|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+y^{2}-2 x y \left (1+y^{2}\right )
\] |
[_rational, _Abel] |
✗ |
1.260 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right ) = x \left (x^{2}+1\right ) \cos \left (y\right )^{2}
\] |
[‘y=_G(x,y’)‘] |
✗ |
16.481 |
|
|
\[
{}\left (x^{2}+1\right ) y^{\prime } = 1+x^{2}-y \,\operatorname {arccot}\left (x \right )
\] |
[_linear] |
✓ |
1.944 |
|
|
\[
{}\left (-x^{2}+4\right ) y^{\prime }+4 y = \left (x +2\right ) y^{2}
\] |
[_rational, _Bernoulli] |
✓ |
1.630 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime } = b +x y
\] |
[_linear] |
✓ |
1.286 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime } = \left (b +y\right ) \left (x +\sqrt {a^{2}+x^{2}}\right )
\] |
[_separable] |
✓ |
3.892 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime }+\left (x -y\right ) y = 0
\] |
[_rational, _Bernoulli] |
✓ |
2.655 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime } = a^{2}+3 x y-2 y^{2}
\] |
[_rational, _Riccati] |
✓ |
153.671 |
|
|
\[
{}\left (a^{2}+x^{2}\right ) y^{\prime }+x y+b x y^{2} = 0
\] |
[_separable] |
✓ |
2.799 |
|
|
\[
{}x \left (1-x \right ) y^{\prime } = a +\left (x +1\right ) y
\] |
[_linear] |
✓ |
1.181 |
|
|
\[
{}x \left (1-x \right ) y^{\prime } = 2+2 x y
\] |
[_linear] |
✓ |
1.317 |
|
|
\[
{}x \left (1-x \right ) y^{\prime } = 2 x y-2
\] |
[_linear] |
✓ |
1.321 |
|
|
\[
{}x \left (x +1\right ) y^{\prime } = \left (1-2 x \right ) y
\] |
[_separable] |
✓ |
1.664 |
|
|
\[
{}x \left (1-x \right ) y^{\prime }+\left (2 x +1\right ) y = a
\] |
[_linear] |
✓ |
1.408 |
|
|
\[
{}x \left (1-x \right ) y^{\prime } = a +2 \left (2-x \right ) y
\] |
[_linear] |
✓ |
1.543 |
|
|
\[
{}x \left (1-x \right ) y^{\prime }+2-3 x y+y = 0
\] |
[_linear] |
✓ |
1.388 |
|
|
\[
{}x \left (x +1\right ) y^{\prime } = \left (x +1\right ) \left (x^{2}-1\right )+\left (x^{2}+x -1\right ) y
\] |
[_linear] |
✓ |
1.690 |
|
|
\[
{}\left (-2+x \right ) \left (x -3\right ) y^{\prime }+x^{2}-8 y+3 x y = 0
\] |
[_linear] |
✓ |
1.714 |
|
|
\[
{}x \left (x +a \right ) y^{\prime } = \left (b +c y\right ) y
\] |
[_separable] |
✓ |
3.011 |
|
|
\[
{}\left (x +a \right )^{2} y^{\prime } = 2 \left (x +a \right ) \left (b +y\right )
\] |
[_separable] |
✓ |
1.188 |
|
|
\[
{}\left (x -a \right )^{2} y^{\prime }+k \left (x +y-a \right )^{2}+y^{2} = 0
\] |
[[_homogeneous, ‘class C‘], _rational, _Riccati] |
✓ |
38.900 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k y = 0
\] |
[_separable] |
✓ |
2.576 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime } = \left (x -a \right ) \left (x -b \right )+\left (2 x -a -b \right ) y
\] |
[_linear] |
✓ |
1.732 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime } = c y^{2}
\] |
[_separable] |
✓ |
2.293 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (y-a \right ) \left (y-b \right ) = 0
\] |
[_separable] |
✓ |
4.096 |
|
|
\[
{}\left (x -a \right ) \left (x -b \right ) y^{\prime }+k \left (x +y-a \right ) \left (x +y-b \right )+y^{2} = 0
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
3.349 |
|
|
\[
{}2 x^{2} y^{\prime } = y
\] |
[_separable] |
✓ |
1.754 |
|
|
\[
{}2 x^{2} y^{\prime }+x \cot \left (x \right )-1+2 x^{2} y \cot \left (x \right ) = 0
\] |
[_linear] |
✓ |
2.206 |
|
|
\[
{}2 x^{2} y^{\prime }+1+2 x y-y^{2} x^{2} = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.906 |
|
|
\[
{}2 x^{2} y^{\prime } = 2 x y+\left (1-x \cot \left (x \right )\right ) \left (x^{2}-y^{2}\right )
\] |
[[_homogeneous, ‘class D‘], _Riccati] |
✓ |
49.169 |
|
|
\[
{}2 \left (-x^{2}+1\right ) y^{\prime } = \sqrt {-x^{2}+1}+\left (x +1\right ) y
\] |
[_linear] |
✓ |
3.551 |
|
|
\[
{}x \left (1-2 x \right ) y^{\prime }+1+\left (1-4 x \right ) y = 0
\] |
[_linear] |
✓ |
1.343 |
|
|
\[
{}x \left (1-2 x \right ) y^{\prime } = 4 x -\left (1+4 x \right ) y+y^{2}
\] |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
2.792 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime }+x +\left (1-2 x \right ) y = 0
\] |
[_linear] |
✓ |
1.491 |
|
|
\[
{}2 x \left (1-x \right ) y^{\prime }+x +\left (1-x \right ) y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
2.469 |
|
|
\[
{}2 \left (x^{2}+x +1\right ) y^{\prime } = 1+8 x^{2}-\left (2 x +1\right ) y
\] |
[_linear] |
✓ |
4.153 |
|
|
\[
{}4 \left (x^{2}+1\right ) y^{\prime }-4 x y-x^{2} = 0
\] |
[_linear] |
✓ |
1.436 |
|
|
\[
{}a \,x^{2} y^{\prime } = x^{2}+y a x +b^{2} y^{2}
\] |
[[_homogeneous, ‘class A‘], _rational, _Riccati] |
✓ |
6.401 |
|
|
\[
{}\left (b \,x^{2}+a \right ) y^{\prime } = A +B y^{2}
\] |
[_separable] |
✓ |
3.626 |
|
|
\[
{}\left (b \,x^{2}+a \right ) y^{\prime } = c x y \ln \left (y\right )
\] |
[_separable] |
✓ |
1.906 |
|
|
\[
{}x \left (a x +1\right ) y^{\prime }+a -y = 0
\] |
[_separable] |
✓ |
0.996 |
|
|
\[
{}\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3} = 0
\] |
[_rational, _Abel] |
✗ |
2.213 |
|
|
\[
{}x^{3} y^{\prime } = a +b \,x^{2} y
\] |
[_linear] |
✓ |
1.239 |
|
|
\[
{}x^{3} y^{\prime } = 3-x^{2}+x^{2} y
\] |
[_linear] |
✓ |
1.388 |
|
|
\[
{}x^{3} y^{\prime } = x^{4}+y^{2}
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
1.425 |
|
|
\[
{}x^{3} y^{\prime } = y \left (x^{2}+y\right )
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.293 |
|
|
\[
{}x^{3} y^{\prime } = x^{2} \left (y-1\right )+y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.320 |
|
|
\[
{}x^{3} y^{\prime } = \left (x +1\right ) y^{2}
\] |
[_separable] |
✓ |
1.804 |
|
|
\[
{}x^{3} y^{\prime }+20+x^{2} y \left (1-x^{2} y\right ) = 0
\] |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
2.095 |
|
|
\[
{}x^{3} y^{\prime }+3+\left (3-2 x \right ) x^{2} y-x^{6} y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
1.608 |
|
|
\[
{}x^{3} y^{\prime } = \left (2 x^{2}+y^{2}\right ) y
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
74.536 |
|
|
\[
{}x^{3} y^{\prime } = \cos \left (y\right ) \left (\cos \left (y\right )-2 x^{2} \sin \left (y\right )\right )
\] |
[‘y=_G(x,y’)‘] |
✗ |
38.220 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = a \,x^{2}+y
\] |
[_linear] |
✓ |
1.149 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime } = a \,x^{2}+y
\] |
[_linear] |
✓ |
1.221 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = a \,x^{3}+y
\] |
[_linear] |
✓ |
1.365 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = a -x^{2} y
\] |
[_linear] |
✓ |
1.159 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = \left (-x^{2}+1\right ) y
\] |
[_separable] |
✓ |
1.691 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime } = \left (x^{2}-x +1\right ) y
\] |
[_separable] |
✓ |
1.980 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime } = a \,x^{3}+\left (-2 x^{2}+1\right ) y
\] |
[_linear] |
✓ |
1.194 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime } = x^{3} \left (-x^{2}+1\right )+\left (-2 x^{2}+1\right ) y
\] |
[_linear] |
✓ |
3.059 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = 2-4 x^{2} y
\] |
[_linear] |
✓ |
1.408 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime } = x -\left (5 x^{2}+3\right ) y
\] |
[_linear] |
✓ |
1.449 |
|
|
\[
{}x \left (-x^{2}+1\right ) y^{\prime }+x^{2}+\left (-x^{2}+1\right ) y^{2} = 0
\] |
[_rational, _Riccati] |
✓ |
139.129 |
|
|
\[
{}x^{2} \left (1-x \right ) y^{\prime } = \left (2-x \right ) x y-y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.488 |
|
|
\[
{}2 x^{3} y^{\prime } = \left (x^{2}-y^{2}\right ) y
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
76.828 |
|
|
\[
{}2 x^{3} y^{\prime } = \left (3 x^{2}+a y^{2}\right ) y
\] |
[[_homogeneous, ‘class A‘], _rational, _Bernoulli] |
✓ |
14.704 |
|
|
\[
{}6 x^{3} y^{\prime } = 4 x^{2} y+\left (1-3 x \right ) y^{4}
\] |
[_rational, _Bernoulli] |
✓ |
2.432 |
|
|
\[
{}x \left (c \,x^{2}+b x +a \right ) y^{\prime }+x^{2}-\left (c \,x^{2}+b x +a \right ) y = y^{2}
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.165 |
|
|
\[
{}x^{4} y^{\prime } = \left (x^{3}+y\right ) y
\] |
[[_homogeneous, ‘class D‘], _rational, _Bernoulli] |
✓ |
2.452 |
|
|
\[
{}x^{4} y^{\prime }+a^{2}+x^{4} y^{2} = 0
\] |
[_rational, [_Riccati, _special]] |
✓ |
3.315 |
|
|
\[
{}x^{4} y^{\prime }+x^{3} y+\csc \left (x y\right ) = 0
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]] |
✓ |
23.645 |
|
|
\[
{}\left (-x^{4}+1\right ) y^{\prime } = 2 x \left (1-y^{2}\right )
\] |
[_separable] |
✓ |
2.729 |
|
|
\[
{}x \left (-x^{3}+1\right ) y^{\prime } = 2 x -\left (-4 x^{3}+1\right ) y
\] |
[_linear] |
✓ |
1.411 |
|
|
\[
{}x \left (-x^{3}+1\right ) y^{\prime } = x^{2}+\left (1-2 x y\right ) y
\] |
[_rational, _Riccati] |
✓ |
1.961 |
|
|
\[
{}x^{2} \left (-x^{2}+1\right ) y^{\prime } = \left (x -3 x^{3} y\right ) y
\] |
[_rational, _Bernoulli] |
✓ |
1.576 |
|
|
\[
{}x \left (-2 x^{3}+1\right ) y^{\prime } = 2 \left (-x^{3}+1\right ) y
\] |
[_separable] |
✓ |
1.766 |
|
|
\[
{}\left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0
\] |
[_rational, _Riccati] |
✓ |
3.628 |
|
|
\[
{}x^{5} y^{\prime } = 1-3 x^{4} y
\] |
[_linear] |
✓ |
1.524 |
|
|
\[
{}x \left (-x^{4}+1\right ) y^{\prime } = 2 x \left (x^{2}-y^{2}\right )+\left (-x^{4}+1\right ) y
\] |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
2.214 |
|
|
\[
{}x^{7} y^{\prime }+5 x^{3} y^{2}+2 \left (x^{2}+1\right ) y^{3} = 0
\] |
[_rational, _Abel] |
✗ |
1.088 |
|
|
\[
{}x^{n} y^{\prime } = a +b \,x^{n -1} y
\] |
[_linear] |
✓ |
1.337 |
|
|
\[
{}x^{n} y^{\prime } = x^{2 n -1}-y^{2}
\] |
[_Riccati] |
✓ |
1.998 |
|
|
\[
{}x^{n} y^{\prime }+x^{-2+2 n}+y^{2}+\left (1-n \right ) x^{n -1} = 0
\] |
[_Riccati] |
✓ |
10.230 |
|
|
\[
{}x^{n} y^{\prime } = a^{2} x^{-2+2 n}+b^{2} y^{2}
\] |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
4.366 |
|
|
\[
{}x^{n} y^{\prime } = x^{n -1} \left (a \,x^{2 n}+n y-b y^{2}\right )
\] |
[_rational, _Riccati] |
✓ |
3.677 |
|
|
\[
{}x^{k} y^{\prime } = a \,x^{m}+b y^{n}
\] |
[_Chini] |
✗ |
1.981 |
|
|
\[
{}y^{\prime } \sqrt {x^{2}+1} = 2 x -y
\] |
[_linear] |
✓ |
1.805 |
|
|
\[
{}y^{\prime } \sqrt {-x^{2}+1} = 1+y^{2}
\] |
[_separable] |
✓ |
2.896 |
|
|
\[
{}\left (x -\sqrt {x^{2}+1}\right ) y^{\prime } = y+\sqrt {1+y^{2}}
\] |
[_separable] |
✓ |
2.720 |
|
|
\[
{}y^{\prime } \sqrt {a^{2}+x^{2}}+x +y = \sqrt {a^{2}+x^{2}}
\] |
[_linear] |
✓ |
1.631 |
|
|
\[
{}y^{\prime } \sqrt {b^{2}+x^{2}} = \sqrt {y^{2}+a^{2}}
\] |
[_separable] |
✓ |
16.850 |
|
|
\[
{}y^{\prime } \sqrt {b^{2}-x^{2}} = \sqrt {a^{2}-y^{2}}
\] |
[_separable] |
✓ |
30.389 |
|
|
\[
{}x y^{\prime } \sqrt {a^{2}+x^{2}} = y \sqrt {b^{2}+y^{2}}
\] |
[_separable] |
✓ |
3.353 |
|
|
\[
{}x y^{\prime } \sqrt {-a^{2}+x^{2}} = y \sqrt {y^{2}-b^{2}}
\] |
[_separable] |
✓ |
21.136 |
|
|
\[
{}y^{\prime } \sqrt {X}+\sqrt {Y} = 0
\] |
[_quadrature] |
✓ |
0.432 |
|
|
\[
{}y^{\prime } \sqrt {X} = \sqrt {Y}
\] |
[_quadrature] |
✓ |
0.431 |
|
|
\[
{}x^{{3}/{2}} y^{\prime } = a +b \,x^{{3}/{2}} y^{2}
\] |
[_rational, [_Riccati, _special]] |
✓ |
1.964 |
|
|
\[
{}y^{\prime } \sqrt {x^{3}+1} = \sqrt {1+y^{3}}
\] |
[_separable] |
✓ |
3.706 |
|
|
\[
{}y^{\prime } \sqrt {x \left (1-x \right ) \left (-a x +1\right )} = \sqrt {y \left (1-y\right ) \left (1-a y\right )}
\] |
[_separable] |
✓ |
3.307 |
|
|
\[
{}y^{\prime } \sqrt {-x^{4}+1} = \sqrt {1-y^{4}}
\] |
[_separable] |
✓ |
4.100 |
|