|
# |
ODE |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}\left (1-4 x +3 x y^{2}\right ) y^{\prime } = \left (2-y^{2}\right ) y \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.402 |
|
|
\[ {}x \left (x -3 y^{2}\right ) y^{\prime }+\left (2 x -y^{2}\right ) y = 0 \] |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _exact, _rational] |
✓ |
✓ |
3.18 |
|
|
\[ {}3 x \left (x +y^{2}\right ) y^{\prime }+x^{3}-3 x y-2 y^{3} = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.22 |
|
|
\[ {}x \left (x^{3}-3 x^{3} y+4 y^{2}\right ) y^{\prime } = 6 y^{3} \] |
unknown |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✗ |
N/A |
1.036 |
|
|
\[ {}6 x y^{2} y^{\prime }+x +2 y^{3} = 0 \] |
exact, bernoulli, first_order_ode_lie_symmetry_lookup |
[[_homogeneous, ‘class G‘], _exact, _rational, _Bernoulli] |
✓ |
✓ |
0.997 |
|
|
\[ {}x \left (x +6 y^{2}\right ) y^{\prime }+x y-3 y^{3} = 0 \] |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.339 |
|
|
\[ {}x \left (x^{2}-6 y^{2}\right ) y^{\prime } = 4 \left (x^{2}+3 y^{2}\right ) y \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.709 |
|
|
\[ {}x \left (3 x -7 y^{2}\right ) y^{\prime }+\left (5 x -3 y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.315 |
|
|
\[ {}x^{2} y^{2} y^{\prime }+1-x +x^{3} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.359 |
|
|
\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = x y^{3} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
0.975 |
|
|
\[ {}\left (1-x^{2} y^{2}\right ) y^{\prime } = \left (x y+1\right ) y^{2} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
0.971 |
|
|
\[ {}x \left (1+x y^{2}\right ) y^{\prime }+y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.185 |
|
|
\[ {}x \left (1+x y^{2}\right ) y^{\prime } = \left (2-3 x y^{2}\right ) y \] |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.484 |
|
|
\[ {}x^{2} \left (y+a \right )^{2} y^{\prime } = \left (x^{2}+1\right ) \left (y^{2}+a^{2}\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.001 |
|
|
\[ {}\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y^{2}\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.552 |
|
|
\[ {}\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }+2 x y \left (1-y\right )^{2} = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.2 |
|
|
\[ {}\left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime } = \left (6+3 x y-4 y^{3}\right ) x \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.378 |
|
|
\[ {}x \left (3+5 x -12 x y^{2}+4 x^{2} y\right ) y^{\prime }+\left (3+10 x -8 x y^{2}+6 x^{2} y\right ) y = 0 \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.655 |
|
|
\[ {}x^{3} \left (1+y^{2}\right ) y^{\prime }+3 x^{2} y = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
1.556 |
|
|
\[ {}x \left (1-x y\right )^{2} y^{\prime }+\left (1+x^{2} y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.223 |
|
|
\[ {}\left (1-x^{4} y^{2}\right ) y^{\prime } = x^{3} y^{3} \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.205 |
|
|
\[ {}\left (3 x -y^{3}\right ) y^{\prime } = x^{2}-3 y \] |
exact, differentialType |
[_exact, _rational] |
✓ |
✓ |
1.135 |
|
|
\[ {}\left (x^{3}-y^{3}\right ) y^{\prime }+x^{2} y = 0 \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.447 |
|
|
\[ {}\left (x^{3}+y^{3}\right ) y^{\prime }+x^{2} \left (x a +3 y\right ) = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
4.654 |
|
|
\[ {}\left (x -x^{2} y-y^{3}\right ) y^{\prime } = x^{3}-y+x y^{2} \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.349 |
|
|
\[ {}\left (x \,a^{2}+\left (x^{2}-y^{2}\right ) y\right ) y^{\prime }+x \left (x^{2}-y^{2}\right ) = a^{2} y \] |
exactByInspection |
[_rational] |
✓ |
✓ |
1.569 |
|
|
\[ {}\left (a +x^{2}+y^{2}\right ) y y^{\prime } = x \left (a -x^{2}-y^{2}\right ) \] |
exact |
[_exact, _rational] |
✓ |
✓ |
1.224 |
|
|
\[ {}\left (y^{2}+3 x^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right ) = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.694 |
|
|
\[ {}\left (a -3 x^{2}-y^{2}\right ) y y^{\prime }+x \left (-x^{2}+y^{2}+a \right ) = 0 \] |
unknown |
[_rational] |
✗ |
N/A |
1.15 |
|
|
\[ {}2 y^{3} y^{\prime } = x^{3}-x y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
6.335 |
|
|
\[ {}y \left (2 y^{2}+1\right ) y^{\prime } = x \left (2 x^{2}+1\right ) \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.901 |
|
|
\[ {}\left (3 x^{2}+2 y^{2}\right ) y y^{\prime }+x^{3} = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.286 |
|
|
\[ {}\left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right ) = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
1.744 |
|
|
\[ {}\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3} = 0 \] |
unknown |
[_rational] |
✗ |
N/A |
1.039 |
|
|
\[ {}\left (3 x^{3}+6 x^{2} y-3 x y^{2}+20 y^{3}\right ) y^{\prime }+4 x^{3}+9 x^{2} y+6 x y^{2}-y^{3} = 0 \] |
exact, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert] |
✓ |
✓ |
3.194 |
|
|
\[ {}\left (x^{3}+a y^{3}\right ) y^{\prime } = x^{2} y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.381 |
|
|
\[ {}x y^{3} y^{\prime } = \left (-x^{2}+1\right ) \left (1+y^{2}\right ) \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.384 |
|
|
\[ {}x \left (x -y^{3}\right ) y^{\prime } = \left (3 x +y^{3}\right ) y \] |
exactByInspection, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.442 |
|
|
\[ {}x \left (2 x^{3}+y^{3}\right ) y^{\prime } = \left (2 x^{3}-x^{2} y+y^{3}\right ) y \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
0.92 |
|
|
\[ {}x \left (2 x^{3}-y^{3}\right ) y^{\prime } = \left (x^{3}-2 y^{3}\right ) y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.336 |
|
|
\[ {}x \left (x^{3}+3 x^{2} y+y^{3}\right ) y^{\prime } = \left (y^{2}+3 x^{2}\right ) y^{2} \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.106 |
|
|
\[ {}x \left (x^{3}-2 y^{3}\right ) y^{\prime } = \left (2 x^{3}-y^{3}\right ) y \] |
homogeneousTypeD2, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
3.612 |
|
|
\[ {}x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
5.208 |
|
|
\[ {}x \left (x +y+2 y^{3}\right ) y^{\prime } = \left (x -y\right ) y \] |
exactByInspection |
[_rational] |
✓ |
✓ |
1.237 |
|
|
\[ {}\left (5 x -y-7 x y^{3}\right ) y^{\prime }+5 y-y^{4} = 0 \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.248 |
|
|
\[ {}x \left (1-2 x y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y\right ) y = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.28 |
|
|
\[ {}x \left (2-x y^{2}-2 x y^{3}\right ) y^{\prime }+1+2 y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
2.363 |
|
|
\[ {}\left (2-10 x^{2} y^{3}+3 y^{2}\right ) y^{\prime } = x \left (1+5 y^{4}\right ) \] |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.213 |
|
|
\[ {}x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.53 |
|
|
\[ {}x \left (1-2 x^{2} y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.236 |
|
|
\[ {}x \left (1-x y\right ) \left (1-x^{2} y^{2}\right ) y^{\prime }+\left (x y+1\right ) \left (1+x^{2} y^{2}\right ) y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.201 |
|
|
\[ {}\left (x^{2}-y^{4}\right ) y^{\prime } = x y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.335 |
|
|
\[ {}\left (x^{3}-y^{4}\right ) y^{\prime } = 3 x^{2} y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.968 |
|
|
\[ {}\left (a^{2} x^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime } = y a^{2} x \] |
first_order_ode_lie_symmetry_calculated |
[_rational] |
✓ |
✓ |
6.356 |
|
|
\[ {}2 \left (x -y^{4}\right ) y^{\prime } = y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.197 |
|
|
\[ {}\left (4 x -x y^{3}-2 y^{4}\right ) y^{\prime } = \left (2+y^{3}\right ) y \] |
exactWithIntegrationFactor |
[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]] |
✓ |
✓ |
1.21 |
|
|
\[ {}\left (a \,x^{3}+\left (x a +b y\right )^{3}\right ) y y^{\prime }+x \left (\left (x a +b y\right )^{3}+b y^{3}\right ) = 0 \] |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
5.631 |
|
|
\[ {}\left (x +2 y+2 x^{2} y^{3}+y^{4} x \right ) y^{\prime }+\left (1+y^{4}\right ) y = 0 \] |
unknown |
[_rational] |
✗ |
N/A |
1.309 |
|
|
\[ {}2 x \left (x^{3}+y^{4}\right ) y^{\prime } = \left (x^{3}+2 y^{4}\right ) y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.709 |
|
|
\[ {}x \left (1-y^{4} x^{2}\right ) y^{\prime }+y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.585 |
|
|
\[ {}\left (x^{2}-y^{5}\right ) y^{\prime } = 2 x y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.212 |
|
|
\[ {}x \left (x^{3}+y^{5}\right ) y^{\prime } = \left (x^{3}-y^{5}\right ) y \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
3.368 |
|
|
\[ {}x^{3} \left (1+5 x^{3} y^{7}\right ) y^{\prime }+\left (3 x^{5} y^{5}-1\right ) y^{3} = 0 \] |
exactWithIntegrationFactor |
[_rational] |
✓ |
✓ |
1.467 |
|
|
\[ {}\left (1+a \left (x +y\right )\right )^{n} y^{\prime }+a \left (x +y\right )^{n} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
1.24 |
|
|
\[ {}x \left (a +x y^{n}\right ) y^{\prime }+b y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational] |
✓ |
✓ |
1.418 |
|
|
\[ {}f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n} = 0 \] |
unknown |
[_Bernoulli] |
✗ |
N/A |
1.502 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}+y^{2}} = \sqrt {a^{2}+x^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.374 |
|
|
\[ {}y^{\prime } \sqrt {b^{2}-y^{2}} = \sqrt {a^{2}-x^{2}} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
2.537 |
|
|
\[ {}y^{\prime } \sqrt {y} = \sqrt {x} \] |
exact, separable, differentialType, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
166.565 |
|
|
\[ {}\left (1+\sqrt {x +y}\right ) y^{\prime }+1 = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.79 |
|
|
\[ {}y^{\prime } \sqrt {x y}+x -y = \sqrt {x y} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
9.322 |
|
|
\[ {}\left (x -2 \sqrt {x y}\right ) y^{\prime } = y \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
5.217 |
|
|
\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
8.955 |
|
|
\[ {}\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime } = 1+y^{2} \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
7.553 |
|
|
\[ {}\left (x -\sqrt {x^{2}+y^{2}}\right ) y^{\prime } = y \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _rational, _dAlembert] |
✓ |
✓ |
1.154 |
|
|
\[ {}x \left (1-\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = y \] |
unknown |
[‘y=_G(x,y’)‘] |
✗ |
N/A |
1.405 |
|
|
\[ {}x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}} = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _dAlembert] |
✓ |
✓ |
78.474 |
|
|
\[ {}x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime } = x y^{2}-\left (x^{2}-y^{2}\right )^{\frac {3}{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _dAlembert] |
✓ |
✓ |
1.333 |
|
|
\[ {}\left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime } = x \left (x^{2}+y^{2}\right )+y \sqrt {1+x^{2}+y^{2}} \] |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.963 |
|
|
\[ {}y^{\prime } \cos \left (y\right ) \left (\cos \left (y\right )-\sin \left (A \right ) \sin \left (x \right )\right )+\cos \left (x \right ) \left (\cos \left (x \right )-\sin \left (A \right ) \sin \left (y\right )\right ) = 0 \] |
exact |
unknown |
✓ |
✓ |
4.592 |
|
|
\[ {}\left (a \cos \left (b x +a y\right )-b \sin \left (x a +b y\right )\right ) y^{\prime }+b \cos \left (b x +a y\right )-a \sin \left (x a +b y\right ) = 0 \] |
exact |
[_exact] |
✓ |
✓ |
2.376 |
|
|
\[ {}\left (x +\cos \left (x \right ) \sec \left (y\right )\right ) y^{\prime }+\tan \left (y\right )-y \sin \left (x \right ) \sec \left (y\right ) = 0 \] |
exactWithIntegrationFactor |
[NONE] |
✓ |
✓ |
18.316 |
|
|
\[ {}\left (1+\left (x +y\right ) \tan \left (y\right )\right ) y^{\prime }+1 = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.122 |
|
|
\[ {}x \left (x -y \tan \left (\frac {y}{x}\right )\right ) y^{\prime }+\left (x +y \tan \left (\frac {y}{x}\right )\right ) y = 0 \] |
exactByInspection, homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.917 |
|
|
\[ {}\left ({\mathrm e}^{x}+x \,{\mathrm e}^{y}\right ) y^{\prime }+{\mathrm e}^{x} y+{\mathrm e}^{y} = 0 \] |
exact |
[_exact] |
✓ |
✓ |
1.407 |
|
|
\[ {}\left (1-2 x -\ln \left (y\right )\right ) y^{\prime }+2 y = 0 \] |
exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries]] |
✓ |
✓ |
1.09 |
|
|
\[ {}\left (\sinh \left (x \right )+x \cosh \left (y\right )\right ) y^{\prime }+y \cosh \left (x \right )+\sinh \left (y\right ) = 0 \] |
exact |
[_exact] |
✓ |
✓ |
35.838 |
|
|
\[ {}y^{\prime } \left (1+\sinh \left (x \right )\right ) \sinh \left (y\right )+\cosh \left (x \right ) \left (\cosh \left (y\right )-1\right ) = 0 \] |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
3.651 |
|
|
\[ {}{y^{\prime }}^{2} = a \,x^{n} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.273 |
|
|
\[ {}{y^{\prime }}^{2} = y \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.28 |
|
|
\[ {}{y^{\prime }}^{2} = x -y \] |
dAlembert, first_order_nonlinear_p_but_linear_in_x_y |
[[_homogeneous, ‘class C‘], _dAlembert] |
✓ |
✓ |
0.408 |
|
|
\[ {}{y^{\prime }}^{2} = y+x^{2} \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.932 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} = 4 y \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.751 |
|
|
\[ {}{y^{\prime }}^{2}+3 x^{2} = 8 y \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
3.812 |
|
|
\[ {}{y^{\prime }}^{2}+x^{2} a +b y = 0 \] |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘]] |
✓ |
✓ |
2.79 |
|
|
\[ {}{y^{\prime }}^{2} = 1+y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.377 |
|
|
\[ {}{y^{\prime }}^{2} = 1-y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.512 |
|
|
\[ {}{y^{\prime }}^{2} = a^{2}-y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.484 |
|
|
\[ {}{y^{\prime }}^{2} = y^{2} a^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.398 |
|
|
\[ {}{y^{\prime }}^{2} = a +b y^{2} \] |
quadrature |
[_quadrature] |
✓ |
✓ |
0.543 |
|
|
|
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