Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.
|
# |
ODE |
A |
B |
C |
Program classification |
CAS classification |
Solved? |
Verified? |
time (sec) |
|
\[ {}y^{\prime } = f \left (x \right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.192 |
|
|
\[ {}y^{\prime } = f \left (y\right ) \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.25 |
|
|
\[ {}y^{\prime } = f \left (x \right ) g \left (y\right ) \] |
1 |
1 |
1 |
exact, separable, first_order_ode_lie_symmetry_lookup |
[_separable] |
✓ |
✓ |
0.701 |
|
|
\[ {}g \left (x \right ) y^{\prime } = f_{1} \left (x \right ) y+f_{0} \left (x \right ) \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
1.224 |
|
|
\[ {}g \left (x \right ) y^{\prime } = f_{1} \left (x \right ) y+f_{n} \left (x \right ) y^{n} \] |
1 |
1 |
1 |
bernoulli, first_order_ode_lie_symmetry_lookup |
[_Bernoulli] |
✓ |
✓ |
0.948 |
|
|
\[ {}y^{\prime } = f \left (\frac {y}{x}\right ) \] |
1 |
1 |
1 |
homogeneousTypeD2, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class A‘], _dAlembert] |
✓ |
✓ |
1.033 |
|
|
\[ {}y^{\prime } = a y^{2}+b x +c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.483 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2} x^{2}+3 a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.68 |
|
|
\[ {}y^{\prime } = y^{2}+a^{2} x^{2}+b x +c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
14.279 |
|
|
\[ {}y^{\prime } = a y^{2}+b \,x^{n} \] |
1 |
1 |
1 |
riccati |
[[_Riccati, _special]] |
✓ |
✓ |
2.231 |
|
|
\[ {}y^{\prime } = y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
118.678 |
|
|
\[ {}y^{\prime } = a y^{2}+b \,x^{2 n}+c \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
36.872 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{-n -2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _Riccati] |
✓ |
✓ |
2.179 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.477 |
|
|
\[ {}y^{\prime } = y^{2}+k \left (x a +b \right )^{n} \left (c x +d \right )^{-n -4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
5.879 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
128.141 |
|
|
\[ {}y^{\prime } = \left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
90.599 |
|
|
\[ {}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
94.806 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, [_Riccati, _special]] |
✓ |
✓ |
4.039 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right ) \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.519 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b \,x^{n}+c \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.32 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}+a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.649 |
|
|
\[ {}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
143.639 |
|
|
\[ {}x^{4} y^{\prime } = -x^{4} y^{2}-a^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_Riccati, _special]] |
✓ |
✓ |
2.489 |
|
|
\[ {}a \,x^{2} \left (-1+x \right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
7.046 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, _Riccati] |
✓ |
✓ |
6.493 |
|
|
\[ {}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+c \,x^{m}+d \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.091 |
|
|
\[ {}\left (a \,x^{n}+b \right ) y^{\prime } = b y^{2}+a \,x^{n -2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
5.682 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
7.027 |
|
|
\[ {}y^{\prime } = a y^{2}+b y+c x +k \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.887 |
|
|
\[ {}y^{\prime } = y^{2}+a \,x^{n} y+a \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
130.475 |
|
|
\[ {}y^{\prime } = y^{2}+a \,x^{n} y+b \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.266 |
|
|
\[ {}y^{\prime } = y^{2}+\left (\alpha x +\beta \right ) y+x^{2} a +b x +c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
67.095 |
|
|
\[ {}y^{\prime } = y^{2}+a \,x^{n} y-a b \,x^{n}-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.001 |
|
|
\[ {}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{1+m +n}-a \,x^{m} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
5.684 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+b c \,x^{m}-a \,c^{2} x^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.307 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,x^{m}+c \right ) y+b m \,x^{m -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
5.657 |
|
|
\[ {}y^{\prime } = -a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
31.73 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b \,x^{m} y+c k \,x^{k -1}-b c \,x^{m +k}-a \,c^{2} x^{n +2 k} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
9.673 |
|
|
\[ {}x y^{\prime } = a y^{2}+b y+c \,x^{2 b} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.538 |
|
|
\[ {}x y^{\prime } = a y^{2}+b y+c \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.43 |
|
|
\[ {}x y^{\prime } = a y^{2}+\left (n +b \,x^{n}\right ) y+c \,x^{2 n} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.291 |
|
|
\[ {}x y^{\prime } = x y^{2}+a y+b \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.086 |
|
|
\[ {}x y^{\prime }+a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
7.144 |
|
|
\[ {}x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{-n} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
3.868 |
|
|
\[ {}x y^{\prime } = a \,x^{n} y^{2}+m y-a \,b^{2} x^{n +2 m} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.948 |
|
|
\[ {}x y^{\prime } = x^{2 n} y^{2}+\left (m -n \right ) y+x^{2 m} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.868 |
|
|
\[ {}x y^{\prime } = a \,x^{n} y^{2}+b y+c \,x^{m} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.839 |
|
|
\[ {}x y^{\prime } = a \,x^{2 n} y^{2}+\left (b \,x^{n}-n \right ) y+c \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.958 |
|
|
\[ {}x y^{\prime } = a \,x^{2 n +m} y^{2}+\left (b \,x^{m +n}-n \right ) y+c \,x^{m} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
5.187 |
|
|
\[ {}\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y+a_{0} x +b_{0} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
33.569 |
|
|
\[ {}\left (x a +c \right ) y^{\prime } = \alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _rational, _Riccati] |
✓ |
✓ |
5.273 |
|
|
\[ {}2 x^{2} y^{\prime } = 2 y^{2}+x y-2 x \,a^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
2.02 |
|
|
\[ {}2 x^{2} y^{\prime } = 2 y^{2}+3 x y-2 x \,a^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.115 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \] |
1 |
1 |
1 |
riccati, exactWithIntegrationFactor, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class G‘], _rational, _Riccati] |
✓ |
✓ |
4.879 |
|
|
\[ {}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (x^{2} a +b x \right ) y+\alpha \,x^{2}+\beta x +\gamma \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
123.02 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{n}+s \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
5.068 |
|
|
\[ {}x^{2} y^{\prime } = a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
7.152 |
|
|
\[ {}x^{2} y^{\prime } = c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
161.362 |
|
|
\[ {}x^{2} y^{\prime } = \left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
70.098 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.589 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha } = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.492 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
57.786 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime }+y^{2}-2 x y+\left (1-a \right ) x^{2}-b = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.773 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime } = y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
6.681 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime } = y^{2}+\left (x a +\mu \right ) y-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +c \lambda \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
145.468 |
|
|
\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
158.893 |
|
|
\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime } = y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
157.181 |
|
|
\[ {}\left (x -a \right ) \left (-b +x \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right ) = 0 \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
5.77 |
|
|
\[ {}\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y+a_{0} = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
35.999 |
|
|
\[ {}x^{3} y^{\prime } = a \,x^{3} y^{2}+\left (b \,x^{2}+c \right ) y+s x \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
35.645 |
|
|
\[ {}x^{3} y^{\prime } = a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
83.606 |
|
|
\[ {}x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
4.635 |
|
|
\[ {}x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y+\alpha x +\beta = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
6.953 |
|
|
\[ {}\left (x^{2} a +b x +e \right ) \left (-y+x y^{\prime }\right )-y^{2}+x^{2} = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
2.059 |
|
|
\[ {}x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y+s = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
7.855 |
|
|
\[ {}a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y+c \,x^{2}+d x +s = 0 \] |
1 |
1 |
0 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
3.535 |
|
|
\[ {}x^{n +1} y^{\prime } = a \,x^{2 n} y^{2}+b \,x^{n} y+c \,x^{m}+d \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.516 |
|
|
\[ {}x \left (a \,x^{k}+b \right ) y^{\prime } = \alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
5.438 |
|
|
\[ {}x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s = 0 \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✗ |
15.242 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = c y^{2}-b \,x^{m -1} y+a \,x^{n -2} \] |
1 |
1 |
1 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
94.876 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = a \,x^{n -2} y^{2}+b \,x^{m -1} y+c \] |
1 |
1 |
0 |
riccati |
[_rational, _Riccati] |
✓ |
✓ |
83.819 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime } = \alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \] |
1 |
0 |
1 |
riccati |
[_rational, _Riccati] |
✗ |
N/A |
105.711 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+x y^{\prime }\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right ) = 0 \] |
1 |
1 |
1 |
riccati, homogeneousTypeD2 |
[[_homogeneous, ‘class D‘], _rational, _Riccati] |
✓ |
✓ |
3.273 |
|
|
\[ {}y^{\prime } = a y^{2}+b \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.061 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.118 |
|
|
\[ {}y^{\prime } = \sigma y^{2}+a +b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.621 |
|
|
\[ {}y^{\prime } = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.375 |
|
|
\[ {}y^{\prime } = y^{2}+b y+a \left (\lambda -b \right ) {\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.973 |
|
|
\[ {}y^{\prime } = y^{2}+a \,{\mathrm e}^{\lambda x} y-a b \,{\mathrm e}^{\lambda x}-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.715 |
|
|
\[ {}y^{\prime } = y^{2}+a \,{\mathrm e}^{2 \lambda x} \left ({\mathrm e}^{\lambda x}+b \right )^{n}-\frac {\lambda ^{2}}{4} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
106.799 |
|
|
\[ {}y^{\prime } = y^{2}+a \,{\mathrm e}^{8 \lambda x}+b \,{\mathrm e}^{6 \lambda x}+c \,{\mathrm e}^{4 \lambda x}-\lambda ^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✗ |
8.615 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b \,{\mathrm e}^{s x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.537 |
|
|
\[ {}y^{\prime } = b \,{\mathrm e}^{\mu x} y^{2}+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} b \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.774 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b y+c \,{\mathrm e}^{-\lambda x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
2.21 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+\lambda y-a \,b^{2} {\mathrm e}^{\left (\mu +2 \lambda \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.217 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y+a \lambda \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.423 |
|
|
\[ {}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\mu x} y-a \,{\mathrm e}^{\left (\mu -\lambda \right ) x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.789 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\mu x} y^{2}+a b \,{\mathrm e}^{x \left (\lambda +\mu \right )} y-b \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
110.655 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.636 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\left (\mu +2 \lambda \right ) x} y^{2}+\left (b \,{\mathrm e}^{x \left (\lambda +\mu \right )}-\lambda \right ) y+c \,{\mathrm e}^{\mu x} \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
2.656 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{k n x}+d \,{\mathrm e}^{k \left (2 n +1\right ) x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.434 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\mu x} \left (y-b \,{\mathrm e}^{\lambda x}\right )^{2}+b \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
1.938 |
|
|
\[ {}\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime } = y^{2}+k \,{\mathrm e}^{x \nu } y-m^{2}+k m \,{\mathrm e}^{x \nu } \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
80.895 |
|
|
\[ {}\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x} = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.68 |
|
|
\[ {}y^{\prime } = y^{2}+a x \,{\mathrm e}^{\lambda x} y+{\mathrm e}^{\lambda x} a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.27 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b \,{\mathrm e}^{-\lambda x} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
2.273 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+b n \,x^{n -1}-a \,b^{2} {\mathrm e}^{\lambda x} x^{2 n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
4.69 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.981 |
|
|
\[ {}y^{\prime } = -\lambda \,{\mathrm e}^{\lambda x} y^{2}+a \,x^{n} {\mathrm e}^{\lambda x} y-a \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.862 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.889 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
4.379 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.848 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.435 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-{\mathrm e}^{\lambda x} a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.784 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}-a \,x^{n} \left (b \,{\mathrm e}^{\lambda x}+c \right ) y+c \,x^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
5.43 |
|
|
\[ {}y^{\prime } = a \,x^{n} {\mathrm e}^{2 \lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-\lambda \right ) y+c \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
10.79 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.992 |
|
|
\[ {}x y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+k y+a \,b^{2} x^{2 k} {\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.052 |
|
|
\[ {}x y^{\prime } = a \,x^{2 n} {\mathrm e}^{\lambda x} y^{2}+\left (b \,x^{n} {\mathrm e}^{\lambda x}-n \right ) y+c \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.039 |
|
|
\[ {}y^{\prime } = y^{2}+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} {\mathrm e}^{2 \lambda \,x^{2}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.316 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{-\lambda \,x^{2}} y^{2}+\lambda x y+a \,b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.102 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}+\lambda x y+a \,b^{2} x^{n} {\mathrm e}^{\lambda \,x^{2}} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.352 |
|
|
\[ {}x^{4} \left (y^{\prime }-y^{2}\right ) = a +b \,{\mathrm e}^{\frac {k}{x}}+c \,{\mathrm e}^{\frac {2 k}{x}} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
25.376 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.966 |
|
|
\[ {}y^{\prime } = y^{2}+a \sinh \left (\beta x \right ) y+a b \sinh \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.682 |
|
|
\[ {}y^{\prime } = y^{2}+a x \sinh \left (b x \right )^{m} y+a \sinh \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
13.319 |
|
|
\[ {}y^{\prime } = \lambda \sinh \left (\lambda x \right ) y^{2}-\lambda \sinh \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.879 |
|
|
\[ {}y^{\prime } = \left (a \sinh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}-a \sinh \left (\lambda x \right )^{2}+\lambda -a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
13.552 |
|
|
\[ {}\left (a \sinh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sinh \left (\mu x \right ) y-d^{2}+c d \sinh \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
109.078 |
|
|
\[ {}\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \sinh \left (\lambda x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.491 |
|
|
\[ {}y^{\prime } = \alpha y^{2}+\beta +\gamma \cosh \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.461 |
|
|
\[ {}y^{\prime } = y^{2}+a \cosh \left (\beta x \right ) y+a b \cosh \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.974 |
|
|
\[ {}y^{\prime } = y^{2}+a x \cosh \left (b x \right )^{m} y+a \cosh \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.689 |
|
|
\[ {}y^{\prime } = \left (a \cosh \left (\lambda x \right )^{2}-\lambda \right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
11.165 |
|
|
\[ {}2 y^{\prime } = \left (a -\lambda +a \cosh \left (\lambda x \right )\right ) y^{2}+a +\lambda -a \cosh \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
17.17 |
|
|
\[ {}y^{\prime } = y^{2}-\lambda ^{2}+a \cosh \left (\lambda x \right )^{n} \sinh \left (\lambda x \right )^{-n -4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
27.444 |
|
|
\[ {}y^{\prime } = a \sinh \left (\lambda x \right ) y^{2}+b \sinh \left (\lambda x \right ) \cosh \left (\lambda x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.098 |
|
|
\[ {}y^{\prime } = a \cosh \left (\lambda x \right ) y^{2}+b \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
11.575 |
|
|
\[ {}\left (a \cosh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cosh \left (\mu x \right ) y-d^{2}+c d \cosh \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
155.652 |
|
|
\[ {}\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.684 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.315 |
|
|
\[ {}y^{\prime } = y^{2}+3 a \lambda -\lambda ^{2}-a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.465 |
|
|
\[ {}y^{\prime } = y^{2}+a x \tanh \left (b x \right )^{m} y+a \tanh \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.38 |
|
|
\[ {}\left (a \tanh \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \tanh \left (\mu x \right ) y-d^{2}+c d \tanh \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
87.704 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.385 |
|
|
\[ {}y^{\prime } = y^{2}-\lambda ^{2}+3 a \lambda -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.603 |
|
|
\[ {}y^{\prime } = y^{2}+a x \coth \left (b x \right )^{m} y+a \coth \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.881 |
|
|
\[ {}\left (a \coth \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \coth \left (\mu x \right ) y-d^{2}+c d \coth \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
109.109 |
|
|
\[ {}y^{\prime } = y^{2}-2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}-2 \lambda ^{2} \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
19.498 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +b \lambda -2 a b -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
38.283 |
|
|
\[ {}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{2 m} \ln \left (x \right )^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
4.673 |
|
|
\[ {}x y^{\prime } = a y^{2}+b \ln \left (x \right )+c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.865 |
|
|
\[ {}x y^{\prime } = a y^{2}+b \ln \left (x \right )^{k}+c \ln \left (x \right )^{2 k +2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
32.25 |
|
|
\[ {}x y^{\prime } = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2}+a \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.323 |
|
|
\[ {}x y^{\prime } = x y^{2}-a^{2} x \ln \left (\beta x \right )^{2 k}+a k \ln \left (\beta x \right )^{k -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.624 |
|
|
\[ {}x y^{\prime } = a \,x^{n} y^{2}+b -a \,b^{2} x^{n} \ln \left (x \right )^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.071 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}+a \ln \left (x \right )^{2}+b \ln \left (x \right )+c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.459 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}+a \left (b \ln \left (x \right )+c \right )^{n}+\frac {1}{4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.644 |
|
|
\[ {}x^{2} \ln \left (x a \right ) \left (y^{\prime }-y^{2}\right ) = 1 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.682 |
|
|
\[ {}y^{\prime } = y^{2}+a \ln \left (\beta x \right ) y-a b \ln \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.299 |
|
|
\[ {}y^{\prime } = y^{2}+a x \ln \left (b x \right )^{m} y+a \ln \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.332 |
|
|
\[ {}y^{\prime } = a \,x^{n} y^{2}-a b \,x^{n +1} \ln \left (x \right ) y+b \ln \left (x \right )+b \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.271 |
|
|
\[ {}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.411 |
|
|
\[ {}y^{\prime } = a \ln \left (x \right )^{n} y-a b x \ln \left (x \right )^{n +1} y+b \ln \left (x \right )+b \] |
1 |
1 |
1 |
linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup |
[_linear] |
✓ |
✓ |
2.198 |
|
|
\[ {}y^{\prime } = a \ln \left (x \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.208 |
|
|
\[ {}y^{\prime } = a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.691 |
|
|
\[ {}x y^{\prime } = \left (a y+b \ln \left (x \right )\right )^{2} \] |
1 |
1 |
1 |
riccati, first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], _Riccati] |
✓ |
✓ |
1.66 |
|
|
\[ {}x y^{\prime } = a \ln \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \ln \left (\lambda x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.96 |
|
|
\[ {}x y^{\prime } = a \,x^{n} \left (y+b \ln \left (x \right )\right )^{2}-b \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.03 |
|
|
\[ {}x y^{\prime } = a \,x^{2 n} \ln \left (x \right ) y^{2}+\left (b \,x^{n} \ln \left (x \right )-n \right ) y+c \ln \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.723 |
|
|
\[ {}x^{2} y^{\prime } = y^{2} a^{2} x^{2}-x y+b^{2} \ln \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.816 |
|
|
\[ {}\left (a \ln \left (x \right )+b \right ) y^{\prime } = y^{2}+c \ln \left (x \right )^{n} y-\lambda ^{2}+\lambda c \ln \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.789 |
|
|
\[ {}\left (a \ln \left (x \right )+b \right ) y^{\prime } = \ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+c \lambda \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
10.08 |
|
|
\[ {}y^{\prime } = \alpha y^{2}+\beta +\gamma \sin \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.474 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.052 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x +a \right )^{n} \sin \left (\lambda x +b \right )^{-n -4} \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
101.086 |
|
|
\[ {}y^{\prime } = y^{2}+a \sin \left (\beta x \right ) y+a b \sin \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.875 |
|
|
\[ {}y^{\prime } = y^{2}+a \sin \left (b x \right )^{m} y+a \sin \left (b x \right )^{m} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
21.685 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+\lambda \sin \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.08 |
|
|
\[ {}2 y^{\prime } = \left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✗ |
165.824 |
|
|
\[ {}y^{\prime } = \left (\lambda +a \sin \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \sin \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.181 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \sin \left (x \right )^{m} y-a \sin \left (x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
20.835 |
|
|
\[ {}y^{\prime } = a \sin \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
4.291 |
|
|
\[ {}x y^{\prime } = a \sin \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \sin \left (\lambda x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
12.188 |
|
|
\[ {}\left (a \sin \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \sin \left (\mu x \right ) y-d^{2}+c d \sin \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
75.029 |
|
|
\[ {}\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \sin \left (\lambda x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.806 |
|
|
\[ {}y^{\prime } = \alpha y^{2}+\beta +\gamma \cos \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.688 |
|
|
\[ {}y^{\prime } = y^{2}-a^{2}+a \lambda \cos \left (\lambda x \right )+\cos \left (\lambda x \right )^{2} a^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.29 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+c \cos \left (\lambda x +a \right )^{n} \cos \left (\lambda x +b \right )^{-n -4} \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
90.988 |
|
|
\[ {}y^{\prime } = y^{2}+a \cos \left (\beta x \right ) y+a b \cos \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.187 |
|
|
\[ {}y^{\prime } = y^{2}+a \cos \left (b x \right )^{m} y+a \cos \left (b x \right )^{m} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
13.831 |
|
|
\[ {}y^{\prime } = \lambda \cos \left (\lambda x \right ) y^{2}+\lambda \cos \left (\lambda x \right )^{3} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
15.277 |
|
|
\[ {}2 y^{\prime } = \left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \cos \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
26.537 |
|
|
\[ {}y^{\prime } = \left (\lambda +a \cos \left (\lambda x \right )^{2}\right ) y^{2}+\lambda -a +a \cos \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.991 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cos \left (x \right )^{m} y-a \cos \left (x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
21.884 |
|
|
\[ {}y^{\prime } = a \cos \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
5.94 |
|
|
\[ {}x y^{\prime } = a \cos \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cos \left (\lambda x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
13.526 |
|
|
\[ {}\left (a \cos \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cos \left (\mu x \right ) y-d^{2}+c d \cos \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
75.735 |
|
|
\[ {}\left (a \cos \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )-a \,\lambda ^{2} \cos \left (\lambda x \right ) = 0 \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.257 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.743 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.959 |
|
|
\[ {}y^{\prime } = a y^{2}+b \tan \left (x \right ) y+c \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.842 |
|
|
\[ {}y^{\prime } = a y^{2}+2 a b \tan \left (x \right ) y+b \left (a b -1\right ) \tan \left (x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
2.634 |
|
|
\[ {}y^{\prime } = y^{2}+a \tan \left (\beta x \right ) y+a b \tan \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
62.734 |
|
|
\[ {}y^{\prime } = y^{2}+a x \tan \left (b x \right )^{m} y+a \tan \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
11.513 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \tan \left (x \right )^{m} y-a \tan \left (x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
19.612 |
|
|
\[ {}y^{\prime } = a \tan \left (\lambda x \right )^{n} y^{2}-a \,b^{2} \tan \left (\lambda x \right )^{n +2}+b \lambda \tan \left (\lambda x \right )^{2}+b \lambda \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
67.178 |
|
|
\[ {}y^{\prime } = a \tan \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
0 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
N/A |
65.097 |
|
|
\[ {}x y^{\prime } = a \tan \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \tan \left (\lambda x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
20.34 |
|
|
\[ {}\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+k \tan \left (\mu x \right ) y-d^{2}+k d \tan \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
74.821 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.776 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+3 a \lambda +a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.389 |
|
|
\[ {}y^{\prime } = y^{2}-2 a b \cot \left (x a \right ) y+b^{2}-a^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
19.195 |
|
|
\[ {}y^{\prime } = y^{2}+a \cot \left (\beta x \right ) y+a b \cot \left (\beta x \right )-b^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.083 |
|
|
\[ {}y^{\prime } = y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.09 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
19.724 |
|
|
\[ {}y^{\prime } = a \cot \left (\lambda x +\mu \right )^{k} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \] |
1 |
0 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✗ |
N/A |
64.715 |
|
|
\[ {}x y^{\prime } = a \cot \left (\lambda x \right )^{m} y^{2}+k y+a \,b^{2} x^{2 k} \cot \left (\lambda x \right )^{m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
13.916 |
|
|
\[ {}\left (a \cot \left (\lambda x \right )+b \right ) y^{\prime } = y^{2}+c \cot \left (\mu x \right ) y-d^{2}+c d \cot \left (\mu x \right ) \] |
1 |
0 |
1 |
riccati |
[_Riccati] |
✗ |
N/A |
76.847 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+c \sin \left (\lambda x \right )^{n} \cos \left (\lambda x \right )^{-n -4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
16.077 |
|
|
\[ {}y^{\prime } = a \sin \left (\lambda x \right ) y^{2}+b \sin \left (\lambda x \right ) \cos \left (\lambda x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.107 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \cos \left (\lambda x \right )^{n} y-a \cos \left (\lambda x \right )^{n -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
43.741 |
|
|
\[ {}y^{\prime } = a \cos \left (\lambda x \right ) y^{2}+b \cos \left (\lambda x \right ) \sin \left (\lambda x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
7.491 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \,x^{n} \cos \left (\lambda x \right ) y-a \,x^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.113 |
|
|
\[ {}\sin \left (2 x \right )^{n +1} y^{\prime } = a y^{2} \sin \left (x \right )^{2 n}+b \cos \left (x \right )^{2 n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
14.986 |
|
|
\[ {}y^{\prime } = y^{2}-y \tan \left (x \right )+a \left (1-a \right ) \cot \left (x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
6.871 |
|
|
\[ {}y^{\prime } = y^{2}-m y \tan \left (x \right )+b^{2} \cos \left (x \right )^{2 m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
117.879 |
|
|
\[ {}y^{\prime } = y^{2}+m y \cot \left (x \right )+b^{2} \sin \left (x \right )^{2 m} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
56.028 |
|
|
\[ {}y^{\prime } = y^{2}-2 \lambda ^{2} \tan \left (x \right )^{2}-2 \lambda ^{2} \cot \left (\lambda x \right )^{2} \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
108.796 |
|
|
\[ {}y^{\prime } = y^{2}+a \lambda +b \lambda +2 a b +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.46 |
|
|
\[ {}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{2}-\frac {3 \lambda ^{2} \tan \left (\lambda x \right )^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
15.905 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+a \sin \left (\lambda x \right ) y-a \tan \left (\lambda x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.273 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda \arcsin \left (x \right )^{n} y-a^{2}+a \lambda \arcsin \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.146 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda x \arcsin \left (x \right )^{n} y+\arcsin \left (x \right )^{n} \lambda \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.367 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arcsin \left (x \right )^{n} \left (x^{k +1} y-1\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
50.332 |
|
|
\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arcsin \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.394 |
|
|
\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arcsin \left (x \right )^{n} y+b m \,x^{m -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
18.415 |
|
|
\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arcsin \left (x \right )^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
35.628 |
|
|
\[ {}y^{\prime } = \lambda \arcsin \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
18.722 |
|
|
\[ {}x y^{\prime } = \lambda \arcsin \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arcsin \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
22.781 |
|
|
\[ {}x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arcsin \left (x \right )^{m}-n y \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
37.218 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda \arccos \left (x \right )^{n} y-a^{2}+a \lambda \arccos \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✗ |
15.007 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda x \arccos \left (x \right )^{n} y+\lambda \arccos \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.089 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{k +1} y-1\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
45.421 |
|
|
\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arccos \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
9.422 |
|
|
\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arccos \left (x \right )^{n} y+b m \,x^{m -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
22.444 |
|
|
\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arccos \left (x \right )^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
42.609 |
|
|
\[ {}y^{\prime } = \lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
24.128 |
|
|
\[ {}x y^{\prime } = \lambda \arccos \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arccos \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
23.808 |
|
|
\[ {}x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arccos \left (x \right )^{m}-n y \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
50.481 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda \arctan \left (x \right )^{n} y-a^{2}+a \lambda \arctan \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.76 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda x \arctan \left (x \right )^{n} y+\arctan \left (x \right )^{n} \lambda \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.543 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \arctan \left (x \right )^{n} \left (x^{k +1} y-1\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
38.991 |
|
|
\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \arctan \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.361 |
|
|
\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}-b \lambda \,x^{m} \arctan \left (x \right )^{n} y+b m \,x^{m -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
37.891 |
|
|
\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \arctan \left (x \right )^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
38.494 |
|
|
\[ {}y^{\prime } = \lambda \arctan \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
17.001 |
|
|
\[ {}x y^{\prime } = \lambda \arctan \left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
34.692 |
|
|
\[ {}x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \arctan \left (x \right )^{m}-n y \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
38.444 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} y-a^{2}+a \lambda \operatorname {arccot}\left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
4.935 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\operatorname {arccot}\left (x \right )^{n} \lambda \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
14.819 |
|
|
\[ {}y^{\prime } = -\left (k +1\right ) x^{k} y^{2}+\lambda \operatorname {arccot}\left (x \right )^{n} \left (x^{k +1} y-1\right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
40.262 |
|
|
\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+a y+a b -b^{2} \lambda \operatorname {arccot}\left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
8.333 |
|
|
\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}-b \lambda \,x^{m} \operatorname {arccot}\left (x \right )^{n} y+b m \,x^{m -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
40.612 |
|
|
\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+\beta m \,x^{m -1}-\lambda \,\beta ^{2} x^{2 m} \operatorname {arccot}\left (x \right )^{n} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
40.046 |
|
|
\[ {}y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
22.6 |
|
|
\[ {}x y^{\prime } = \lambda \operatorname {arccot}\left (x \right )^{n} y^{2}+k y+\lambda \,b^{2} x^{2 k} \operatorname {arccot}\left (x \right )^{n} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
35.312 |
|
|
\[ {}x y^{\prime } = \left (a \,x^{2 m} y^{2}+b \,x^{n} y+c \right ) \operatorname {arccot}\left (x \right )^{m}-n y \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
56.868 |
|
|
\[ {}y^{\prime } = y^{2}+f \left (x \right ) y-a^{2}-a f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.848 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a y-a b -b^{2} f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.316 |
|
|
\[ {}y^{\prime } = y^{2}+x f \left (x \right ) y+f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.094 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \,x^{n} f \left (x \right ) y+a n \,x^{n -1} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.922 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+a n \,x^{n -1}-a^{2} x^{2 n} f \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.168 |
|
|
\[ {}y^{\prime } = -\left (n +1\right ) x^{n} y^{2}+x^{n +1} f \left (x \right ) y-f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.671 |
|
|
\[ {}x y^{\prime } = y^{2} f \left (x \right )+n y+a \,x^{2 n} f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.434 |
|
|
\[ {}x y^{\prime } = x^{2 n} f \left (x \right ) y^{2}+\left (a \,x^{n} f \left (x \right )-n \right ) y+f \left (x \right ) b \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.541 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y-a^{2} f \left (x \right )-a g \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.256 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+a n \,x^{n -1}-a \,x^{n} g \left (x \right )-a^{2} x^{2 n} f \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.632 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \,x^{n} g \left (x \right ) y+a n \,x^{n -1}+a^{2} x^{2 n} \left (g \left (x \right )-f \left (x \right )\right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.948 |
|
|
\[ {}y^{\prime } = a \,{\mathrm e}^{\lambda x} y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+\lambda f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.315 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
4.577 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+a \lambda \,{\mathrm e}^{\lambda x}-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.252 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+\lambda y+a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
2.203 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-f \left (x \right ) \left ({\mathrm e}^{\lambda x} a +b \right ) y+a \lambda \,{\mathrm e}^{\lambda x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
5.406 |
|
|
\[ {}y^{\prime } = {\mathrm e}^{\lambda x} f \left (x \right ) y^{2}+\left (a f \left (x \right )-\lambda \right ) y+b \,{\mathrm e}^{-\lambda x} f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
3.375 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{\lambda x} g \left (x \right )-a^{2} {\mathrm e}^{2 \lambda x} f \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.322 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \,{\mathrm e}^{\lambda x} g \left (x \right ) y+a \lambda \,{\mathrm e}^{\lambda x}+a^{2} {\mathrm e}^{2 \lambda x} \left (g \left (x \right )-f \left (x \right )\right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
3.463 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+2 a \lambda x \,{\mathrm e}^{\lambda \,x^{2}}-a^{2} f \left (x \right ) {\mathrm e}^{2 \lambda \,x^{2}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.914 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+\lambda x y+a f \left (x \right ) {\mathrm e}^{\lambda x} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
2.235 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \tanh \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
266.436 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \coth \left (\lambda x \right )^{2} \left (a f \left (x \right )+\lambda \right )+a \lambda \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
166.462 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \sinh \left (\lambda x \right )-a^{2} f \left (x \right ) \sinh \left (\lambda x \right )^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
49.226 |
|
|
\[ {}x y^{\prime } = y^{2} f \left (x \right )+a -a^{2} f \left (x \right ) \ln \left (x \right )^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
4.026 |
|
|
\[ {}x y^{\prime } = f \left (x \right ) \left (y+a \ln \left (x \right )\right )^{2}-a \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
5.224 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a x \ln \left (x \right ) f \left (x \right ) y+a \ln \left (x \right )+a \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.733 |
|
|
\[ {}y^{\prime } = -a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
5.499 |
|
|
\[ {}y^{\prime } = \lambda \sin \left (\lambda x \right ) y^{2}+f \left (x \right ) \cos \left (\lambda x \right ) y-f \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
10.557 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \sin \left (\lambda x \right )+a^{2} f \left (x \right ) \sin \left (\lambda x \right )^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
53.092 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a^{2} f \left (x \right )+a \lambda \cos \left (\lambda x \right )+a^{2} f \left (x \right ) \cos \left (\lambda x \right )^{2} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
53.049 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \tan \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
176.561 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-a \cot \left (\lambda x \right )^{2} \left (a f \left (x \right )-\lambda \right )+a \lambda \] |
1 |
0 |
0 |
riccati |
[_Riccati] |
❇ |
N/A |
151.456 |
|
|
\[ {}y^{\prime } = y^{2}-f \left (x \right )^{2}+f^{\prime }\left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.621 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )-f \left (x \right ) g \left (x \right ) y+g^{\prime }\left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.8 |
|
|
\[ {}y^{\prime } = -f^{\prime }\left (x \right ) y^{2}+f \left (x \right ) g \left (x \right ) y-g \left (x \right ) \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.904 |
|
|
\[ {}y^{\prime } = g \left (x \right ) \left (y-f \left (x \right )\right )^{2}+f^{\prime }\left (x \right ) \] |
1 |
1 |
1 |
riccati |
[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati] |
✓ |
✓ |
0.834 |
|
|
\[ {}y^{\prime } = \frac {f^{\prime }\left (x \right ) y^{2}}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.332 |
|
|
\[ {}f \left (x \right )^{2} y^{\prime }-f^{\prime }\left (x \right ) y^{2}+g \left (x \right ) \left (y-f \left (x \right )\right ) = 0 \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.255 |
|
|
\[ {}y^{\prime } = f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+{\mathrm e}^{\lambda x} a \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
1.279 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+g^{\prime }\left (x \right ) y+a f \left (x \right ) {\mathrm e}^{2 g \left (x \right )} \] |
1 |
1 |
1 |
riccati |
[_Riccati] |
✓ |
✓ |
0.686 |
|
|
\[ {}y^{\prime } = y^{2}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )} \] |
1 |
0 |
1 |
unknown |
[_Riccati] |
✗ |
N/A |
0.245 |
|
|
\[ {}y^{\prime } = y^{2}+a^{2} f \left (x a +b \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.684 |
|
|
\[ {}y^{\prime } = y^{2}+\frac {f \left (\frac {1}{x}\right )}{x^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.774 |
|
|
\[ {}y^{\prime } = y^{2}+\frac {f \left (\frac {x a +b}{c x +d}\right )}{\left (c x +d \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.447 |
|
|
\[ {}x^{2} y^{\prime } = x^{4} f \left (x \right ) y^{2}+1 \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.27 |
|
|
\[ {}x^{2} y^{\prime } = x^{4} y^{2}+x^{2 n} f \left (a \,x^{n}+b \right )-\frac {n^{2}}{4}+\frac {1}{4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.606 |
|
|
\[ {}y^{\prime } = y^{2} f \left (x \right )+g \left (x \right ) y+h \left (x \right ) \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
0.952 |
|
|
\[ {}y^{\prime } = y^{2}+{\mathrm e}^{2 \lambda x} f \left ({\mathrm e}^{\lambda x}\right )-\frac {\lambda ^{2}}{4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.012 |
|
|
\[ {}y^{\prime } = y^{2}-\frac {\lambda ^{2}}{4}+\frac {{\mathrm e}^{2 \lambda x} f \left (\frac {{\mathrm e}^{\lambda x} a +b}{c \,{\mathrm e}^{\lambda x}+d}\right )}{\left (c \,{\mathrm e}^{\lambda x}+d \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
14.124 |
|
|
\[ {}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\coth \left (\lambda x \right )\right )}{\sinh \left (\lambda x \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
28.228 |
|
|
\[ {}y^{\prime } = y^{2}-\lambda ^{2}+\frac {f \left (\tanh \left (\lambda x \right )\right )}{\cosh \left (\lambda x \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
10.156 |
|
|
\[ {}x^{2} y^{\prime } = x^{2} y^{2}+f \left (a \ln \left (x \right )+b \right )+\frac {1}{4} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
1.028 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\cot \left (\lambda x \right )\right )}{\sin \left (\lambda x \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
29.443 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\tan \left (\lambda x \right )\right )}{\cos \left (\lambda x \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
9.927 |
|
|
\[ {}y^{\prime } = y^{2}+\lambda ^{2}+\frac {f \left (\frac {\sin \left (\lambda x +a \right )}{\sin \left (\lambda x +b \right )}\right )}{\sin \left (\lambda x +b \right )^{4}} \] |
1 |
1 |
0 |
riccati |
[_Riccati] |
✓ |
✓ |
61.176 |
|
|
\[ {}y y^{\prime }-y = A \] |
1 |
1 |
1 |
quadrature |
[_quadrature] |
✓ |
✓ |
0.431 |
|
|
\[ {}y y^{\prime }-y = A x +B \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
19.496 |
|
|
\[ {}y y^{\prime }-y = -\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.157 |
|
|
\[ {}y y^{\prime }-y = 2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.533 |
|
|
\[ {}y y^{\prime }-y = A x +\frac {B}{x}-\frac {B^{2}}{x^{3}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.716 |
|
|
\[ {}y y^{\prime }-y = A \,x^{k -1}-k B \,x^{k}+k \,B^{2} x^{2 k -1} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.948 |
|
|
\[ {}y y^{\prime }-y = \frac {A}{x}-\frac {A^{2}}{x^{3}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.587 |
|
|
\[ {}y y^{\prime }-y = A +B \,{\mathrm e}^{-\frac {2 x}{A}} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.711 |
|
|
\[ {}y y^{\prime }-y = A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right ) \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.691 |
|
|
\[ {}y y^{\prime }-y = -\frac {2 \left (1+m \right )}{\left (3+m \right )^{2}}+A \,x^{m} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.085 |
|
|
\[ {}y y^{\prime }-y = -\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.686 |
|
|
\[ {}y y^{\prime }-y = \frac {2 m -2}{\left (m -3\right )^{2}}+\frac {2 A \left (m \left (3+m \right ) \sqrt {x}+\left (4 m^{2}+3 m +9\right ) A +\frac {3 m \left (3+m \right ) A^{2}}{\sqrt {x}}\right )}{\left (m -3\right )^{2}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
4.787 |
|
|
\[ {}y y^{\prime }-y = \frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.888 |
|
|
\[ {}y y^{\prime }-y = \frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.655 |
|
|
\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {5 A}{x^{\frac {1}{3}}}-\frac {12 A^{2}}{x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.621 |
|
|
\[ {}y y^{\prime }-y = \frac {A}{x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.483 |
|
|
\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
4.425 |
|
|
\[ {}y y^{\prime }-y = \frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.202 |
|
|
\[ {}y y^{\prime }-y = 2 x +\frac {A}{x^{2}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.499 |
|
|
\[ {}y y^{\prime }-y = -\frac {6 X}{25}+\frac {2 A \left (2 \sqrt {x}+19 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{25} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.227 |
|
|
\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {a^{2}+x^{2}}}{8}-\frac {a^{2}}{16 \sqrt {a^{2}+x^{2}}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.723 |
|
|
\[ {}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.241 |
|
|
\[ {}y y^{\prime }-y = -\frac {9 x}{100}+\frac {A}{x^{\frac {5}{3}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.227 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.263 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (25 \sqrt {x}+41 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{98} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.296 |
|
|
\[ {}y y^{\prime }-y = -\frac {2 x}{9}+\frac {A}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.135 |
|
|
\[ {}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{\frac {7}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.399 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {6 A \left (-3 \sqrt {x}+23 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.559 |
|
|
\[ {}y y^{\prime }-y = -\frac {30 x}{121}+\frac {3 A \left (21 \sqrt {x}+35 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{242} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.225 |
|
|
\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.228 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {4 A \left (-10 \sqrt {x}+27 A +\frac {10 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.65 |
|
|
\[ {}y y^{\prime }-y = \frac {A}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.033 |
|
|
\[ {}y y^{\prime }-y = \frac {A}{x^{2}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.48 |
|
|
\[ {}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.729 |
|
|
\[ {}y y^{\prime }-y = A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (3+2 n \right ) A^{2}}{\sqrt {x}}\right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.452 |
|
|
\[ {}y y^{\prime }-y = A \sqrt {x}+2 A^{2}+\frac {B}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.726 |
|
|
\[ {}y y^{\prime }-y = 2 A^{2}-A \sqrt {x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.938 |
|
|
\[ {}y y^{\prime }-y = -\frac {x}{4}+\frac {6 A \left (\sqrt {x}+8 A +\frac {5 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.223 |
|
|
\[ {}y y^{\prime }-y = -\frac {6 x}{25}+\frac {6 A \left (2 \sqrt {x}+7 A +\frac {4 A^{2}}{\sqrt {x}}\right )}{25} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.659 |
|
|
\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {3 A}{x^{\frac {1}{3}}}-\frac {12 A^{2}}{x^{\frac {5}{3}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.962 |
|
|
\[ {}y y^{\prime }-y = \frac {3 x}{8}+\frac {3 \sqrt {b^{2}+x^{2}}}{8}+\frac {3 b^{2}}{16 \sqrt {b^{2}+x^{2}}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.742 |
|
|
\[ {}y y^{\prime }-y = \frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.853 |
|
|
\[ {}y y^{\prime }-y = -\frac {3 x}{32}-\frac {3 \sqrt {a^{2}+x^{2}}}{32}+\frac {15 a^{2}}{64 \sqrt {a^{2}+x^{2}}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.006 |
|
|
\[ {}y y^{\prime }-y = A \,x^{2}-\frac {9}{625 A} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.58 |
|
|
\[ {}y y^{\prime }-y = -\frac {6}{25} x -A \,x^{2} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.482 |
|
|
\[ {}y y^{\prime }-y = \frac {6}{25} x -A \,x^{2} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.459 |
|
|
\[ {}y y^{\prime }-y = 12 x +\frac {A}{x^{\frac {5}{2}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.59 |
|
|
\[ {}y y^{\prime }-y = \frac {63 x}{4}+\frac {A}{x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.24 |
|
|
\[ {}y y^{\prime }-y = 2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.744 |
|
|
\[ {}y y^{\prime }-y = 2 x +2 A \left (-10 \sqrt {x}+19 A +\frac {30 A^{2}}{\sqrt {x}}\right ) \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.258 |
|
|
\[ {}y y^{\prime }-y = -\frac {28 x}{121}+\frac {2 A \left (5 \sqrt {x}+106 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{121} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.323 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {A \left (5 \sqrt {x}+262 A +\frac {65 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.257 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+A \sqrt {x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
2.723 |
|
|
\[ {}y y^{\prime }-y = 6 x +\frac {A}{x^{4}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.524 |
|
|
\[ {}y y^{\prime }-y = 20 x +\frac {A}{\sqrt {x}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.193 |
|
|
\[ {}y y^{\prime }-y = \frac {15 x}{4}+\frac {A}{x^{7}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.547 |
|
|
\[ {}y y^{\prime }-y = -\frac {10 x}{49}+\frac {2 A \left (4 \sqrt {x}+61 A +\frac {12 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.206 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.283 |
|
|
\[ {}y y^{\prime }-y = -\frac {4 x}{25}+\frac {A \left (7 \sqrt {x}+49 A +\frac {6 A^{2}}{\sqrt {x}}\right )}{50} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.3 |
|
|
\[ {}y y^{\prime }-y = \frac {15 x}{4}+\frac {6 A}{x^{\frac {1}{3}}}-\frac {3 A^{2}}{x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.967 |
|
|
\[ {}y y^{\prime }-y = -\frac {3 x}{16}+\frac {A}{x^{\frac {1}{3}}}+\frac {B}{x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
5.272 |
|
|
\[ {}y y^{\prime }-y = -\frac {5 x}{36}+\frac {A}{x^{\frac {3}{5}}}-\frac {B}{x^{\frac {7}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.936 |
|
|
\[ {}y y^{\prime }-y = \frac {k}{\sqrt {A \,x^{2}+B x +c}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.929 |
|
|
\[ {}y y^{\prime }-y = -\frac {12 x}{49}+3 A \left (\frac {1}{49}+B \right ) \sqrt {x}+3 A^{2} \left (\frac {4}{49}-\frac {5 B}{2}\right )+\frac {15 A^{3} \left (\frac {1}{49}-\frac {5 B}{4}\right )}{4 \sqrt {x}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.879 |
|
|
\[ {}y y^{\prime }-y = -\frac {6 x}{25}+\frac {4 B^{2} \left (\left (2-A \right ) x^{\frac {1}{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{\frac {1}{3}}}-\frac {A \,B^{3}}{x^{\frac {2}{3}}}\right )}{75} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.215 |
|
|
\[ {}y y^{\prime }-y = \frac {3 x}{4}-\frac {3 A \,x^{\frac {1}{3}}}{2}+\frac {3 A^{2}}{4 x^{\frac {1}{3}}}-\frac {27 A^{4}}{625 x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.771 |
|
|
\[ {}y y^{\prime }-y = -\frac {6 x}{25}+\frac {7 A \,x^{\frac {1}{3}}}{5}+\frac {31 A^{2}}{3 x^{\frac {1}{3}}}-\frac {100 A^{4}}{3 x^{\frac {5}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.717 |
|
|
\[ {}y y^{\prime }-y = -\frac {10 x}{49}+\frac {13 A^{2}}{5 x^{\frac {1}{5}}}-\frac {7 A^{3}}{20 x^{\frac {4}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.009 |
|
|
\[ {}y y^{\prime }-y = -\frac {33 x}{169}+\frac {286 A^{2}}{3 x^{\frac {5}{11}}}-\frac {770 A^{3}}{9 x^{\frac {13}{11}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.461 |
|
|
\[ {}y y^{\prime }-y = -\frac {21 x}{100}+\frac {7 A^{2} \left (\frac {123}{x^{\frac {1}{7}}}+\frac {280 A}{x^{\frac {5}{7}}}-\frac {400 A^{2}}{x^{\frac {9}{7}}}\right )}{9} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.716 |
|
|
\[ {}y y^{\prime }-y = x a +b \,x^{m} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.724 |
|
|
\[ {}y y^{\prime }-y = -\frac {\left (1+m \right ) x}{\left (m +2\right )^{2}}+A \,x^{2 m +1}+B \,x^{3 m +1} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.661 |
|
|
\[ {}y y^{\prime }-y = a^{2} \lambda \,{\mathrm e}^{2 \lambda x}-a \left (b \lambda +1\right ) {\mathrm e}^{\lambda x}+b \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.615 |
|
|
\[ {}y y^{\prime }-y = a^{2} \lambda \,{\mathrm e}^{2 \lambda x}+a \lambda x \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\lambda x} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.072 |
|
|
|
|||||||||
|
\[ {}y y^{\prime }-y = 2 a^{2} \lambda \sin \left (2 \lambda x \right )+2 a \sin \left (\lambda x \right ) \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.978 |
|
|
\[ {}y y^{\prime }-y = a^{2} f^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )-\frac {\left (f \left (x \right )+b \right )^{2} f^{\prime \prime }\left (x \right )}{{f^{\prime }\left (x \right )}^{3}} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.249 |
|
|
\[ {}y y^{\prime } = \left (x a +b \right ) y+1 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.652 |
|
|
\[ {}y y^{\prime } = \frac {y}{\left (x a +b \right )^{2}}+1 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.056 |
|
|
\[ {}y y^{\prime } = \left (a -\frac {1}{a x}\right ) y+1 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.592 |
|
|
\[ {}y y^{\prime } = \frac {y}{\sqrt {x a +b}}+1 \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_1st_order, _with_linear_symmetries], [_Abel, ‘2nd type‘, ‘class B‘]] |
✓ |
✓ |
18.811 |
|
|
\[ {}y y^{\prime } = \frac {3 y}{\sqrt {a \,x^{\frac {3}{2}}+8 x}}+1 \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.617 |
|
|
\[ {}y y^{\prime } = \left (\frac {a}{x^{\frac {2}{3}}}-\frac {2}{3 a \,x^{\frac {1}{3}}}\right ) y+1 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.618 |
|
|
\[ {}y y^{\prime } = a \,{\mathrm e}^{\lambda x} y+1 \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.558 |
|
|
\[ {}y y^{\prime } = \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{-\lambda x}\right ) y+1 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.842 |
|
|
\[ {}y y^{\prime } = a y \cosh \left (x \right )+1 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.768 |
|
|
\[ {}y y^{\prime } = a y \sinh \left (x \right )+1 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.849 |
|
|
\[ {}y y^{\prime } = a \cos \left (\lambda x \right ) y+1 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.878 |
|
|
\[ {}y y^{\prime } = a \sin \left (\lambda x \right ) y+1 \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.937 |
|
|
\[ {}y y^{\prime } = \left (x a +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.943 |
|
|
\[ {}y y^{\prime } = \left (3 x a +b \right ) y-a^{2} x^{3}-a b \,x^{2}+c x \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
4.573 |
|
|
\[ {}2 y y^{\prime } = \left (7 x a +5 b \right ) y-3 a^{2} x^{3}-2 c \,x^{2}-3 b^{2} x \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.571 |
|
|
\[ {}y y^{\prime } = \left (\left (3-m \right ) x -1\right ) y-\left (m -1\right ) x a \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.776 |
|
|
\[ {}y y^{\prime }+x \left (x^{2} a +b \right ) y+x = 0 \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.535 |
|
|
\[ {}y y^{\prime }+a \left (1-\frac {1}{x}\right ) y = a^{2} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.596 |
|
|
\[ {}y y^{\prime }-a \left (1-\frac {b}{x}\right ) y = a^{2} b \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.626 |
|
|
\[ {}y y^{\prime } = x^{n -1} \left (\left (2 n +1\right ) x +a n \right ) y-n \,x^{2 n} \left (x +a \right ) \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.243 |
|
|
\[ {}y y^{\prime } = a \left (-n b +x \right ) x^{n -1} y+c \left (x^{2}-\left (2 n +1\right ) b x +n \left (n +1\right ) b^{2}\right ) x^{2 n -1} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.938 |
|
|
\[ {}y y^{\prime } = \left (a \left (2 n +k \right ) x^{k}+b \right ) x^{n -1} y+\left (-a^{2} n \,x^{2 k}-a b \,x^{k}+c \right ) x^{2 n -1} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.699 |
|
|
\[ {}y y^{\prime } = \left (a \left (2 n +k \right ) x^{2 k}+b \left (2 m -k \right )\right ) x^{m -k -1} y-\frac {a^{2} m \,x^{4 k}+c \,x^{2 k}+b^{2} m}{x} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.989 |
|
|
\[ {}y y^{\prime } = \frac {\left (\left (m +2 L -3\right ) x +n -2 L +3\right ) y}{x}+\left (\left (m -L -1\right ) x^{2}+\left (n -m -2 L +3\right ) x -n +L -2\right ) x^{1-2 L} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.241 |
|
|
\[ {}y y^{\prime } = \left (a \left (2 n +1\right ) x^{2}+c x +b \left (2 n -1\right )\right ) x^{n -2} y-\left (n \,a^{2} x^{4}+a c \,x^{3}+n \,b^{2}+c b x +d \,x^{2}\right ) x^{-3+2 n} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.134 |
|
|
\[ {}y y^{\prime } = \left (a \left (n -1\right ) x +b \left (2 \lambda +n \right )\right ) x^{\lambda -1} \left (x a +b \right )^{-\lambda -2} y-\left (a x n +b \left (\lambda +n \right )\right ) x^{2 \lambda -1} \left (x a +b \right )^{-2 \lambda -3} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.847 |
|
|
\[ {}y y^{\prime }-\frac {a \left (\left (m -1\right ) x +1\right ) y}{x} = \frac {a^{2} \left (m x +1\right ) \left (-1+x \right )}{x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.837 |
|
|
\[ {}y y^{\prime }-a \left (1-\frac {b}{\sqrt {x}}\right ) y = \frac {a^{2} b}{\sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.307 |
|
|
\[ {}y y^{\prime } = \frac {3 y}{\left (x a +b \right )^{\frac {1}{3}} x^{\frac {5}{3}}}+\frac {3}{\left (x a +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.938 |
|
|
\[ {}3 y y^{\prime } = \frac {\left (-7 \lambda s \left (3 s +4 \lambda \right ) x +6 s -2 \lambda \right ) y}{x^{\frac {1}{3}}}+\frac {6 \lambda s x -6}{x^{\frac {2}{3}}}+2 \left (\lambda s \left (3 s +4 \lambda \right ) x +5 \lambda \right ) \left (-\lambda s \left (3 s +4 \lambda \right ) x +3 s +4 \lambda \right ) x^{\frac {1}{3}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.431 |
|
|
\[ {}y y^{\prime }+\frac {a \left (6 x -1\right ) y}{2 x} = -\frac {a^{2} \left (-1+x \right ) \left (4 x -1\right )}{2 x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.909 |
|
|
\[ {}y y^{\prime }-\frac {a \left (1+\frac {2 b}{x^{2}}\right ) y}{2} = \frac {a^{2} \left (3 x +\frac {4 b}{x}\right )}{16} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.853 |
|
|
\[ {}y y^{\prime }+\frac {a \left (13 x -20\right ) y}{14 x^{\frac {9}{7}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (x -8\right )}{14 x^{\frac {11}{17}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
30.817 |
|
|
\[ {}y y^{\prime }+\frac {5 a \left (23 x -16\right ) y}{56 x^{\frac {9}{7}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (25 x -32\right )}{56 x^{\frac {11}{17}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
8.818 |
|
|
\[ {}y y^{\prime }+\frac {a \left (19 x +85\right ) y}{26 x^{\frac {18}{13}}} = -\frac {3 a^{2} \left (-1+x \right ) \left (x +25\right )}{26 x^{\frac {23}{13}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.283 |
|
|
\[ {}y y^{\prime }+\frac {a \left (13 x -18\right ) y}{15 x^{\frac {7}{5}}} = -\frac {4 a^{2} \left (-1+x \right ) \left (-6+x \right )}{15 x^{\frac {9}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.372 |
|
|
\[ {}y y^{\prime }+\frac {a \left (1+5 x \right ) y}{2 \sqrt {x}} = a^{2} \left (-x^{2}+1\right ) \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.818 |
|
|
\[ {}y y^{\prime }+\frac {3 a \left (19 x -14\right ) x^{\frac {7}{5}} y}{35} = -\frac {4 a^{2} \left (-1+x \right ) \left (9 x -14\right ) x^{\frac {9}{5}}}{35} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.545 |
|
|
\[ {}y y^{\prime }+\frac {3 a \left (3 x +7\right ) y}{10 x^{\frac {13}{10}}} = -\frac {a^{2} \left (-1+x \right ) \left (x +9\right )}{5 x^{\frac {8}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
36.145 |
|
|
\[ {}y y^{\prime }+\frac {a \left (7 x -12\right ) y}{10 x^{\frac {7}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -16\right )}{10 x^{\frac {9}{5}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.353 |
|
|
\[ {}y y^{\prime }+\frac {3 a \left (13 x -8\right ) y}{20 x^{\frac {7}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (27 x -32\right )}{20 x^{\frac {9}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.264 |
|
|
\[ {}y y^{\prime }+\frac {3 a \left (3 x +11\right ) y}{14 x^{\frac {10}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -27\right )}{14 x^{\frac {13}{7}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.253 |
|
|
\[ {}y y^{\prime }-\frac {a \left (1+x \right ) y}{2 x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +5\right )}{4 x^{\frac {5}{2}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
8.809 |
|
|
\[ {}y y^{\prime }-\frac {a \left (1+x \right ) y}{2 x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (x +5\right )}{4 x^{\frac {5}{2}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
8.52 |
|
|
\[ {}y y^{\prime }-\frac {a \left (3+4 x \right ) y}{14 x^{\frac {8}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (16 x +5\right )}{14 x^{\frac {9}{7}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.339 |
|
|
\[ {}y y^{\prime }+\frac {a \left (13 x -3\right ) y}{6 x^{\frac {2}{3}}} = -\frac {a^{2} \left (-1+x \right ) \left (5 x -1\right )}{6 x^{\frac {1}{3}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
5.437 |
|
|
\[ {}y y^{\prime }-\frac {a \left (8 x -1\right ) y}{28 x^{\frac {8}{7}}} = \frac {a^{2} \left (-1+x \right ) \left (32 x +3\right )}{28 x^{\frac {9}{7}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.278 |
|
|
\[ {}y y^{\prime }-\frac {a \left (5 x -4\right ) y}{x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (3 x -1\right )}{x^{7}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.916 |
|
|
\[ {}y y^{\prime }-\frac {2 a \left (3 x -10\right ) y}{5 x^{4}} = \frac {a^{2} \left (-1+x \right ) \left (8 x -5\right )}{5 x^{7}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.917 |
|
|
\[ {}y y^{\prime }+\frac {a \left (39 x -4\right ) y}{42 x^{\frac {9}{7}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -1\right )}{42 x^{\frac {11}{7}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.349 |
|
|
\[ {}y y^{\prime }+\frac {a \left (-2+x \right ) y}{x} = \frac {2 a^{2} \left (-1+x \right )}{x} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
0.649 |
|
|
\[ {}y y^{\prime }+\frac {a \left (3 x -2\right ) y}{x} = -\frac {2 a^{2} \left (-1+x \right )^{2}}{x} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.349 |
|
|
\[ {}y y^{\prime }+\frac {a \left (1-\frac {b}{x^{2}}\right ) y}{x} = \frac {a^{2} b}{x} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.694 |
|
|
\[ {}y y^{\prime }-\frac {a \left (3 x -4\right ) y}{4 x^{\frac {5}{2}}} = \frac {a^{2} \left (-1+x \right ) \left (2+x \right )}{4 x^{4}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
3.134 |
|
|
\[ {}y y^{\prime }+\frac {a \left (33 x +2\right ) y}{30 x^{\frac {6}{5}}} = -\frac {a^{2} \left (-1+x \right ) \left (9 x -4\right )}{30 x^{\frac {7}{5}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.35 |
|
|
\[ {}y y^{\prime }-\frac {a \left (x -8\right ) y}{8 x^{\frac {5}{2}}} = -\frac {a^{2} \left (-1+x \right ) \left (3 x -4\right )}{8 x^{4}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.372 |
|
|
\[ {}y y^{\prime }+\frac {a \left (17 x +18\right ) y}{30 x^{\frac {22}{15}}} = -\frac {a^{2} \left (-1+x \right ) \left (x +4\right )}{30 x^{\frac {29}{15}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
22.907 |
|
|
\[ {}y y^{\prime }-\frac {a \left (6 x -13\right ) y}{13 x^{\frac {5}{2}}} = -\frac {a^{2} \left (-1+x \right ) \left (x -13\right )}{26 x^{4}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.505 |
|
|
\[ {}y y^{\prime }+\frac {a \left (24 x +11\right ) x^{\frac {27}{20}} y}{30} = -\frac {a^{2} \left (-1+x \right ) \left (9 x +1\right )}{60 x^{\frac {17}{10}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
30.746 |
|
|
\[ {}y y^{\prime }-\frac {2 a \left (3 x +2\right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (8 x +1\right )}{5 x^{\frac {11}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.498 |
|
|
\[ {}y y^{\prime }-\frac {6 a \left (1+4 x \right ) y}{5 x^{\frac {7}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (27 x +8\right )}{5 x^{\frac {9}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.306 |
|
|
\[ {}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{\frac {3}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
1.065 |
|
|
\[ {}y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{\frac {8}{5}}} = \frac {a^{2} \left (-1+x \right ) \left (3 x +7\right )}{5 x^{\frac {11}{5}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.294 |
|
|
\[ {}y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{\frac {5}{2}}} = \frac {a^{2} \left (-1+x \right ) \left (1+3 x \right )}{2 x^{4}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
3.13 |
|
|
\[ {}y y^{\prime }+\frac {a \left (-6+x \right ) y}{5 x^{\frac {7}{5}}} = \frac {2 a^{2} \left (-1+x \right ) \left (x +4\right )}{5 x^{\frac {9}{5}}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
2.274 |
|
|
\[ {}y y^{\prime }+\frac {a \left (21 x +19\right ) y}{5 x^{\frac {7}{5}}} = -\frac {2 a^{2} \left (-1+x \right ) \left (9 x -4\right )}{5 x^{\frac {9}{5}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
2.439 |
|
|
\[ {}y y^{\prime }-\frac {3 a y}{x^{\frac {7}{4}}} = \frac {a^{2} \left (-1+x \right ) \left (x -9\right )}{4 x^{\frac {5}{2}}} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
7.648 |
|
|
\[ {}y y^{\prime }-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}} = \frac {a^{2} \left (k +1\right ) \left (-1+x \right )}{x^{2}} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.105 |
|
|
\[ {}y y^{\prime }-a \left (\left (k -2\right ) x +2 k -3\right ) x^{-k} y = a^{2} \left (k -2\right ) \left (-1+x \right )^{2} x^{1-2 k} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.757 |
|
|
\[ {}y y^{\prime }-\frac {a \left (\left (4 k -7\right ) x -4 k +5\right ) x^{-k} y}{2} = \frac {a^{2} \left (2 k -3\right ) \left (-1+x \right )^{2} x^{1-2 k}}{2} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.814 |
|
|
\[ {}y y^{\prime }-\left (\left (2 n -1\right ) x -a n \right ) x^{-n -1} y = n \left (x -a \right ) x^{-2 n} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.284 |
|
|
\[ {}y y^{\prime }-\left (\left (n +1\right ) x -a n \right ) x^{n -1} \left (x -a \right )^{-n -2} y = n \,x^{2 n} \left (x -a \right )^{-2 n -3} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.543 |
|
|
\[ {}y y^{\prime }-a \left (\left (2 k -3\right ) x +1\right ) x^{-k} y = a^{2} \left (k -2\right ) \left (\left (k -1\right ) x +1\right ) x^{2-2 k} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.473 |
|
|
\[ {}y y^{\prime }-a \left (\left (n +2 k -3\right ) x +3-2 k \right ) x^{-k} y = a^{2} \left (\left (n +k -1\right ) x^{2}-\left (n +2 k -3\right ) x +k -2\right ) x^{1-2 k} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.709 |
|
|
\[ {}y y^{\prime }-\frac {a \left (\left (n +2\right ) x -2\right ) x^{-\frac {2 n +1}{n}} y}{n} = \frac {a^{2} \left (\left (n +1\right ) x^{2}-2 x -n +1\right ) x^{-\frac {3 n +2}{n}}}{n} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.898 |
|
|
\[ {}y y^{\prime }-\frac {a \left (\frac {\left (n +4\right ) x}{n +2}-2\right ) x^{-\frac {2 n +1}{n}} y}{n} = \frac {a^{2} \left (2 x^{2}+\left (n^{2}+n -4\right ) x -\left (n -1\right ) \left (n +2\right )\right ) x^{-\frac {3 n +2}{n}}}{n \left (n +2\right )} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.18 |
|
|
\[ {}y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3} = -\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {n +5}{n +3}}}{2 n +6} \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.401 |
|
|
\[ {}y y^{\prime }-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y = -\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n} \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.323 |
|
|
\[ {}y y^{\prime } = \left (a \,{\mathrm e}^{x}+b \right ) y+c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.419 |
|
|
\[ {}y y^{\prime } = \left (a \left (2 \mu +\lambda \right ) {\mathrm e}^{\lambda x}+b \right ) {\mathrm e}^{\mu x} y+\left (-a^{2} \mu \,{\mathrm e}^{2 \lambda x}-a b \,{\mathrm e}^{\lambda x}+c \right ) {\mathrm e}^{2 \mu x} \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.862 |
|
|
\[ {}y y^{\prime } = \left ({\mathrm e}^{\lambda x} a +b \right ) y+c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right ) \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.864 |
|
|
\[ {}y y^{\prime } = {\mathrm e}^{\lambda x} \left (2 a \lambda x +a +b \right ) y-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right ) \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.523 |
|
|
\[ {}y y^{\prime } = {\mathrm e}^{x a} \left (2 x^{2} a +b +2 x \right ) y+{\mathrm e}^{2 x a} \left (-a \,x^{4}-b \,x^{2}+c \right ) \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.412 |
|
|
\[ {}y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y = -a^{2} b \,x^{2} {\mathrm e}^{2 b x} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.095 |
|
|
\[ {}y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y = -a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
1.917 |
|
|
\[ {}y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y = -a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}} \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
3.027 |
|
|
\[ {}y y^{\prime } = \left (a \cosh \left (x \right )+b \right ) y-a b \sinh \left (x \right )+c \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.025 |
|
|
\[ {}y y^{\prime } = \left (a \sinh \left (x \right )+b \right ) y-a b \cosh \left (x \right )+c \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
2.021 |
|
|
\[ {}y y^{\prime } = \left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \] |
1 |
0 |
1 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
✗ |
N/A |
0.799 |
|
|
\[ {}y y^{\prime } = \left (2 \ln \left (x \right )^{2}+2 \ln \left (x \right )+a \right ) y+x \left (-\ln \left (x \right )^{4}-a \ln \left (x \right )^{2}+b \right ) \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
0.852 |
|
|
\[ {}y y^{\prime } = a x \cos \left (\lambda \,x^{2}\right ) y+x \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.209 |
|
|
\[ {}y y^{\prime } = a x \sin \left (\lambda \,x^{2}\right ) y+x \] |
1 |
0 |
0 |
unknown |
[[_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.153 |
|
|
\[ {}\left (y A +B x +a \right ) y^{\prime }+B y+k x +b = 0 \] |
1 |
1 |
1 |
exact, first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.462 |
|
|
\[ {}\left (y+x a +b \right ) y^{\prime } = \alpha y+\beta x +\gamma \] |
1 |
1 |
1 |
first_order_ode_lie_symmetry_calculated |
[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
9.039 |
|
|
\[ {}\left (y+a k \,x^{2}+b x +c \right ) y^{\prime } = -a y^{2}+2 a k x y+m y+k \left (k +b -m \right ) x +s \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
3.752 |
|
|
\[ {}\left (y+A \,x^{n}+a \right ) y^{\prime }+n A \,x^{n -1} y+k \,x^{m}+b = 0 \] |
1 |
1 |
2 |
exact |
[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class A‘]] |
✓ |
✓ |
1.797 |
|
|
\[ {}\left (y+x^{n +1} a +b \,x^{n}\right ) y^{\prime } = \left (a n \,x^{n}+c \,x^{n -1}\right ) y \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class A‘]] |
❇ |
N/A |
1.812 |
|
|
\[ {}x y y^{\prime } = a y^{2}+b y+c \,x^{n}+s \] |
1 |
0 |
0 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
❇ |
N/A |
0.885 |
|
|
\[ {}x y y^{\prime } = -n y^{2}+a \left (2 n +1\right ) x y+b y-a^{2} n \,x^{2}-a b x +c \] |
1 |
0 |
1 |
unknown |
[_rational, [_Abel, ‘2nd type‘, ‘class B‘]] |
✗ |
N/A |
1.083 |
|
|
\[ {}y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff, second_order_ode_can_be_made_integrable |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
1.442 |
|
|
\[ {}y^{\prime \prime }-\left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.558 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.899 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2} a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.903 |
|
|
\[ {}y^{\prime \prime }+a^{3} x \left (-x a +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.855 |
|
|
\[ {}y^{\prime \prime }-\left (x^{2} a +c b x \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.778 |
|
|
\[ {}y^{\prime \prime }-a \,x^{n} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.545 |
|
|
\[ {}y^{\prime \prime }-a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}y^{\prime \prime }-a \,x^{n -2} \left (a \,x^{n}+n +1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.806 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.959 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_linear_constant_coeff |
[[_2nd_order, _missing_x]] |
✓ |
✓ |
0.337 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.405 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.848 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2}+x a +1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.715 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b x \left (-b \,x^{3}+x a +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.838 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}+a \,x^{n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
N/A |
0.554 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \left (-b \,x^{2 n}-a \,x^{n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.72 |
|
|
\[ {}y^{\prime \prime }+x y^{\prime }+\left (n -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.701 |
|
|
\[ {}y^{\prime \prime }-2 x y^{\prime }+2 n y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.64 |
|
|
\[ {}y^{\prime \prime }+a x y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.744 |
|
|
\[ {}y^{\prime \prime }+a x y^{\prime }+b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.579 |
|
|
\[ {}y^{\prime \prime }+a x y^{\prime }+\left (b x +c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.644 |
|
|
\[ {}y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.033 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }-a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.144 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.445 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (x a +b -c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.835 |
|
|
\[ {}y^{\prime \prime }+\left (x a +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.914 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.722 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.161 |
|
|
\[ {}y^{\prime \prime }+2 \left (x a +b \right ) y^{\prime }+\left (a^{2} x^{2}+2 a b x +c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.745 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.847 |
|
|
\[ {}y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (-c \,x^{2 n}+x^{n +1} a +b \,x^{n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.873 |
|
|
\[ {}y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (x +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.428 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c \left (x^{2} a +b -c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.267 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +2 b \right ) y^{\prime }+\left (a b \,x^{2}-x a +b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.108 |
|
|
\[ {}y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+x^{2} a +b +2 x \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.814 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.248 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.575 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.523 |
|
|
\[ {}y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.381 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-x^{2} a +b^{2}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.183 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 x^{2} a +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.715 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{3}+1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.619 |
|
|
\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime } = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
0.786 |
|
|
\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.218 |
|
|
\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (a \,x^{2 n}+n \,x^{n -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.923 |
|
|
\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.162 |
|
|
\[ {}y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{m +n}+b \,x^{2 m}+m \,x^{m -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.09 |
|
|
\[ {}y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.517 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.55 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}+b^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.268 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.256 |
|
|
\[ {}y^{\prime \prime }+\left (a b \,x^{n}+2 b \,x^{n -1}-x \,a^{2}\right ) y^{\prime }+a \left (a b \,x^{n}+b \,x^{n -1}-x \,a^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.27 |
|
|
\[ {}y^{\prime \prime }+x^{n} \left (x^{2} a +\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (x a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.884 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }-\left (a \,x^{n -1}+b \,x^{m -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.305 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
13.078 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a \left (n +1\right ) x^{n -1}+b \left (1+m \right ) x^{m -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
6.463 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+c \left (a \,x^{n}+b \,x^{m}-c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
52.521 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a b \,x^{m +n}+b \left (1+m \right ) x^{m -1}-a \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.757 |
|
|
\[ {}y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a b \,x^{m +n}+b c \,x^{m}+a n \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.428 |
|
|
\[ {}x y^{\prime \prime }+\frac {y^{\prime }}{2}+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.139 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.493 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b x y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.823 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+\left (b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.064 |
|
|
\[ {}x y^{\prime \prime }+n y^{\prime }+b \,x^{-2 n +1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.723 |
|
|
\[ {}x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0 \] |
1 |
0 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_Emden, _Fowler]] |
✗ |
N/A |
0.932 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.612 |
|
|
\[ {}x y^{\prime \prime }+a y^{\prime }+b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.442 |
|
|
\[ {}x y^{\prime \prime }+a x y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
1.615 |
|
|
\[ {}x y^{\prime \prime }+\left (-x +b \right ) y^{\prime }-a y = 0 \] |
1 |
0 |
1 |
unknown |
[_Laguerre] |
✗ |
N/A |
0.988 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.25 |
|
|
\[ {}x y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a \left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.948 |
|
|
\[ {}x y^{\prime \prime }+\left (x \left (a +b \right )+n +m \right ) y^{\prime }+\left (a b x +a n +b m \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.513 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.671 |
|
|
\[ {}x y^{\prime \prime }-\left (x a +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.168 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x a +1\right ) y^{\prime }+\left (b \,x^{3}+x \,a^{2}+a \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.494 |
|
|
\[ {}x y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c x \left (-c \,x^{2}+x a +b +1\right ) = 0 \] |
1 |
1 |
1 |
second_order_ode_missing_y |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
72.239 |
|
|
\[ {}x y^{\prime \prime }-\left (2 x a +1\right ) y^{\prime }+b \,x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.076 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.941 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }-\left (a c \,x^{2}+\left (b c +c^{2}+a \right ) x +b +2 c \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.257 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +2\right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.163 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (2 x a +b \right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.134 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (c -1\right ) \left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.681 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.52 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{2} a +b x +2\right ) y^{\prime }+\left (c \,x^{2}+d x +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.083 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.037 |
|
|
\[ {}x y^{\prime \prime }+x \left (x^{2} a +b \right ) y^{\prime }+\left (3 x^{2} a +b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.438 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+2\right ) y^{\prime }+b x y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.859 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+x a -1\right ) y^{\prime }+a^{2} b \,x^{3} y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.772 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (x^{2} a +b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.062 |
|
|
\[ {}x y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n}-a \,x^{n -1}-b^{2} x +2 b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.726 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+2\right ) y^{\prime }+a \,x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.139 |
|
|
\[ {}x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.747 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a n \,x^{n -1} y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
11.605 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b -1\right ) x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.28 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+a \left (b +n -1\right ) x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.348 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}-c x +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.408 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{n}+b -3 n +1\right ) y^{\prime }+a^{2} n \left (b -n \right ) x^{2 n -1} y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.289 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{2 n -1}+d \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.921 |
|
|
|
|||||||||
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{n -1}+2\right ) y^{\prime }+b \,x^{n -2} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.186 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b x \right ) y^{\prime }+\left (a b \,x^{n}+a n \,x^{n -1}-b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.628 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{n}+b \,x^{n -1}+x a -1\right ) y^{\prime }+a^{2} b \,x^{n} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.165 |
|
|
\[ {}x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (c -1\right ) \left (a \,x^{n -1}+b \,x^{m -1}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.428 |
|
|
\[ {}x y^{\prime \prime }+\left (a b \,x^{m +n}+a n \,x^{n}+b \,x^{m}+1-2 n \right ) y^{\prime }+a^{2} b n \,x^{2 n +m -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.014 |
|
|
\[ {}\left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.398 |
|
|
\[ {}\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.706 |
|
|
\[ {}\left (x a +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.187 |
|
|
\[ {}\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.505 |
|
|
\[ {}\left (x +\gamma \right ) y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
14.594 |
|
|
\[ {}x^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.561 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.433 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n \left (n +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.912 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+n \left (n +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.778 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2}+2 a b x +b^{2}-b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.13 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.375 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.743 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{4}+a \left (2 b -1\right ) x^{2}+b \left (b +1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
13.386 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.288 |
|
|
\[ {}x^{2} y^{\prime \prime }-\left (a^{2} x^{2 n}+a \left (2 b +n -1\right ) x^{n}+b \left (b -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.938 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2 n}+b \,x^{n}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.935 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.48 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.911 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_euler_ode, second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_2 |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.983 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\left (n +\frac {1}{2}\right )^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.195 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+\left (n +\frac {1}{2}\right )^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.858 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (-\nu ^{2}+x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[_Bessel] |
✓ |
✓ |
0.868 |
|
|
\[ {}x^{2} y^{\prime \prime }+x y^{\prime }-\left (\nu ^{2}+x^{2}\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Bessel, _modified]] |
✓ |
✓ |
0.782 |
|
|
\[ {}x^{2} y^{\prime \prime }+2 x y^{\prime }-\left (a^{2} x^{2}+2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.952 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (b^{2} x^{2}+a \left (1+a \right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.911 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 a x y^{\prime }+\left (-b^{2} x^{2}+a \left (1+a \right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.858 |
|
|
\[ {}x^{2} y^{\prime \prime }+\lambda x y^{\prime }+\left (x^{2} a +b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
5.585 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+\left (b \,x^{n}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.399 |
|
|
\[ {}x^{2} y^{\prime \prime }+a x y^{\prime }+x^{n} \left (b \,x^{n}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.82 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.07 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{2} y^{\prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.283 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.783 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }-b y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.629 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 k n \right ) x +n \left (-n +b -1\right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.49 |
|
|
\[ {}a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.445 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (x^{2} a +\left (a b -1\right ) x +b \right ) y^{\prime }+a^{2} b x y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.062 |
|
|
\[ {}x^{2} y^{\prime \prime }-2 x \left (x^{2}-a \right ) y^{\prime }+\left (2 n \,x^{2}+\left (\left (-1\right )^{n}-1\right ) a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.608 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (x^{2} a +b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.797 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a b \,x^{n}+a c \,x^{n -1}+b^{2} x^{2}+2 c b x +c^{2}-c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.049 |
|
|
\[ {}x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{4 m +2}+a m \,x^{n -1}-m^{2}-m \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.735 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+b \left (a \,x^{n}-1\right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.252 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }+\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.913 |
|
|
\[ {}x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.336 |
|
|
\[ {}x^{2} y^{\prime \prime }+\left (a \,x^{n +2}+b \,x^{2}+c \right ) y^{\prime }+\left (a n \,x^{n +1}+a c \,x^{n}+b c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.882 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (n -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
0.887 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.639 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.435 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+n^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.413 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+n \left (n +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.165 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\nu \left (\nu +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.163 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-3 x y^{\prime }+n \left (n +2\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[_Gegenbauer] |
✓ |
✓ |
1.242 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (n +1\right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.769 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }-2 \left (n -1\right ) x y^{\prime }-\left (\nu -n +1\right ) \left (\nu +n \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.716 |
|
|
\[ {}\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.345 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (2 a -3\right ) x y^{\prime }+\left (n +1\right ) \left (n +2 a -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Gegenbauer] |
✗ |
N/A |
1.642 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\beta -\alpha -\left (\alpha +\beta +2\right ) x \right ) y^{\prime }+n \left (n +\alpha +\beta +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.938 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (\alpha -\beta +\left (\alpha +\beta -2\right ) x \right ) y^{\prime }+\left (n +1\right ) \left (n +\alpha +\beta \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.772 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime \prime }+a x y^{\prime }+c y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
146.921 |
|
|
\[ {}\left (x^{2}+a \right ) y^{\prime \prime }+2 b x y^{\prime }+2 \left (b -1\right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.75 |
|
|
\[ {}\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
66.496 |
|
|
\[ {}\left (a^{2}+x^{2}\right ) y^{\prime \prime }+2 b x y^{\prime }+b \left (b -1\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.525 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (2 n +1\right ) a x y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
233.141 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+\left (2 x^{2} a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.616 |
|
|
\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.621 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.193 |
|
|
\[ {}\left (x^{2} a +b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 b \lambda \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.278 |
|
|
\[ {}x \left (-1+x \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
1.944 |
|
|
\[ {}x \left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+d y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.954 |
|
|
\[ {}2 x \left (-1+x \right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (x a +b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[_Jacobi] |
✗ |
N/A |
1.126 |
|
|
\[ {}\left (2 x a +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-m^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
2.022 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
25.05 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y = 0 \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
70.093 |
|
|
\[ {}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+\left (x a +b \right ) y^{\prime }+d y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
4.695 |
|
|
\[ {}\left (x^{2} a +2 b x +c \right ) y^{\prime \prime }+3 \left (x a +b \right ) y^{\prime }+d y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.126 |
|
|
\[ {}\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.867 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
8.194 |
|
|
\[ {}\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.444 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x a +b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.318 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2} a +b \right ) y^{\prime }+c x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.263 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.944 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.894 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (x^{2} a +b x \right ) y^{\prime }+\left (c x +d \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.191 |
|
|
\[ {}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.567 |
|
|
\[ {}x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.01 |
|
|
\[ {}x \left (x^{2} a +b \right ) y^{\prime \prime }+2 \left (x^{2} a +b \right ) y^{\prime }-2 y a x = 0 \] |
1 |
1 |
1 |
kovacic, exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
2.434 |
|
|
\[ {}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.221 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.557 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }-2 x \left (x a +2 b \right ) y^{\prime }+2 \left (x a +3 b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, second_order_change_of_variable_on_y_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.109 |
|
|
\[ {}x^{2} \left (x a +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (1+m \right )\right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.351 |
|
|
\[ {}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.182 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
8.702 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
4.763 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 x^{2} a -\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (x a +1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
35.024 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
64.142 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
366.355 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
83.315 |
|
|
\[ {}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
16.608 |
|
|
\[ {}2 x \left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (x^{2} a -c \right ) y^{\prime }+\lambda \,x^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
190.64 |
|
|
\[ {}x \left (x^{2} a +b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
64.799 |
|
|
\[ {}x \left (-1+x \right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
3.546 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
166.434 |
|
|
\[ {}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 x^{2} a +2 b x +c \right ) y^{\prime }+\lambda y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✗ |
61.431 |
|
|
\[ {}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 x^{2} a +2 b x +c \right ) y^{\prime }+\left (6 x a +2 b +\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
8.111 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
419.376 |
|
|
\[ {}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
517.49 |
|
|
\[ {}2 x \left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.216 |
|
|
\[ {}x^{4} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.735 |
|
|
\[ {}x^{4} y^{\prime \prime }+\left (x^{2} a +b x +c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.313 |
|
|
\[ {}x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (x \left (a +b \right )+a b \right ) y = 0 \] |
1 |
1 |
1 |
kovacic |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.055 |
|
|
\[ {}x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.009 |
|
|
\[ {}x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.933 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.817 |
|
|
\[ {}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = c \,x^{2} \left (x -a \right )^{2} \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
24.363 |
|
|
\[ {}a \,x^{2} \left (-1+x \right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
28.888 |
|
|
\[ {}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.129 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
1.322 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.63 |
|
|
\[ {}\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
1.669 |
|
|
\[ {}\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.929 |
|
|
\[ {}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (x^{2} a +a -3\right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[_Halm] |
✓ |
✓ |
3.16 |
|
|
\[ {}\left (x^{2} a +b \right )^{2} y^{\prime \prime }+2 a x \left (x^{2} a +b \right ) y^{\prime }+c y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.989 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.765 |
|
|
\[ {}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.744 |
|
|
\[ {}a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.72 |
|
|
\[ {}\left (x^{2} a +b \right )^{2} y^{\prime \prime }+\left (2 x a +c \right ) \left (x^{2} a +b \right ) y^{\prime }+k y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
1.594 |
|
|
\[ {}\left (x^{2} a +b \right )^{2} y^{\prime \prime }+\left (x^{2} a +b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +b c \right ) x y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.871 |
|
|
\[ {}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-\left (b \,x^{n +1}+a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.716 |
|
|
\[ {}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-m \left (b \,x^{n +1}+\left (m -1\right ) x^{2}+a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.651 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }-c y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.921 |
|
|
\[ {}\left (x -a \right )^{2} \left (-b +x \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (-b +x \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.727 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} y^{\prime \prime }+y A = 0 \] |
1 |
1 |
1 |
kovacic, second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
3.69 |
|
|
\[ {}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.513 |
|
|
\[ {}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.656 |
|
|
\[ {}\left (x^{2} a +b x +c \right )^{2} y^{\prime \prime }+\left (2 x a +k \right ) \left (x^{2} a +b x +c \right ) y^{\prime }+m y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
4.501 |
|
|
\[ {}x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_Emden, _Fowler]] |
✓ |
✓ |
0.546 |
|
|
\[ {}x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+b y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.778 |
|
|
\[ {}x^{n} y^{\prime \prime }+c \left (x a +b \right )^{n -4} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.253 |
|
|
\[ {}x^{n} y^{\prime \prime }+a x y^{\prime }-\left (b^{2} x^{n}+2 b \,x^{n -1}+a b x +a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.317 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (x a +b \right ) y^{\prime }-a y = 0 \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
3.0 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.408 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (2 x^{n -1}+x^{2} a +b x \right ) y^{\prime }+b y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.732 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{n}+b \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.721 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (a \,x^{n}-x^{n -1}+a b x +b \right ) y^{\prime }+a^{2} b x y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
5.811 |
|
|
\[ {}x^{n} y^{\prime \prime }+\left (a \,x^{m +n}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.237 |
|
|
\[ {}\left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
3.653 |
|
|
\[ {}\left (a \,x^{n}+b x +c \right ) y^{\prime \prime } = a n \left (n -1\right ) x^{n -2} y \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
53.536 |
|
|
\[ {}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (-b +a \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.2 |
|
|
\[ {}x \left (x^{2 n}+a \right ) y^{\prime \prime }+\left (x^{2 n}+a -a n \right ) y^{\prime }-b^{2} x^{2 n -1} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_1 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
1.605 |
|
|
\[ {}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.823 |
|
|
\[ {}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
25.5 |
|
|
\[ {}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
71.168 |
|
|
\[ {}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +b c \right ) x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
8.43 |
|
|
\[ {}\left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{n -2} \left (b \,x^{1+m}+a n -a \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
4.524 |
|
|
\[ {}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-a n \,x^{n -1}-1\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
5.746 |
|
|
\[ {}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.183 |
|
|
\[ {}\left (x^{n +1} a +b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-n b +\beta \right ) x^{n -2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
42.981 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda -x \right ) y^{\prime }+y = 0 \] |
1 |
1 |
1 |
second_order_ode_non_constant_coeff_transformation_on_B |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
2.92 |
|
|
\[ {}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
2.868 |
|
|
\[ {}2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+d y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
6.141 |
|
|
\[ {}\left (a \,x^{n}+b \right )^{1+m} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }-a n m \,x^{n -1} y = 0 \] |
1 |
1 |
1 |
second_order_ode_lagrange_adjoint_equation_method |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
69.079 |
|
|
\[ {}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.382 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}-b \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.369 |
|
|
\[ {}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.367 |
|
|
\[ {}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.657 |
|
|
\[ {}y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.771 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.735 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.173 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.573 |
|
|
\[ {}y^{\prime \prime }-a y^{\prime }+b \,{\mathrm e}^{2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A, second_order_change_of_variable_on_x_method_1, second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
0.893 |
|
|
\[ {}y^{\prime \prime }+a y^{\prime }+\left (b \,{\mathrm e}^{\lambda x}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_bessel_ode_form_A |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.703 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.756 |
|
|
\[ {}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.355 |
|
|
\[ {}y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\lambda x} a +\lambda \right ) y = 0 \] |
1 |
1 |
1 |
kovacic, second_order_change_of_variable_on_y_method_1, linear_second_order_ode_solved_by_an_integrating_factor |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.663 |
|
|
\[ {}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
0.825 |
|
|
\[ {}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+\mu \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.651 |
|
|
\[ {}y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.574 |
|
|
\[ {}y^{\prime \prime }-\left (a +2 b \,{\mathrm e}^{x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{2 x a} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2, second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.909 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.309 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a -\lambda \right ) y^{\prime }+b \,{\mathrm e}^{2 \lambda x} y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_x_method_2 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.789 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \right ) y^{\prime }+c \left ({\mathrm e}^{\lambda x} a +b -c \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.085 |
|
|
\[ {}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y = 0 \] |
1 |
0 |
0 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
0.976 |
|
|
\[ {}y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.493 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+c \,{\mathrm e}^{\mu x}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.232 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a +b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y = 0 \] |
1 |
1 |
1 |
second_order_change_of_variable_on_y_method_1 |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
0.825 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +2 b -\lambda \right ) y^{\prime }+\left (c \,{\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+b^{2}-b \lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.39 |
|
|
\[ {}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.309 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.314 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.483 |
|
|
\[ {}y^{\prime \prime }+\left (2 \,{\mathrm e}^{\lambda x} a +b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+c \,{\mathrm e}^{2 \mu x}+d \,{\mathrm e}^{\mu x}+k \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
2.063 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}\right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\mu x}+\lambda \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.641 |
|
|
\[ {}y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
N/A |
1.911 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right ) y = 0 \] |
1 |
1 |
1 |
exact linear second order ode, second_order_integrable_as_is |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
3.144 |
|
|
\[ {}y^{\prime \prime }+\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+{\mathrm e}^{\lambda x} a c +b \mu \,{\mathrm e}^{\mu x}\right ) y = 0 \] |
1 |
0 |
1 |
unknown |
[[_2nd_order, _with_linear_symmetries]] |
❇ |
N/A |
1.794 |
|
|
|
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|
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|
|