|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}x^{2} \left (x +1\right ) y^{\prime \prime }+2 x \left (3 x +2\right ) y^{\prime } = 0
\] |
[[_2nd_order, _missing_y]] |
✓ |
1.065 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (-2+x \right ) y^{\prime }}{x \left (-1+x \right )}+\frac {2 \left (x +1\right ) y}{x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.031 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (5 x -4\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (9 x -6\right ) y}{x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.313 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1+b \right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.856 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x +1}-\frac {y}{x \left (x +1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.961 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y^{\prime }}{x \left (-2+x \right )}-\frac {y}{x^{2} \left (-2+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
147.952 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{x \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.783 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (-1+x \right ) \left (x -a \right )}-\frac {\left (\alpha \beta x -q \right ) y}{x \left (-1+x \right ) \left (x -a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.707 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}-\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.353 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x -4\right ) y^{\prime }}{2 x \left (-2+x \right )}-\frac {\left (x -3\right ) y}{2 x^{2} \left (-2+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.051 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x +1}-\frac {\left (3 x +1\right ) y}{4 x^{2} \left (x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.006 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}+\frac {v \left (v +1\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.113 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (-1+x \right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (-1+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.361 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (a x +b \right ) y}{4 x \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.952 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-3 x +1\right ) y}{\left (-1+x \right ) \left (2 x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.030 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}-\frac {\left (a -b \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.227 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (-2+x \right )}+\frac {y}{3 x^{2} \left (-2+x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.247 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.761 |
|
|
\[
{}y^{\prime \prime } = \frac {2 \left (a x +2 b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (2 a x +6 b \right ) y}{\left (a x +b \right ) x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.898 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
143.371 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
3.602 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} a \left (1-a \right )-b \left (x +b \right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.645 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.355 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.230 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (a +b \right ) y^{\prime }}{x^{2}}-\frac {\left (\left (a +b \right ) x +a b \right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.499 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {y}{x^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
0.665 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.975 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.745 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {a^{2} y}{x^{4}}
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.188 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.250 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x +a \right ) y^{\prime }}{x^{2}}-\frac {b y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.780 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.911 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}-\frac {2 y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.510 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.397 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.296 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}-\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.388 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.237 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.150 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {v \left (v +1\right ) y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.152 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.654 |
|
|
\[
{}x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.363 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.797 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (a -1\right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (a -1\right )-v \left (v +1\right )\right )-a \left (a +1\right )\right ) y}{x^{2} \left (x^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.164 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.474 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.918 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}+1}-\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (n +1\right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.271 |
|
|
\[
{}y^{\prime \prime } = -\frac {a x y^{\prime }}{x^{2}+1}-\frac {b y}{\left (x^{2}+1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.233 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.311 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}+\frac {a^{2} y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.358 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.115 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (\left (x^{2}-1\right ) \left (a \,x^{2}+b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.292 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (n +1\right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.288 |
|
|
\[
{}y^{\prime \prime } = \frac {2 x \left (2 a -1\right ) y^{\prime }}{x^{2}-1}-\frac {\left (x^{2} \left (2 a \left (2 a -1\right )-v \left (v +1\right )\right )+2 a +v \left (v +1\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.449 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.434 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
4.000 |
|
|
\[
{}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.679 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (-1+x \right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.180 |
|
|
\[
{}y^{\prime \prime } = \frac {12 y}{\left (x +1\right )^{2} \left (x^{2}+2 x +3\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.307 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.523 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.137 |
|
|
\[
{}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.517 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.805 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.093 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.860 |
|
|
\[
{}y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.175 |
|
|
\[
{}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.927 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (v \left (v +1\right ) \left (-1+x \right )-a^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.201 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (-1+x \right )}-\frac {\left (-v \left (v +1\right ) \left (-1+x \right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.192 |
|
|
\[
{}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.792 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.030 |
|
|
\[
{}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.443 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (-1+x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.910 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (a x +b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.632 |
|
|
\[
{}y^{\prime \prime } = -\frac {y}{\left (a x +b \right )^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.583 |
|
|
\[
{}y^{\prime \prime } = -\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
2.540 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.441 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.197 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (3 x +1\right ) y^{\prime }}{\left (-1+x \right ) \left (x +1\right )}-\frac {36 \left (x +1\right )^{2} y}{\left (-1+x \right )^{2} \left (3 x +5\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.480 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x}-\frac {a y}{x^{6}}
\] |
[[_Emden, _Fowler]] |
✓ |
2.391 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.109 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.041 |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
8.112 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.754 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.226 |
|
|
\[
{}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.744 |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
10.058 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
8.419 |
|
|
\[
{}y^{\prime \prime } = -a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.792 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.124 |
|
|
\[
{}y^{\prime \prime } = \frac {y}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
0.310 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x \ln \left (x \right )}+\ln \left (x \right )^{2} y
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.676 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x \left (\ln \left (x \right )-1\right )}-\frac {y}{x^{2} \left (\ln \left (x \right )-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.693 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.429 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.086 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.901 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}-y \sin \left (x \right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.792 |
|
|
\[
{}y^{\prime \prime } = -\frac {x \sin \left (x \right ) y^{\prime }}{\cos \left (x \right ) x -\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{\cos \left (x \right ) x -\sin \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
10.481 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} \sin \left (x \right )-2 \cos \left (x \right ) x \right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.273 |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.506 |
|
|
\[
{}y^{\prime \prime } = -\frac {a \left (n -1\right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.523 |
|