|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}y^{\prime \prime } = -\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.082 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}-\frac {b y}{x^{2} \left (x^{2}+a \right )}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
3.433 |
|
|
\[
{}y^{\prime \prime } = -\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.616 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (x -1\right )^{2}}-\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.261 |
|
|
\[
{}y^{\prime \prime } = \frac {12 y}{\left (x +1\right )^{2} \left (x^{2}+2 x +3\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.023 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.775 |
|
|
\[
{}y^{\prime \prime } = -\frac {b y}{x^{2} \left (x -a \right )^{2}}+c
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.125 |
|
|
\[
{}y^{\prime \prime } = \frac {c y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.510 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (\alpha +\beta +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (1-\alpha -\beta \right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}-\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.040 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-x^{2} \left (a^{2}-1\right )+2 \left (a +3\right ) b x -b^{2}\right ) y}{4 x^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.767 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \,x^{2}+a -3\right ) y}{4 \left (x^{2}+1\right )^{2}}
\] |
[_Halm] |
✓ |
1.494 |
|
|
\[
{}y^{\prime \prime } = \frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.975 |
|
|
\[
{}y^{\prime \prime } = \frac {3 y}{4 \left (x^{2}+x +1\right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.766 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (v \left (v +1\right ) \left (x -1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.899 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.930 |
|
|
\[
{}y^{\prime \prime } = -\frac {3 y}{16 x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.865 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}-\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.724 |
|
|
\[
{}y^{\prime \prime } = -\frac {b x y^{\prime }}{\left (x^{2}-1\right ) a}-\frac {\left (c \,x^{2}+d x +e \right ) y}{a \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.177 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (b \,x^{2}+c x +d \right ) y}{a \,x^{2} \left (x -1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.693 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 y^{\prime }}{x}-\frac {c y}{x^{2} \left (a x +b \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.856 |
|
|
\[
{}y^{\prime \prime } = -\frac {y}{\left (a x +b \right )^{4}}
\] |
[[_Emden, _Fowler]] |
✓ |
1.323 |
|
|
\[
{}y^{\prime \prime } = -\frac {A y}{\left (a \,x^{2}+b x +c \right )^{2}}
\] |
[[_Emden, _Fowler]] |
✓ |
3.556 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x^{4}}+\frac {y}{x^{5}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.362 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.941 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (3 x +1\right ) y^{\prime }}{\left (x -1\right ) \left (x +1\right )}-\frac {36 \left (x +1\right )^{2} y}{\left (x -1\right )^{2} \left (3 x +5\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.425 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x}-\frac {a y}{x^{6}}
\] |
[[_Emden, _Fowler]] |
✓ |
2.128 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (3 x^{2}+a \right ) y^{\prime }}{x^{3}}-\frac {b y}{x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.974 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.588 |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
6.430 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (-2 x^{2}+1\right ) y}{4 x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.830 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (2 x^{2}+1\right ) y^{\prime }}{x^{3}}-\frac {\left (a \,x^{4}+10 x^{2}+1\right ) y}{4 x^{6}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.213 |
|
|
\[
{}y^{\prime \prime } = -\frac {27 x y}{16 \left (x^{3}-1\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.194 |
|
|
\[
{}y^{\prime \prime } = -\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
7.965 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}-\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
7.714 |
|
|
\[
{}y^{\prime \prime } = -a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y
\] |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.783 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a p \,x^{b}+q \right ) y^{\prime }}{x \left (a \,x^{b}-1\right )}-\frac {\left (a r \,x^{b}+s \right ) y}{x^{2} \left (a \,x^{b}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.532 |
|
|
\[
{}y^{\prime \prime } = \frac {y}{{\mathrm e}^{x}+1}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✗ |
0.160 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x \ln \left (x \right )}+\ln \left (x \right )^{2} y
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
0.779 |
|
|
\[
{}y^{\prime \prime } = \frac {y^{\prime }}{x \left (\ln \left (x \right )-1\right )}-\frac {y}{x^{2} \left (\ln \left (x \right )-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.480 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \sinh \left (x \right )^{2}-n \left (n -1\right )\right ) y}{\sinh \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.507 |
|
|
\[
{}y^{\prime \prime } = -\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}-\left (-a^{2}+n^{2}\right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.221 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 n +1\right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\left (v +n +1\right ) \left (v -n \right ) y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.609 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\sin \left (x \right )^{2}-\cos \left (x \right )\right ) y^{\prime }}{\sin \left (x \right )}-y \sin \left (x \right )^{2}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.321 |
|
|
\[
{}y^{\prime \prime } = -\frac {x \sin \left (x \right ) y^{\prime }}{\cos \left (x \right ) x -\sin \left (x \right )}+\frac {\sin \left (x \right ) y}{\cos \left (x \right ) x -\sin \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.153 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\sin \left (x \right ) x^{2}-2 \cos \left (x \right ) x \right ) y^{\prime }}{x^{2} \cos \left (x \right )}-\frac {\left (2 \cos \left (x \right )-x \sin \left (x \right )\right ) y}{x^{2} \cos \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.267 |
|
|
\[
{}\cos \left (x \right )^{2} y^{\prime \prime }-\left (a \cos \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.595 |
|
|
\[
{}y^{\prime \prime } = -\frac {a \left (n -1\right ) \sin \left (2 a x \right ) y^{\prime }}{\cos \left (a x \right )^{2}}-\frac {n \,a^{2} \left (\left (n -1\right ) \sin \left (a x \right )^{2}+\cos \left (a x \right )^{2}\right ) y}{\cos \left (a x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.827 |
|
|
\[
{}y^{\prime \prime } = \frac {2 y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.319 |
|
|
\[
{}y^{\prime \prime } = -\frac {a y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.318 |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a \sin \left (x \right )^{2}+n \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.606 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a^{2} \cos \left (x \right )^{2}-\left (3-2 a \right ) \cos \left (x \right )-3+3 a \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.320 |
|
|
\[
{}\sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.578 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \sin \left (x \right )^{2}-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.006 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (a \cos \left (x \right )^{2}+b \sin \left (x \right )^{2}+c \right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.128 |
|
|
\[
{}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.251 |
|
|
\[
{}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (v \left (v +1\right ) \sin \left (x \right )^{2}-n^{2}\right ) y}{\sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.428 |
|
|
\[
{}y^{\prime \prime } = \frac {\cos \left (2 x \right ) y^{\prime }}{\sin \left (2 x \right )}-2 y
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.220 |
|
|
\[
{}y^{\prime \prime } = -\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}-\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.106 |
|
|
\[
{}y^{\prime \prime } = -\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}-\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}+\sqrt {\cos \left (x \right )}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✗ |
1.636 |
|
|
\[
{}y^{\prime \prime } = -\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}-\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.437 |
|
|
\[
{}y^{\prime \prime } = -\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.079 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.406 |
|
|
\[
{}y^{\prime \prime } = \frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}+\frac {\sin \left (x \right )^{2} y}{\cos \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.624 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \left (m -1\right ) \sin \left (x \right )^{2}-n \left (n -1\right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.065 |
|
|
\[
{}y^{\prime \prime } = \frac {\phi ^{\prime }\left (x \right ) y^{\prime }}{\phi \left (x \right )-\phi \left (a \right )}-\frac {\left (-n \left (n +1\right ) \left (\phi \left (x \right )-\phi \left (a \right )\right )^{2}+D^{\left (2\right )}\left (\phi \right )\left (a \right )\right ) y}{\phi \left (x \right )-\phi \left (a \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.561 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (\phi \left (x^{3}\right )-\phi \left (x \right ) \phi ^{\prime }\left (x \right )-\phi ^{\prime \prime }\left (x \right )\right ) y^{\prime }}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}}-\frac {\left ({\phi ^{\prime }\left (x \right )}^{2}-\phi \left (x \right )^{2} \phi ^{\prime }\left (x \right )-\phi \left (x \right ) \phi ^{\prime \prime }\left (x \right )\right ) y}{\phi ^{\prime }\left (x \right )+\phi \left (x \right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.091 |
|
|
\[
{}y^{\prime \prime } = \frac {2 \,\operatorname {JacobiSN}\left (x , k\right ) \operatorname {JacobiCN}\left (x , k\right ) \operatorname {JacobiDN}\left (x , k\right ) y^{\prime }-2 \left (1-2 \left (k^{2}+1\right ) \operatorname {JacobiSN}\left (a , k\right )^{2}+3 k^{2} \operatorname {JacobiSN}\left (a , k\right )^{4}\right ) y}{\operatorname {JacobiSN}\left (x , k\right )^{2}-\operatorname {JacobiSN}\left (a , k\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.962 |
|
|
\[
{}y^{\prime \prime } = -\frac {x y^{\prime }}{f \left (x \right )}+\frac {y}{f \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.827 |
|
|
\[
{}y^{\prime \prime } = -\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}-\frac {g \left (x \right ) y}{f \left (x \right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.378 |
|
|
\[
{}y^{\prime \prime } = -\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.301 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (x -1\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.010 |
|
|
\[
{}y^{\prime \prime } = -\frac {y^{\prime }}{x}-\frac {\left (-x -1\right ) y}{x^{4}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.974 |
|
|
\[
{}y^{\prime \prime } = -\frac {b^{2} y}{\left (-a^{2}+x^{2}\right )^{2}}
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.766 |
|
|
\[
{}y^{\prime \prime \prime }-\lambda y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.095 |
|
|
\[
{}y^{\prime \prime \prime }+y a \,x^{3}-b x = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✗ |
0.040 |
|
|
\[
{}y^{\prime \prime \prime }-a \,x^{b} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.039 |
|
|
\[
{}y^{\prime \prime \prime }+3 y^{\prime }-4 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.069 |
|
|
\[
{}y^{\prime \prime \prime }-a^{2} y^{\prime }-{\mathrm e}^{2 a x} \sin \left (x \right )^{2} = 0
\] |
[[_3rd_order, _missing_y]] |
✓ |
0.220 |
|
|
\[
{}y^{\prime \prime \prime }+2 a x y^{\prime }+a y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.070 |
|
|
\[
{}y^{\prime \prime \prime }-3 \left (2 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+b y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.049 |
|
|
\[
{}y^{\prime \prime \prime }+\left (-n^{2}+1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\frac {\left (\left (-n^{2}+1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right )-a \right ) y}{2} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}y^{\prime \prime \prime }-\left (4 n \left (n +1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }-2 n \left (n +1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.047 |
|
|
\[
{}y^{\prime \prime \prime }+\left (A \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+B \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.043 |
|
|
\[
{}y^{\prime \prime \prime }-\left (3 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime }+\left (b +c \operatorname {JacobiSN}\left (z , x\right )^{2}-3 k^{2} \operatorname {JacobiSN}\left (z , x\right ) \operatorname {JacobiCN}\left (z , x\right ) \operatorname {JacobiDN}\left (z , x\right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.049 |
|
|
\[
{}y^{\prime \prime \prime }-\left (6 k^{2} \sin \left (x \right )^{2}+a \right ) y^{\prime }+b y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.050 |
|
|
\[
{}y^{\prime \prime \prime }+2 f \left (x \right ) y^{\prime }+f^{\prime }\left (x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.043 |
|
|
\[
{}y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime }+10 y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.067 |
|
|
\[
{}y^{\prime \prime \prime }-2 y^{\prime \prime }-a^{2} y^{\prime }+2 a^{2} y-\sinh \left (x \right ) = 0
\] |
[[_3rd_order, _linear, _nonhomogeneous]] |
✓ |
0.558 |
|
|
\[
{}y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y-{\mathrm e}^{a x} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.130 |
|
|
\[
{}y^{\prime \prime \prime }+\operatorname {a2} y^{\prime \prime }+\operatorname {a1} y^{\prime }+\operatorname {a0} y = 0
\] |
[[_3rd_order, _missing_x]] |
✓ |
0.203 |
|
|
\[
{}y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 a x y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}y^{\prime \prime \prime }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.041 |
|
|
\[
{}y^{\prime \prime \prime }-y^{\prime \prime } \sin \left (x \right )-2 y^{\prime } \cos \left (x \right )+\sin \left (x \right ) y-\ln \left (x \right ) = 0
\] |
[[_3rd_order, _fully, _exact, _linear]] |
✗ |
0.045 |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime }+f \left (x \right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.040 |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) \left (x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y\right ) = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.043 |
|
|
\[
{}y^{\prime \prime \prime }+f \left (x \right ) y^{\prime \prime }+g \left (x \right ) y^{\prime }+\left (f \left (x \right ) g \left (x \right )+g^{\prime }\left (x \right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}y^{\prime \prime \prime }+3 f \left (x \right ) y^{\prime \prime }+\left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}+4 g \left (x \right )\right ) y^{\prime }+\left (4 f \left (x \right ) g \left (x \right )+2 g^{\prime }\left (x \right )\right ) y = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✗ |
0.046 |
|
|
\[
{}4 y^{\prime \prime \prime }-8 y^{\prime \prime }-11 y^{\prime }-3 y+18 \,{\mathrm e}^{x} = 0
\] |
[[_3rd_order, _with_linear_symmetries]] |
✓ |
0.110 |
|