|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left (x +y\right ) y^{\prime }+x = y \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.139 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.856 |
|
|
\[ {}y^{\prime } = \frac {2 x -y}{x +4 y} \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.781 |
|
|
\[ {}-y+x y^{\prime } = \sqrt {x^{2}-y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.621 |
|
|
\[ {}x +y y^{\prime } = 2 y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.059 |
|
|
\[ {}x y^{\prime }-y+\sqrt {-x^{2}+y^{2}} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.514 |
|
|
\[ {}x^{2}+y^{2} = x y y^{\prime } \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
1.809 |
|
|
\[ {}\left (x y-x^{2}\right ) y^{\prime }-y^{2} = 0 \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.395 |
|
|
\[ {}y+x y^{\prime } = 2 \sqrt {x y} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
18.583 |
|
|
\[ {}x +y+\left (x -y\right ) y^{\prime } = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.725 |
|
|
\[ {}y \left (x^{2}-x y+y^{2}\right )+x y^{\prime } \left (x^{2}+x y+y^{2}\right ) = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
2.396 |
|
|
\[ {}x y^{\prime }-y-x \sin \left (\frac {y}{x}\right ) = 0 \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.111 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\cosh \left (\frac {y}{x}\right ) \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.426 |
|
|
\[ {}x^{2}+y^{2} = 2 x y y^{\prime } \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
✓ |
2.511 |
|
|
\[ {}\left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }+1 = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.8 |
|
|
\[ {}x \,{\mathrm e}^{\frac {y}{x}}+y = x y^{\prime } \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
2.766 |
|
|
\[ {}y^{\prime } = \frac {x +y}{x -y} \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.561 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\tan \left (\frac {y}{x}\right ) \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
4.964 |
|
|
\[ {}\left (3 x y-2 x^{2}\right ) y^{\prime } = 2 y^{2}-x y \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
6.517 |
|
|
\[ {}y^{\prime } = \frac {y}{x -k \sqrt {x^{2}+y^{2}}} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
6.158 |
|
|
\[ {}y^{2} \left (y y^{\prime }-x \right )+x^{3} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.408 |
|
|
\[ {}y^{\prime } = \frac {y}{x}+\tanh \left (\frac {y}{x}\right ) \] |
homogeneousTypeD, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
3.241 |
|
|
\[ {}x +y-\left (x -y+2\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.943 |
|
|
\[ {}x +\left (x -2 y+2\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert] |
✓ |
✓ |
5.287 |
|
|
\[ {}2 x -y+1+\left (x +y\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.047 |
|
|
\[ {}x -y+2+\left (x +y-1\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.987 |
|
|
\[ {}x -y+\left (-x +y+1\right ) y^{\prime } = 0 \] |
exact, differentialType, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.164 |
|
|
\[ {}y^{\prime } = \frac {x +y-1}{x -y-1} \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.024 |
|
|
\[ {}x +y+\left (2 x +2 y-1\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.849 |
|
|
\[ {}x -y+1+\left (x -y-1\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.875 |
|
|
\[ {}x +2 y+\left (3 x +6 y+3\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.882 |
|
|
\[ {}x +2 y+2 = \left (2 x +y-1\right ) y^{\prime } \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
5.508 |
|
|
\[ {}3 x -y+1+\left (x -3 y-5\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
6.16 |
|
|
\[ {}6 x -3 y+6+\left (2 x -y+5\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.849 |
|
|
\[ {}2 x +3 y+2+\left (y-x \right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
10.464 |
|
|
\[ {}x +y+4 = \left (2 x +2 y-1\right ) y^{\prime } \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.935 |
|
|
\[ {}2 x +3 y-1+\left (2 x +3 y+2\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.653 |
|
|
\[ {}3 x -y+2+\left (x +2 y+1\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
9.152 |
|
|
\[ {}3 x +2 y+3-\left (x +2 y-1\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
13.745 |
|
|
\[ {}x -2 y+3+\left (1-x +2 y\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.68 |
|
|
\[ {}2 x +y+\left (4 x +2 y+1\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.239 |
|
|
\[ {}2 x +y+\left (4 x -2 y+1\right ) y^{\prime } = 0 \] |
homogeneousTypeMapleC, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
42.992 |
|
|
\[ {}x +y+\left (x -2 y\right ) y^{\prime } = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.179 |
|
|
\[ {}3 x +y+\left (x +3 y\right ) y^{\prime } = 0 \] |
exact, differentialType, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.376 |
|
|
\[ {}a_{1} x +b_{1} y+c_{1} +\left (b_{1} x +b_{2} y+c_{2} \right ) y^{\prime } = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.795 |
|
|
\[ {}x \left (6 x y+5\right )+\left (2 x^{3}+3 y\right ) y^{\prime } = 0 \] |
exact, differentialType |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.313 |
|
|
\[ {}3 x^{2} y+x y^{2}+{\mathrm e}^{x}+\left (x^{3}+x^{2} y+\sin \left (y\right )\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
3.523 |
|
|
\[ {}2 x y-\left (x^{2}+y^{2}\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
4.753 |
|
|
\[ {}y \cos \left (x \right )-2 \sin \left (y\right ) = \left (2 x \cos \left (y\right )-\sin \left (x \right )\right ) y^{\prime } \] |
exact |
[_exact] |
✓ |
✓ |
16.003 |
|
|
\[ {}\frac {2 x y-1}{y}+\frac {\left (x +3 y\right ) y^{\prime }}{y^{2}} = 0 \] |
exact, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.958 |
|
|
\[ {}{\mathrm e}^{x} y-2 x +{\mathrm e}^{x} y^{\prime } = 0 \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.329 |
|
|
\[ {}3 \sin \left (x \right ) y-\cos \left (y\right )+\left (\sin \left (y\right ) x -3 \cos \left (x \right )\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
27.461 |
|
|
\[ {}x y^{2}+2 y+\left (2 y^{3}-x^{2} y+2 x \right ) y^{\prime } = 0 \] |
unknown |
[_rational] |
❇ |
N/A |
1.437 |
|
|
\[ {}\frac {2}{y}-\frac {y}{x^{2}}+\left (\frac {1}{x}-\frac {2 x}{y^{2}}\right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.707 |
|
|
\[ {}\frac {x y+1}{y}+\frac {\left (2 y-x \right ) y^{\prime }}{y^{2}} = 0 \] |
exact, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.969 |
|
|
\[ {}\frac {y \left (2+x^{3} y\right )}{x^{3}} = \frac {\left (1-2 x^{3} y\right ) y^{\prime }}{x^{2}} \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.761 |
|
|
\[ {}y^{2} \csc \left (x \right )^{2}+6 x y-2 = \left (2 y \cot \left (x \right )-3 x^{2}\right ) y^{\prime } \] |
exact |
[_exact, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
58.102 |
|
|
\[ {}\frac {2 y}{x^{3}}+\frac {2 x}{y^{2}} = \left (\frac {1}{x^{2}}+\frac {2 x^{2}}{y^{3}}\right ) y^{\prime } \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
5.387 |
|
|
\[ {}\cos \left (y\right )-\left (\sin \left (y\right ) x -y^{2}\right ) y^{\prime } = 0 \] |
exact |
[_exact, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
19.427 |
|
|
\[ {}2 y \sin \left (x y\right )+\left (2 x \sin \left (x y\right )+y^{3}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
88.614 |
|
|
\[ {}\frac {x \cos \left (\frac {x}{y}\right )}{y}+\sin \left (\frac {x}{y}\right )+\cos \left (x \right )-\frac {x^{2} \cos \left (\frac {x}{y}\right ) y^{\prime }}{y^{2}} = 0 \] |
exact |
[_exact] |
✓ |
✓ |
6.221 |
|
|
\[ {}y \,{\mathrm e}^{x y}+2 x y+\left (x \,{\mathrm e}^{x y}+x^{2}\right ) y^{\prime } = 0 \] |
exact |
[_exact] |
✓ |
✓ |
5.852 |
|
|
\[ {}\frac {x^{2}+3 y^{2}}{x \left (3 x^{2}+4 y^{2}\right )}+\frac {\left (2 x^{2}+y^{2}\right ) y^{\prime }}{y \left (3 x^{2}+4 y^{2}\right )} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
18.233 |
|
|
\[ {}\frac {x^{2}-y^{2}}{x \left (2 x^{2}+y^{2}\right )}+\frac {\left (x^{2}+2 y^{2}\right ) y^{\prime }}{y \left (2 x^{2}+y^{2}\right )} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
23.792 |
|
|
\[ {}\frac {2 x^{2}}{x^{2}+y^{2}}+\ln \left (x^{2}+y^{2}\right )+\frac {2 x y y^{\prime }}{x^{2}+y^{2}} = 0 \] |
exact |
[_exact] |
✓ |
✓ |
3.178 |
|
|
\[ {}x y^{\prime }+\ln \left (x \right )-y = 0 \] |
linear, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.441 |
|
|
\[ {}x y+\left (y+x^{2}\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
3.876 |
|
|
\[ {}\left (x -2 x y\right ) y^{\prime }+2 y = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.566 |
|
|
\[ {}x^{2} y+y^{2}+x^{3} y^{\prime } = 0 \] |
riccati, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.413 |
|
|
\[ {}x y^{3}-1+x^{2} y^{2} y^{\prime } = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
2.648 |
|
|
\[ {}\left (x^{3} y^{3}-1\right ) y^{\prime }+y^{4} x^{2} = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
2.335 |
|
|
\[ {}y \left (-x^{2}+y\right )+x^{3} y^{\prime } = 0 \] |
riccati, bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
✓ |
1.716 |
|
|
\[ {}y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
2.576 |
|
|
\[ {}\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y = 0 \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
2.782 |
|
|
\[ {}2 x y+\left (-x^{2}+y\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
2.267 |
|
|
\[ {}y = x \left (x^{2} y-1\right ) y^{\prime } \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.134 |
|
|
\[ {}{\mathrm e}^{x} y^{\prime } = 2 x y^{2}+{\mathrm e}^{x} y \] |
riccati, bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
1.677 |
|
|
\[ {}\left (x^{2}+y^{2}+x \right ) y^{\prime } = y \] |
exactByInspection |
[_rational] |
✓ |
✓ |
2.014 |
|
|
\[ {}\left (2 x +3 x^{2} y\right ) y^{\prime }+y+2 x y^{2} = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
4.182 |
|
|
\[ {}2 x^{2} y y^{\prime }+{\mathrm e}^{x} x^{4}-2 x y^{2} = 0 \] |
bernoulli, homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class D‘], _Bernoulli] |
✓ |
✓ |
2.105 |
|
|
\[ {}y \left (1-x^{4} y^{2}\right )+x y^{\prime } = 0 \] |
bernoulli, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _rational, _Bernoulli] |
✓ |
✓ |
1.967 |
|
|
\[ {}y \left (x^{2}-1\right )+x \left (x^{2}+1\right ) y^{\prime } = 0 \] |
exact, linear, separable, homogeneousTypeD2, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.358 |
|
|
\[ {}x^{2} y^{2}-y+\left (2 x^{3} y+x \right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
5.316 |
|
|
\[ {}\left (x^{2}+y^{2}-2 y\right ) y^{\prime } = 2 x \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.453 |
|
|
\[ {}y-x^{2} \sqrt {x^{2}-y^{2}}-x y^{\prime } = 0 \] |
unknown |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
❇ |
N/A |
2.97 |
|
|
\[ {}y \left (x +y^{2}\right )+x \left (x -y^{2}\right ) y^{\prime } = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
18.458 |
|
|
\[ {}x y^{\prime }+2 y = x^{2} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.292 |
|
|
\[ {}y^{\prime }-x y = {\mathrm e}^{\frac {x^{2}}{2}} \cos \left (x \right ) \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.634 |
|
|
\[ {}y^{\prime }+2 x y = 2 x \,{\mathrm e}^{-x^{2}} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.285 |
|
|
\[ {}y^{\prime } = y+3 x^{2} {\mathrm e}^{x} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.171 |
|
|
\[ {}x^{\prime }+x = {\mathrm e}^{-y} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[[_linear, ‘class A‘]] |
✓ |
✓ |
1.164 |
|
|
\[ {}y x^{\prime }+\left (1+y \right ) x = {\mathrm e}^{y} \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.446 |
|
|
\[ {}y+\left (2 x -3 y\right ) y^{\prime } = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.38 |
|
|
\[ {}x y^{\prime }-2 x^{4}-2 y = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.245 |
|
|
\[ {}1 = \left ({\mathrm e}^{y}+x \right ) y^{\prime } \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_exponential_symmetries]] |
✓ |
✓ |
1.615 |
|
|
\[ {}y^{2} x^{\prime }+\left (y^{2}+2 y \right ) x = 1 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.394 |
|
|
\[ {}x y^{\prime } = 5 y+x +1 \] |
linear, homogeneousTypeMapleC, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.558 |
|
|
\[ {}x^{2} y^{\prime }+y-2 x y-2 x^{2} = 0 \] |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (1+x \right ) y^{\prime }+2 y = \frac {{\mathrm e}^{x}}{1+x} \] |
exact, linear, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.405 |
|
|
\[ {}\cos \left (y\right )^{2}+\left (x -\tan \left (y\right )\right ) y^{\prime } = 0 \] |
exactWithIntegrationFactor |
[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
8.594 |
|
|
|
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