✓ |
✓ |
0.92 |
|
|
\[ {}x \left (2 x +3 y\right ) y^{\prime } = y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.437 |
|
|
\[ {}x \left (2 x +3 y\right ) y^{\prime }+3 \left (x +y\right )^{2} = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.763 |
|
|
\[ {}\left (3+6 x y+x^{2}\right ) y^{\prime }+2 x +2 x y+3 y^{2} = 0 \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.997 |
|
|
\[ {}3 x \left (2 y+x \right ) y^{\prime }+x^{3}+3 y \left (y+2 x \right ) = 0 \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.068 |
|
|
\[ {}a x y y^{\prime } = x^{2}+y^{2} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.403 |
|
|
\[ {}a x y y^{\prime }+x^{2}-y^{2} = 0 \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.078 |
|
|
\[ {}x \left (a +b y\right ) y^{\prime } = c y \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.72 |
|
|
\[ {}x \left (x -a y\right ) y^{\prime } = y \left (-x a +y\right ) \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
3.023 |
|
|
\[ {}x \left (x^{n}+a y\right ) y^{\prime }+\left (b +c y\right ) y^{2} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
6.516 |
|
|
\[ {}\left (1-x^{2} y\right ) y^{\prime }+1-x y^{2} = 0 \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.925 |
|
|
\[ {}\left (1-x^{2} y\right ) y^{\prime }-1+x y^{2} = 0 \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.747 |
|
|
\[ {}x \left (1-x y\right ) y^{\prime }+\left (x y+1\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.041 |
|
|
\[ {}x \left (x y+2\right ) y^{\prime } = 3+2 x^{3}-2 y-x y^{2} \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.267 |
|
|
\[ {}x \left (2-x y\right ) y^{\prime }+2 y-x y^{2} \left (x y+1\right ) = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.848 |
|
|
\[ {}x \left (3-x y\right ) y^{\prime } = y \left (x y-1\right ) \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.184 |
|
|
\[ {}x^{2} \left (1-y\right ) y^{\prime }+\left (1-x \right ) y = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.03 |
|
|
\[ {}x^{2} \left (1-y\right ) y^{\prime }+\left (1+x \right ) y^{2} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.244 |
|
|
\[ {}\left (x^{2}+1\right ) y y^{\prime }+x \left (1-y^{2}\right ) = 0 \] |
exact, bernoulli, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.071 |
|
|
\[ {}\left (-x^{2}+1\right ) y y^{\prime }+2 x^{2}+x y^{2} = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
0.832 |
|
|
\[ {}2 x^{2} y y^{\prime } = x^{2} \left (2 x +1\right )-y^{2} \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.65 |
|
|
\[ {}x \left (1-2 x y\right ) y^{\prime }+y \left (2 x y+1\right ) = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.144 |
|
|
\[ {}x \left (2 x y+1\right ) y^{\prime }+\left (2+3 x y\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.707 |
|
|
\[ {}x \left (2 x y+1\right ) y^{\prime }+\left (1+2 x y-x^{2} y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘]] |
✓ |
✓ |
0.883 |
|
|
\[ {}x^{2} \left (x -2 y\right ) y^{\prime } = 2 x^{3}-4 x y^{2}+y^{3} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.968 |
|
|
\[ {}2 \left (1+x \right ) x y y^{\prime } = 1+y^{2} \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.658 |
|
|
\[ {}3 x^{2} y y^{\prime }+1+2 x y^{2} = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.836 |
|
|
\[ {}x^{2} \left (4 x -3 y\right ) y^{\prime } = \left (6 x^{2}-3 x y+2 y^{2}\right ) y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.125 |
|
|
\[ {}\left (1-x^{3} y\right ) y^{\prime } = x^{2} y^{2} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.39 |
|
|
\[ {}2 x^{3} y y^{\prime }+a +3 x^{2} y^{2} = 0 \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.829 |
|
|
\[ {}x \left (3-2 x^{2} y\right ) y^{\prime } = 4 x -3 y+3 x^{2} y^{2} \] |
exact |
[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
1.14 |
|
|
\[ {}x \left (3+2 x^{2} y\right ) y^{\prime }+\left (4+3 x^{2} y\right ) y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
6.575 |
|
|
\[ {}8 x^{3} y y^{\prime }+3 x^{4}-6 x^{2} y^{2}-y^{4} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.09 |
|
|
\[ {}x y \left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2} \] |
exact, bernoulli, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.143 |
|
|
\[ {}3 x^{4} y y^{\prime } = 1-2 x^{3} y^{2} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
0.97 |
|
|
\[ {}x^{7} y y^{\prime } = 2 x^{2}+2+5 x^{3} y \] |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.408 |
|
|
\[ {}y y^{\prime } \sqrt {x^{2}+1}+x \sqrt {1+y^{2}} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.895 |
|
|
\[ {}\left (y+1\right ) y^{\prime } \sqrt {x^{2}+1} = y^{3} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.23 |
|
|
\[ {}\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime } = \operatorname {f0} \left (x \right )+\operatorname {f1} \left (x \right ) y+\operatorname {f2} \left (x \right ) y^{2}+\operatorname {f3} \left (x \right ) y^{3} \] |
unknown |
[[_Abel, ‘2nd type‘, ‘class C‘]] |
❇ |
N/A |
1.832 |
|
|
\[ {}y^{2} y^{\prime }+x \left (2-y\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.56 |
|
|
\[ {}y^{2} y^{\prime } = x \left (1+y^{2}\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.704 |
|
|
\[ {}\left (x +y^{2}\right ) y^{\prime }+y = b x +a \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.427 |
|
|
\[ {}\left (x -y^{2}\right ) y^{\prime } = x^{2}-y \] |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
9.654 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+x y = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.749 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime } = x y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.924 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.559 |
|
|
\[ {}\left (x^{2}-y^{2}\right ) y^{\prime }+x \left (2 y+x \right ) = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
7.746 |
|
|
\[ {}\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (y+2 x \right ) = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
9.385 |
|
|
\[ {}\left (1-x^{2}+y^{2}\right ) y^{\prime } = 1+x^{2}-y^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
0.654 |
|
|
\[ {}\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+2 x y = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
8.592 |
|
|
\[ {}\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+b^{2}+x^{2}+2 x y = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.213 |
|
|
\[ {}\left (x +x^{2}+y^{2}\right ) y^{\prime } = y \] |
exactByInspection |
[_rational] |
✓ |
✓ |
0.873 |
|
|
\[ {}\left (3 x^{2}-y^{2}\right ) y^{\prime } = 2 x y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.833 |
|
|
\[ {}\left (x^{4}+y^{2}\right ) y^{\prime } = 4 x^{3} y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.571 |
|
|
\[ {}y \left (y+1\right ) y^{\prime } = \left (1+x \right ) x \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
165.359 |
|
|
\[ {}\left (x +2 y+y^{2}\right ) y^{\prime }+y \left (y+1\right )+\left (x +y\right )^{2} y^{2} = 0 \] |
unknown |
[_rational] |
✗ |
N/A |
1.139 |
|
|
\[ {}\left (x^{2}+2 y+y^{2}\right ) y^{\prime }+2 x = 0 \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
0.951 |
|
|
\[ {}\left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 x^{2} y = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
10.477 |
|
|
\[ {}\left (1+y+x y+y^{2}\right ) y^{\prime }+1+y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.296 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = a^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.724 |
|
|
\[ {}\left (x -y\right )^{2} y^{\prime } = a^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
2.306 |
|
|
\[ {}\left (x^{2}+2 x y-y^{2}\right ) y^{\prime }+x^{2}-2 x y+y^{2} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.846 |
|
|
\[ {}\left (x +y\right )^{2} y^{\prime } = x^{2}-2 x y+5 y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.983 |
|
|
\[ {}\left (a +b +x +y\right )^{2} y^{\prime } = 2 \left (y+a \right )^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
1.479 |
|
|
\[ {}\left (2 x^{2}+4 x y-y^{2}\right ) y^{\prime } = x^{2}-4 x y-2 y^{2} \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.232 |
|
|
\[ {}\left (3 x +y\right )^{2} y^{\prime } = 4 \left (3 x +2 y\right ) y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.04 |
|
|
\[ {}\left (1-3 x -y\right )^{2} y^{\prime } = \left (1-2 y\right ) \left (3-6 x -4 y\right ) \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
3.851 |
|
|
\[ {}\left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime } = y^{3} \csc \left (x \right ) \sec \left (x \right ) \] |
unknown |
[‘y=_G(x,y’)‘] |
❇ |
N/A |
37.644 |
|
|
\[ {}3 y^{2} y^{\prime } = 1+x +a y^{3} \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_rational, _Bernoulli] |
✓ |
✓ |
1.469 |
|
|
\[ {}\left (x^{2}-3 y^{2}\right ) y^{\prime }+1+2 x y = 0 \] |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
9.799 |
|
|
\[ {}\left (2 x^{2}+3 y^{2}\right ) y^{\prime }+x \left (3 x +y\right ) = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.871 |
|
|
\[ {}3 \left (x^{2}-y^{2}\right ) y^{\prime }+3 \,{\mathrm e}^{x}+6 x y \left (1+x \right )-2 y^{3} = 0 \] |
exactWithIntegrationFactor |
[‘y=_G(x,y’)‘] |
✓ |
✓ |
1.554 |
|
|
\[ {}\left (3 x^{2}+2 x y+4 y^{2}\right ) y^{\prime }+2 x^{2}+6 x y+y^{2} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.765 |
|
|
\[ {}\left (1-3 x +2 y\right )^{2} y^{\prime } = \left (4+2 x -3 y\right )^{2} \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational] |
✓ |
✓ |
5.158 |
|
|
\[ {}\left (1-3 x^{2} y+6 y^{2}\right ) y^{\prime }+x^{2}-3 x y^{2} = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.418 |
|
|
\[ {}\left (x -6 y\right )^{2} y^{\prime }+a +2 x y-6 y^{2} = 0 \] |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘]] |
✓ |
✓ |
1.193 |
|
|
\[ {}\left (x^{2}+a y^{2}\right ) y^{\prime } = x y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.386 |
|
|
\[ {}\left (x^{2}+x y+a y^{2}\right ) y^{\prime } = x^{2} a +x y+y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.31 |
|
|
\[ {}\left (x^{2} a +2 x y-a y^{2}\right ) y^{\prime }+x^{2}-2 a x y-y^{2} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
10.048 |
|
|
\[ {}\left (x^{2} a +2 b x y+c y^{2}\right ) y^{\prime }+k \,x^{2}+2 a x y+b y^{2} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.106 |
|
|
\[ {}x \left (1-y^{2}\right ) y^{\prime } = \left (x^{2}+1\right ) y \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.766 |
|
|
\[ {}x \left (3 x -y^{2}\right ) y^{\prime }+\left (5 x -2 y^{2}\right ) y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.765 |
|
|
\[ {}x \left (x^{2}+y^{2}\right ) y^{\prime } = \left (x^{2}+x^{4}+y^{2}\right ) y \] |
homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational] |
✓ |
✓ |
1.078 |
|
|
\[ {}x \left (1-x^{2}+y^{2}\right ) y^{\prime }+\left (1+x^{2}-y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.356 |
|
|
\[ {}x \left (a -x^{2}-y^{2}\right ) y^{\prime }+\left (a +x^{2}+y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
✓ |
5.602 |
|
|
\[ {}x \left (2 x^{2}+y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left (x \left (a -x^{2}-y^{2}\right )+y\right ) y^{\prime }+x -\left (a -x^{2}-y^{2}\right ) y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational] |
✓ |
✓ |
2.066 |
|
|
\[ {}x \left (y+a \right )^{2} y^{\prime } = b y^{2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.206 |
|
|
\[ {}x \left (x^{2}-x y+y^{2}\right ) y^{\prime }+\left (x^{2}+x y+y^{2}\right ) y = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.157 |
|
|
\[ {}x \left (x^{2}-x y-y^{2}\right ) y^{\prime } = \left (x^{2}+x y-y^{2}\right ) y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.309 |
|
|
\[ {}x \left (x^{2}+a x y+y^{2}\right ) y^{\prime } = \left (x^{2}+b x y+y^{2}\right ) y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.513 |
|
|
\[ {}x \left (x^{2}-2 y^{2}\right ) y^{\prime } = \left (2 x^{2}-y^{2}\right ) y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.329 |
|
|
\[ {}x \left (x^{2}+2 y^{2}\right ) y^{\prime } = \left (2 x^{2}+3 y^{2}\right ) y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.257 |
|
|
\[ {}2 x \left (5 x^{2}+y^{2}\right ) y^{\prime } = x^{2} y-y^{3} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.391 |
|
|
\[ {}x \left (x^{2}+a x y+2 y^{2}\right ) y^{\prime } = \left (x a +2 y\right ) y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.135 |
|
|
\[ {}3 x y^{2} y^{\prime } = 2 x -y^{3} \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
1.135 |
|
|
|
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