|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
11.971 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
4.466 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
59.839 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
17.706 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
19.818 |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
25.952 |
|
|
\[
{}y^{\prime } x = a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
25.520 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right )
\] |
[_Riccati] |
✗ |
101.369 |
|
|
\[
{}\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
5.150 |
|
|
\[
{}y^{\prime } = \alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
2.479 |
|
|
\[
{}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+a^{2} \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
4.580 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4}
\] |
[_Riccati] |
✗ |
108.372 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
3.729 |
|
|
\[
{}y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m}
\] |
[_Riccati] |
✗ |
10.513 |
|
|
\[
{}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3}
\] |
[_Riccati] |
✓ |
14.186 |
|
|
\[
{}2 y^{\prime } = \left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
26.317 |
|
|
\[
{}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
14.431 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
16.975 |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
25.453 |
|
|
\[
{}y^{\prime } x = a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
92.521 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right )
\] |
[_Riccati] |
✗ |
100.527 |
|
|
\[
{}\left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right ) = 0
\] |
[_Riccati] |
✓ |
4.860 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
15.864 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
23.948 |
|
|
\[
{}y^{\prime } = a y^{2}+b \tan \left (x \right ) y+c
\] |
[_Riccati] |
✓ |
4.362 |
|
|
\[
{}y^{\prime } = a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
3.006 |
|
|
\[
{}y^{\prime } = y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
73.807 |
|
|
\[
{}y^{\prime } = y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
6.034 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
12.858 |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda
\] |
[_Riccati] |
✓ |
25.046 |
|
|
\[
{}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
84.914 |
|
|
\[
{}y^{\prime } x = a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
92.592 |
|
|
\[
{}\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right )
\] |
[_Riccati] |
✗ |
237.474 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
15.819 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
23.152 |
|
|
\[
{}y^{\prime } = y^{2}-2 a b \cot \left (a x \right ) y+b^{2}-a^{2}
\] |
[_Riccati] |
✗ |
93.078 |
|
|
\[
{}y^{\prime } = y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2}
\] |
[_Riccati] |
✓ |
4.918 |
|
|
\[
{}y^{\prime } = y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m}
\] |
[_Riccati] |
✓ |
6.355 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m}
\] |
[_Riccati] |
✓ |
13.477 |
|
|
\[
{}y^{\prime } = a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
97.260 |
|
|
\[
{}y^{\prime } x = a \cot \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cot \left (\lambda x \right )^{m}
\] |
[_Riccati] |
✓ |
99.822 |
|
|
\[
{}\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right )
\] |
[_Riccati] |
✗ |
280.050 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x \right )^{n} \cos \left (\lambda x \right )^{-n -4}
\] |
[_Riccati] |
✗ |
14.944 |
|
|
\[
{}y^{\prime } = a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
7.021 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \cos \left (\lambda x \right )^{n} y-a \cos \left (\lambda x \right )^{n -1}
\] |
[_Riccati] |
✗ |
47.887 |
|
|
\[
{}y^{\prime } = a \cos \left (\lambda x \right ) y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
9.479 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-a \,x^{n}
\] |
[_Riccati] |
✓ |
10.792 |
|
|
\[
{}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n}
\] |
[_Riccati] |
✗ |
287.704 |
|
|
\[
{}y^{\prime } = y^{2}-y \tan \left (x \right )+a \left (1-a \right ) \cot \left (x \right )^{2}
\] |
[_Riccati] |
✓ |
6.592 |
|
|
\[
{}y^{\prime } = y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m}
\] |
[_Riccati] |
✓ |
35.292 |
|
|
\[
{}y^{\prime } = y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m}
\] |
[_Riccati] |
✓ |
35.433 |
|
|
\[
{}y^{\prime } = y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✗ |
106.466 |
|
|
\[
{}y^{\prime } = y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2}
\] |
[_Riccati] |
✓ |
11.368 |
|
|
\[
{}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}
\] |
[_Riccati] |
✓ |
31.030 |
|
|
\[
{}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \sin \left (\lambda x \right ) y-a \tan \left (\lambda x \right )
\] |
[_Riccati] |
✓ |
7.462 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arcsin \left (x \right )^{n} y-a^{2}+a \lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
3.338 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
5.117 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
51.674 |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
6.068 |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
20.775 |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
31.914 |
|
|
\[
{}y^{\prime } = \lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
24.796 |
|
|
\[
{}y^{\prime } x = \lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
22.942 |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arcsin \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
131.577 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
4.907 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arccos \left (x \right )^{n} y+\lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
5.895 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
58.148 |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
10.404 |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
58.992 |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
41.680 |
|
|
\[
{}y^{\prime } = \lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
35.747 |
|
|
\[
{}y^{\prime } x = \lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
139.812 |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
180.499 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
4.412 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \arctan \left (x \right )^{n} y+\lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
6.741 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
39.721 |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
5.434 |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
44.769 |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✗ |
40.020 |
|
|
\[
{}y^{\prime } = \lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
84.521 |
|
|
\[
{}y^{\prime } x = \lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n}
\] |
[_Riccati] |
✓ |
35.664 |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
125.419 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} y-a^{2}+a \lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
4.657 |
|
|
\[
{}y^{\prime } = y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
5.567 |
|
|
\[
{}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{k +1} y-1\right )
\] |
[_Riccati] |
✓ |
40.459 |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✗ |
8.592 |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1}
\] |
[_Riccati] |
✗ |
58.897 |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✗ |
40.909 |
|
|
\[
{}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1}
\] |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
106.923 |
|
|
\[
{}y^{\prime } x = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n}
\] |
[_Riccati] |
✓ |
36.476 |
|
|
\[
{}y^{\prime } x = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \operatorname {arccot}\left (x \right )^{m}-n y
\] |
[_Riccati] |
✗ |
271.845 |
|
|
\[
{}y^{\prime } = y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right )
\] |
[_Riccati] |
✓ |
1.641 |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a y-a b -b^{2} f \left (x \right )
\] |
[_Riccati] |
✓ |
2.164 |
|
|
\[
{}y^{\prime } = y^{2}+x f \left (x \right ) y+f \left (x \right )
\] |
[_Riccati] |
✓ |
1.733 |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1}
\] |
[_Riccati] |
✓ |
2.749 |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right )
\] |
[_Riccati] |
✗ |
2.958 |
|
|
\[
{}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right )
\] |
[_Riccati] |
✓ |
2.960 |
|
|
\[
{}y^{\prime } x = y^{2} f \left (x \right )+n y+a \,x^{2 n} f \left (x \right )
\] |
[_Riccati] |
✓ |
2.187 |
|
|
\[
{}y^{\prime } x = x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+b f \left (x \right )
\] |
[_Riccati] |
✗ |
5.442 |
|
|
\[
{}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y-a^{2} f \left (x \right )-a g \left (x \right )
\] |
[_Riccati] |
✓ |
2.100 |
|