|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.380 |
|
|
\[
{}\left (a \,x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
305.698 |
|
|
\[
{}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.355 |
|
|
\[
{}\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y = 0
\] |
[_Gegenbauer] |
✓ |
1.638 |
|
|
\[
{}\left (a \,x^{2}+b x \right ) y^{\prime \prime }+2 b y^{\prime }-2 a y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.506 |
|
|
\[
{}\operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.674 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.655 |
|
|
\[
{}x^{3} y^{\prime \prime }+y^{\prime } x -\left (2 x +3\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.431 |
|
|
\[
{}x^{3} y^{\prime \prime }+2 y^{\prime } x -y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.511 |
|
|
\[
{}x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (a \,x^{2}+b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.493 |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-2 y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.383 |
|
|
\[
{}x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y-\ln \left (x \right )^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.835 |
|
|
\[
{}x^{3} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.657 |
|
|
\[
{}x^{3} y^{\prime \prime }+3 x^{2} y^{\prime }+x y-1 = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.615 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 x^{2}+1\right ) y^{\prime }-v \left (v +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.044 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+2 \left (x^{2}-1\right ) y^{\prime }-2 x y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.219 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (n +1\right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.313 |
|
|
\[
{}x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (-v +n -1\right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.293 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime }+y a \,x^{3} = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.820 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-x y = 0
\] |
[[_elliptic, _class_II]] |
✗ |
59.205 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+x y = 0
\] |
[[_elliptic, _class_I]] |
✗ |
0.887 |
|
|
\[
{}x \left (x^{2}-1\right ) y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.484 |
|
|
\[
{}x \left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime }-6 x y = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
1.916 |
|
|
\[
{}x \left (x^{2}-2\right ) y^{\prime \prime }-\left (x^{3}+3 x^{2}-2 x -2\right ) y^{\prime }+\left (x^{2}+4 x +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.585 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+\left (2 x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.161 |
|
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+2 x \left (3 x +2\right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.074 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x -2\right ) y^{\prime }}{x \left (x -1\right )}+\frac {2 \left (x +1\right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.104 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.263 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.620 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x +1}-\frac {y}{x \left (x +1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.947 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (x -2\right )}-\frac {y}{x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
109.095 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{x \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.950 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.411 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
68.917 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (x -2\right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.191 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x +1}-\frac {\left (3 x +1\right ) y}{4 x^{2} \left (x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.174 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {v \left (v +1\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.978 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.272 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (a x +b \right ) y}{4 x \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.931 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-3 x +1\right ) y}{\left (x -1\right ) \left (2 x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.158 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}-\frac {\left (a -b \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.102 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}+\frac {y}{3 x^{2} \left (x -2\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.321 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.533 |
|
|
\[
{}y^{\prime \prime } = \frac {2 \left (a x +2 b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (2 a x +6 b \right ) y}{\left (a x +b \right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.078 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
107.684 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
3.408 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.604 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.293 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.365 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (\left (a +b \right ) x +a b \right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.704 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.816 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.869 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.658 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}}
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.268 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.392 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.928 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.494 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.792 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.433 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.214 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.254 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.318 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.003 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.010 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.750 |
|
|
\[
{}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.152 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.550 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.141 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.445 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.085 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.171 |
|
|
\[
{}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.060 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.589 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.429 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.962 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.176 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.168 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.290 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.319 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
4.359 |
|
|
\[
{}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.706 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (x -1\right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.349 |
|
|
\[
{}y^{\prime \prime } = \frac {12 y}{\left (x +1\right )^{2} \left (x^{2}+2 x +3\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.082 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.924 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.148 |
|
|
\[
{}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.587 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.286 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.886 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.671 |
|
|
\[
{}y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.108 |
|
|
\[
{}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.881 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (v \left (v +1\right ) \left (x -1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.056 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.165 |
|
|
\[
{}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.946 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.892 |
|
|
\[
{}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.321 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.811 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (a x +b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.934 |
|
|
\[
{}y^{\prime \prime } = -\frac {y}{\left (a x +b \right )^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.796 |
|
|
\[
{}y^{\prime \prime } = -\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
3.576 |
|