| # |
ODE |
CAS classification |
Solved |
Maple |
Mma |
Sympy |
time(sec) |
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=4 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.046 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
y \left (1\right ) &= 3 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
1.791 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 7 \\
y^{\prime }\left (1\right ) &= 2 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.525 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
1.528 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
1.602 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
y \left (1\right ) &= 3 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
1.779 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 7 \\
y^{\prime }\left (1\right ) &= 2 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.618 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
1.618 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=72 x^{5} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.168 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=x^{2}-1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.052 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+3 t y^{\prime }-3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.971 |
|
| \begin{align*}
2 t^{2} y^{\prime \prime }-5 t y^{\prime }+5 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.614 |
|
| \begin{align*}
t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y&={\mathrm e}^{2 t} t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.533 |
|
| \begin{align*}
\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y&=2 \left (t -1\right )^{2} {\mathrm e}^{-t} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.284 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y&=4 t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
5.702 |
|
| \begin{align*}
t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y&={\mathrm e}^{2 t} t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.835 |
|
| \begin{align*}
\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y&=2 \left (t -1\right ) {\mathrm e}^{-t} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.235 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
1.899 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.589 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.089 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=2 x^{2}+2 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.635 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=2 x^{4} \sin \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
10.961 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=2 \left (x -1\right )^{2} {\mathrm e}^{x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.381 |
|
| \begin{align*}
\left (x -1\right )^{2} y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+2 y&=\left (x -1\right )^{2} \\
y \left (0\right ) &= 3 \\
y^{\prime }\left (0\right ) &= -6 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
30.496 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y&=-2 x^{2} \\
y \left (1\right ) &= 1 \\
y^{\prime }\left (1\right ) &= -1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.901 |
|
| \begin{align*}
\left (x +1\right ) \left (2 x +3\right ) y^{\prime \prime }+2 \left (x +2\right ) y^{\prime }-2 y&=\left (2 x +3\right )^{2} \\
y \left (0\right ) &= 0 \\
y^{\prime }\left (0\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.591 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.234 |
|
| \begin{align*}
y^{\prime \prime }-\frac {2 \left (t +1\right ) y^{\prime }}{t^{2}+2 t -1}+\frac {2 y}{t^{2}+2 t -1}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.324 |
|
| \begin{align*}
\left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y&=0 \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✗ |
1.168 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.092 |
|
| \begin{align*}
\left (2 t +1\right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.927 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.713 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.255 |
|
| \begin{align*}
\left (t -1\right )^{2} y^{\prime \prime }-2 \left (t -1\right ) y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
2.694 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.731 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.804 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+5 t y^{\prime }-5 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.211 |
|
| \begin{align*}
\left (t -1\right )^{2} y^{\prime \prime }-2 \left (t -1\right ) y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
2.668 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.783 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=4 x +\sin \left (\ln \left (x \right )\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
11.332 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
0.851 |
|
| \begin{align*}
x y^{\prime \prime }+x&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.016 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
4.734 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-x y^{\prime }+y&=9 x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
2.598 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
1.799 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x^{2}+2 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
16.851 |
|
| \begin{align*}
x y^{\prime \prime }&=x +y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.477 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=\ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
10.459 |
|
| \begin{align*}
-y+x y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.819 |
|
| \begin{align*}
-\csc \left (x \right )^{2} y+\cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
8.854 |
|
| \begin{align*}
\left (\cot \left (x \right )+\csc \left (x \right )\right ) y^{\prime }+y^{\prime \prime }&=1+a \csc \left (x \right ) \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
5.123 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.434 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=x^{n} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
3.901 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.730 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.678 |
|
| \begin{align*}
a y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.369 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Laguerre] |
✓ |
✓ |
✓ |
✗ |
2.030 |
|
| \begin{align*}
-y-\left (2-x \right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.548 |
|
| \begin{align*}
y-\left (x +3\right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Laguerre] |
✓ |
✓ |
✓ |
✗ |
2.097 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.899 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (1-x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.109 |
|
| \begin{align*}
-2 y^{\prime }+\left (-x +a \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.991 |
|
| \begin{align*}
y^{\prime }+2 x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.858 |
|
| \begin{align*}
c y^{\prime }+\left (b x +a \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
2.960 |
|
| \begin{align*}
y-x y^{\prime }+\left (-x \cot \left (x \right )+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
5.387 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.867 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
4.104 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=a \,x^{2} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
5.549 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=x^{2} \left (x +3\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
4.690 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=3 x^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
4.468 |
|
| \begin{align*}
-y+\left (x +a \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.783 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
6.052 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=4 x^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
7.009 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x^{3} \sin \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
39.598 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=2 x \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
38.697 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x^{5} \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
38.556 |
|
| \begin{align*}
y-x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.651 |
|
| \begin{align*}
-y+x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✗ |
2.983 |
|
| \begin{align*}
y-x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
6.795 |
|
| \begin{align*}
-y+x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=x \left (-x^{2}+1\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
6.760 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.404 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.630 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=\left (-x^{2}+1\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.061 |
|
| \begin{align*}
x \left (x +1\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.902 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x \left (1-x \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.201 |
|
| \begin{align*}
-2 y+2 x y^{\prime }+x \left (1-x \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.619 |
|
| \begin{align*}
2 y+2 \left (1-x \right ) y^{\prime }+\left (2-x \right ) x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.086 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-x y^{\prime }+y&=x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
8.175 |
|
| \begin{align*}
-3 y+3 x y^{\prime }+\left (2 x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
7.368 |
|
| \begin{align*}
-y+\left (x +1\right ) y^{\prime }+2 x \left (1-x \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[_Jacobi] |
✓ |
✓ |
✓ |
✗ |
2.493 |
|
| \begin{align*}
y+\left (1-x \right ) y^{\prime }+2 x \left (1-x \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[_Jacobi] |
✓ |
✓ |
✓ |
✗ |
1.868 |
|
| \begin{align*}
4 y+2 \left (1-2 x \right ) y^{\prime }+\left (1-2 x \right ) \left (1-x \right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.954 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+2 \left (x +1\right )^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
6.731 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+2 \left (x +1\right )^{2} y^{\prime \prime }&=x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
45.450 |
|
| \begin{align*}
-9 y-3 \left (-3 x +1\right ) y^{\prime }+\left (-3 x +1\right )^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
6.044 |
|
| \begin{align*}
2 a^{2} y-2 a^{2} x y^{\prime }+\left (-a^{2} x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✓ |
3.126 |
|
| \begin{align*}
-y+x y^{\prime }+x^{3} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.918 |
|
| \begin{align*}
-y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.546 |
|
| \begin{align*}
x^{3}-y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.671 |
|
| \begin{align*}
2 y x -2 \left (x^{2}+1\right ) y^{\prime }+x \left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.707 |
|
| \begin{align*}
-2 y x -2 \left (-x^{2}+1\right ) y^{\prime }+x \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
1.852 |
|
| \begin{align*}
2 y+2 \left (2-x \right ) y^{\prime }+\left (2-x \right )^{2} x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.082 |
|
| \begin{align*}
-y+x y^{\prime }+x^{5} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.826 |
|
| \begin{align*}
a^{2} x^{a -1} y+\left (1-2 a \right ) x^{a} y^{\prime }+x^{a +1} y^{\prime \prime }&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✗ |
2.628 |
|
| \begin{align*}
-x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=-x^{2}+3 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.717 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
2.402 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x^{2} {\mathrm e}^{-x} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.401 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.922 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=x^{2} \\
y \left (1\right ) &= 0 \\
y^{\prime }\left (1\right ) &= -1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.072 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
1.335 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.939 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x -\frac {1}{x} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.061 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
3.728 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=4 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.943 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.401 |
|
| \begin{align*}
\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.809 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.839 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.383 |
|
| \begin{align*}
t^{2} N^{\prime \prime }-2 t N^{\prime }+2 N&=t \ln \left (t \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
7.934 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}}&=\ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.292 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.036 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=\ln \left (x \right )^{2}-\ln \left (x^{2}\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
13.503 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime \prime }+\left (x +1\right ) y^{\prime }-y&=\ln \left (x +1\right )^{2}+x -1 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.385 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=2 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.467 |
|
| \begin{align*}
\left (x^{2}+4\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=8 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.519 |
|
| \begin{align*}
\left (x +1\right ) y^{\prime \prime }-\left (3 x +4\right ) y^{\prime }+3 y&=\left (3 x +2\right ) {\mathrm e}^{3 x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.180 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=\frac {-x^{2}+1}{x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.428 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=-\frac {2}{x}-\ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.432 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.849 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.912 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}}&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.737 |
|
| \begin{align*}
y-x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.385 |
|
| \begin{align*}
x y^{\prime \prime }+x y^{\prime }-y&=x^{2}+2 x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.629 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x^{2}+2 x \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.061 |
|
| \begin{align*}
-y+x y^{\prime }+x^{3} y^{\prime \prime }&=\cos \left (\frac {1}{x}\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.598 |
|
| \begin{align*}
x \left (x +1\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y&=x +\frac {1}{x} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.174 |
|
| \begin{align*}
2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y&=x^{2}-1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.307 |
|
| \begin{align*}
x^{2} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x y^{\prime }+y&=x \left (1-\ln \left (x \right )\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.625 |
|
| \begin{align*}
\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y&=\left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x} \\
\end{align*} |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
5.938 |
|
| \begin{align*}
y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1}&=0 \\
\end{align*} |
[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
3.113 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}}&=0 \\
y \left (1\right ) &= 1 \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
22.941 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}}&=0 \\
y \left (1\right ) &= 0 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
22.786 |
|
| \begin{align*}
\left (3 x -1\right )^{2} y^{\prime \prime }+\left (9 x -3\right ) y^{\prime }-9 y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.011 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.428 |
|
| \begin{align*}
-2 y^{\prime }+x y^{\prime \prime }&=x^{3} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.376 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=4 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.658 |
|
| \begin{align*}
x y^{\prime \prime }-3 y^{\prime }&=5 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.579 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=\left (x^{2}-1\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.884 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (1-x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.041 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x y^{\prime \prime }&=x^{2} {\mathrm e}^{2 x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.507 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
19.096 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
y \left (0\right ) &= 0 \\
y^{\prime }\left (0\right ) &= 0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
8.188 |
|
| \begin{align*}
t y^{\prime \prime }-y^{\prime }&=2 t^{2} \\
y \left (0\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.649 |
|
| \begin{align*}
x y^{\prime \prime }&=y^{\prime }+x^{5} \\
y \left (1\right ) &= {\frac {1}{2}} \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.891 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }+x&=0 \\
y \left (2\right ) &= -1 \\
y^{\prime }\left (2\right ) &= -{\frac {1}{2}} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.123 |
|
| \begin{align*}
\cos \left (x \right ) y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.363 |
|
| \begin{align*}
-x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=-x^{2}+3 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.217 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.694 |
|
| \begin{align*}
t y^{\prime \prime }+4 y^{\prime }&=t^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.957 |
|
| \begin{align*}
t y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.710 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-2 y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.596 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=2 x^{3}-x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
17.243 |
|
| \begin{align*}
-y+x y^{\prime }+y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.368 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.133 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Laguerre] |
✓ |
✓ |
✓ |
✗ |
0.944 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y-a \,x^{2}&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.319 |
|
| \begin{align*}
-y+\left (x +a \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.321 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y-3 x^{3}&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
2.572 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y-x^{5} \ln \left (x \right )&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
12.440 |
|
| \begin{align*}
y-x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.342 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.294 |
|
| \begin{align*}
y^{\prime \prime }&=-\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {y}{\sin \left (x \right )^{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
3.009 |
|
| \begin{align*}
y^{\prime \prime }&=-\frac {x y^{\prime }}{f \left (x \right )}+\frac {y}{f \left (x \right )} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.573 |
|
| \begin{align*}
y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.810 |
|
| \begin{align*}
\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
5.809 |
|
| \begin{align*}
x^{n} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
1.595 |
|
| \begin{align*}
\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda -x \right ) y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
1.305 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.749 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (1-x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.480 |
|
| \begin{align*}
\sin \left (x \right ) y^{\prime \prime }+2 \cos \left (x \right ) y^{\prime }+3 y \sin \left (x \right )&={\mathrm e}^{x} \\
\end{align*} |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.164 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.654 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.103 |
|
| \begin{align*}
t x^{\prime \prime }+x^{\prime }&=1 \\
x \left (1\right ) &= 0 \\
x^{\prime }\left (1\right ) &= 2 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.342 |
|
| \begin{align*}
t^{2} x^{\prime \prime }-3 x^{\prime } t +3 x&=4 t^{7} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
19.352 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
y \left (1\right ) &= 3 \\
y^{\prime }\left (1\right ) &= 2 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
4.766 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y&=1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
5.922 |
|
| \begin{align*}
\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y&=\left (x +2\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.643 |
|
| \begin{align*}
\left (2 x +1\right ) \left (x +1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y&=\left (2 x +1\right )^{2} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.203 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
4.291 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y&=-6 x^{3}+4 x^{2} \\
y \left (2\right ) &= 4 \\
y^{\prime }\left (2\right ) &= -1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
5.752 |
|
| \begin{align*}
\left (2 x -3\right )^{2} y^{\prime \prime }-6 \left (2 x -3\right ) y^{\prime }+12 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
5.661 |
|
| \begin{align*}
t^{2} x^{\prime \prime }+x^{\prime } t -x&=0 \\
x \left (1\right ) &= 1 \\
x^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.783 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y&=0 \\
y \left (1\right ) &= -2 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
4.684 |
|
| \begin{align*}
u^{\prime \prime }+\frac {2 u^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.098 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.160 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y&=1-2 x \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.475 |
|
| \begin{align*}
-\csc \left (x \right )^{2} y+\cot \left (x \right ) y^{\prime }+y^{\prime \prime }&=\cos \left (x \right ) \\
\end{align*} |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
27.840 |
|
| \begin{align*}
y^{\prime \prime }+\frac {2 y^{\prime }}{x}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.956 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&={\mathrm e}^{x} x^{2} \\
y \left (0\right ) &= -1 \\
y^{\prime }\left (0\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.368 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.071 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
2.077 |
|
| \begin{align*}
x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime }&=x^{2} \\
y \left (1\right ) &= 0 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.066 |
|
| \begin{align*}
x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime }&=x^{2} \\
y \left (5\right ) &= 0 \\
y^{\prime }\left (5\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.091 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
0.977 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 2 \\
y^{\prime }\left (1\right ) &= -1 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.484 |
|
| \begin{align*}
x y^{\prime \prime }+4 y^{\prime }&=18 x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.272 |
|
| \begin{align*}
x y^{\prime \prime }&=2 y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.902 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=6 x^{5} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.116 |
|
| \begin{align*}
x y^{\prime \prime }+4 y^{\prime }&=18 x^{2} \\
y \left (1\right ) &= 8 \\
y^{\prime }\left (1\right ) &= -3 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.375 |
|
| \begin{align*}
x y^{\prime \prime }&=2 y^{\prime } \\
y \left (-1\right ) &= 4 \\
y^{\prime }\left (-1\right ) &= 12 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.108 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=6 \\
y \left (1\right ) &= 4 \\
y^{\prime }\left (1\right ) &= 5 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.286 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 5 \\
y^{\prime }\left (1\right ) &= 3 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.876 |
|
| \begin{align*}
\left (x +1\right )^{2} y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y&=0 \\
y \left (0\right ) &= 0 \\
y^{\prime }\left (0\right ) &= 4 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
2.242 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
2.225 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 3 \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.871 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-x y^{\prime }+y&=\frac {50}{x^{3}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
11.088 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=3 \sqrt {x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
11.391 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=\sqrt {x} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.263 |
|
| \begin{align*}
\left (x +1\right ) y^{\prime \prime }+x y^{\prime }-y&=\left (x +1\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.470 |
|
| \begin{align*}
y^{\prime }+2 x y^{\prime \prime }&=\sqrt {x} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.128 |
|
| \begin{align*}
x y^{\prime \prime }&=3 y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.940 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=\frac {1}{x^{2}+1} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.474 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+7 t y^{\prime }-7 y&=0 \\
y \left (1\right ) &= 2 \\
y^{\prime }\left (1\right ) &= -22 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.954 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+t y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.868 |
|
| \begin{align*}
3 t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.329 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
2.413 |
|
| \begin{align*}
t^{2} \left (\ln \left (t \right )-1\right ) y^{\prime \prime }-t y^{\prime }+y&=-\frac {3 \left (1+\ln \left (t \right )\right )}{4 \sqrt {t}} \\
y \left (1\right ) &= 0 \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.015 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-8 x y^{\prime }+8 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.106 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-7 x y^{\prime }+7 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.108 |
|
| \begin{align*}
2 x^{2} y^{\prime \prime }-7 x y^{\prime }+7 y&=0 \\
y \left (1\right ) &= -1 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.503 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (-1\right ) &= 0 \\
y^{\prime }\left (-1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
2.349 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-5 t y^{\prime }+5 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
3.061 |
|
| \begin{align*}
x y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.240 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.525 |
|
| \begin{align*}
x y^{\prime \prime }&=y^{\prime }+x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.418 |
|
| \begin{align*}
x \ln \left (x \right ) y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.118 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.395 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.555 |
|
| \begin{align*}
\left (x +2\right )^{2} y^{\prime \prime }+3 \left (x +2\right ) y^{\prime }-3 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.771 |
|
| \begin{align*}
\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }+4 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.436 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x^{m} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.974 |
|
| \begin{align*}
\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y&=0 \\
\end{align*} |
[_Jacobi] |
✓ |
✓ |
✓ |
✗ |
1.648 |
|
| \begin{align*}
\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (x +1\right ) y^{\prime }+6 y&=6 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.065 |
|
| \begin{align*}
y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }&=1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.250 |
|
| \begin{align*}
x \ln \left (x \right ) y^{\prime \prime }-y^{\prime }&=\ln \left (x \right )^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
1.299 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (x -1\right )^{2} {\mathrm e}^{x} \\
y \left (-\infty \right ) &= 0 \\
y^{\prime }\left (0\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✗ |
✗ |
3.297 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.576 |
|
| \begin{align*}
y-x y^{\prime }+\left (-x \cot \left (x \right )+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
4.333 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=3 x^{2}+2 \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
30.970 |
|
| \begin{align*}
t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y&={\mathrm e}^{2 t} t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.656 |
|
| \begin{align*}
\left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y&=2 \left (t -1\right )^{2} {\mathrm e}^{-t} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.560 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=g \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.902 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y&=4 t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
25.901 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=2 x^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
2.932 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=4 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.362 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=3 x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.116 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.953 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.146 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
y \left (1\right ) &= 3 \\
y^{\prime }\left (1\right ) &= 5 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
2.257 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[_Laguerre] |
✓ |
✓ |
✓ |
✗ |
0.844 |
|
| \begin{align*}
\left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=\left (x^{2}-1\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.302 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (1-x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.454 |
|
| \begin{align*}
y-\left (x +1\right ) y^{\prime }+x y^{\prime \prime }&=x^{2} {\mathrm e}^{2 x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.013 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
11.555 |
|
| \begin{align*}
t^{2} x^{\prime \prime }-2 x^{\prime } t +2 x&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
2.172 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y&=\frac {1}{x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
10.003 |
|
| \begin{align*}
v^{\prime \prime }+\frac {2 v^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.587 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=2 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.061 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=\ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
6.732 |
|
| \begin{align*}
x y^{\prime \prime }+3 y^{\prime }&=3 x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.181 |
|
| \begin{align*}
V^{\prime \prime }+\frac {2 V^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.590 |
|
| \begin{align*}
V^{\prime \prime }+\frac {V^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.732 |
|
| \begin{align*}
v^{\prime \prime }+\frac {2 v^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.533 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=2 \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
4.191 |
|
| \begin{align*}
\left (2 x -1\right )^{3} y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
0.610 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x^{m} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.554 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.168 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.374 |
|
| \begin{align*}
-y+x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=x \left (-x^{2}+1\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.133 |
|
| \begin{align*}
-y+x y^{\prime }+y^{\prime \prime }&=f \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.289 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-5 x y^{\prime }+5 y&=\frac {1}{x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
7.020 |
|
| \begin{align*}
y^{\prime \prime }+\frac {2 y^{\prime }}{r}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.414 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+2 x y^{\prime }-2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.044 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=2 \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
6.533 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=x^{m} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.937 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.661 |
|
| \begin{align*}
y-x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=2 x \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
5.128 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.677 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=x \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.139 |
|
| \begin{align*}
-y+x y^{\prime }+y^{\prime \prime }&=X \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.594 |
|
| \begin{align*}
\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y&={\mathrm e}^{x} \left (x +1\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.170 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=\left (1-x \right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.498 |
|
| \begin{align*}
x y^{\prime }-y&=\left (x -1\right ) \left (y^{\prime \prime }-x +1\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.437 |
|
| \begin{align*}
\left (x^{2}+a \right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.069 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=8 x^{3} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
2.529 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
1.831 |
|
| \begin{align*}
\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y&={\mathrm e}^{x} \left (x +1\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.079 |
|
| \begin{align*}
-y+x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=x \left (-x^{2}+1\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.395 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.977 |
|
| \begin{align*}
-y+x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=x \left (-x^{2}+1\right )^{{3}/{2}} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.553 |
|
| \begin{align*}
\left (x +2\right ) y^{\prime \prime }-\left (5+2 x \right ) y^{\prime }+2 y&={\mathrm e}^{x} \left (x +1\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.257 |
|
| \begin{align*}
x y^{\prime \prime }+\left (x -1\right ) y^{\prime }-y&=x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.296 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&={\mathrm e}^{x} x^{2} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
3.753 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✗ |
1.274 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.348 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=0 \\
y \left (1\right ) &= 1 \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
2.453 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.317 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.290 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=3 x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.893 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=3 x^{2}-x \\
y \left (1\right ) &= \pi \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
44.911 |
|
| \begin{align*}
x^{\prime \prime }-\frac {x^{\prime }}{t}&=0 \\
x \left (1\right ) &= 0 \\
x^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.003 |
|
| \begin{align*}
t^{2} x^{\prime \prime }+3 x^{\prime } t -3 x&=t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
2.123 |
|
| \begin{align*}
2 y-2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=6 \left (x^{2}+1\right )^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.968 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x^{3} \sin \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
16.966 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=3 x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.638 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
3.398 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=16 x^{3} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.725 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.533 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
2.876 |
|
| \begin{align*}
x y^{\prime \prime }-3 y^{\prime }&=4 x^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.145 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
2.834 |
|
| \begin{align*}
y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[_Hermite] |
✓ |
✓ |
✓ |
✗ |
3.070 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=x \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✗ |
4.182 |
|
| \begin{align*}
\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[_Gegenbauer] |
✓ |
✓ |
✓ |
✗ |
2.386 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
39.411 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✓ |
6.379 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+3 x y^{\prime }-3 y&=x^{2}-4 x +2 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
3.289 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
4.190 |
|
| \begin{align*}
\left (2 x +3\right )^{2} y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }-2 y&=24 x^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
3.268 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=3 x -2 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
46.418 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-4 x y^{\prime }+4 y&=24 x +24 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
71.193 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.065 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+y^{\prime }&=2 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.009 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=3 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.993 |
|
| \begin{align*}
y-x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.915 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.872 |
|
| \begin{align*}
x y^{\prime \prime }+4 y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.857 |
|
| \begin{align*}
y^{\prime \prime }+\frac {y^{\prime }}{x}&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.621 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }+3 y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
0.911 |
|
| \begin{align*}
5 x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.765 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.342 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+4 x y^{\prime }-4 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.027 |
|
| \begin{align*}
\left (x -2\right )^{2} y^{\prime \prime }-\left (x -2\right ) y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.146 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (1\right ) &= 1 \\
y^{\prime }\left (1\right ) &= 0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.623 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=0 \\
y \left (-1\right ) &= 1 \\
y^{\prime }\left (-1\right ) &= 0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.546 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=\tan \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
13.521 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+4 x y^{\prime }-4 y&=x^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.569 |
|
| \begin{align*}
5 x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=\sqrt {x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
8.813 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+4 x y^{\prime }-4 y&=x^{{1}/{4}} \ln \left (x \right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.773 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=\ln \left (x \right ) \\
y \left (1\right ) &= A \\
y \left (2\right ) &= B \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
23.132 |
|
| \begin{align*}
\left (2 x +1\right ) y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
0.986 |
|
| \begin{align*}
\sin \left (x \right ) y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.727 |
|
| \begin{align*}
x y^{\prime \prime }&=y^{\prime }+x^{5} \\
y \left (1\right ) &= {\frac {1}{2}} \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.530 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }+x&=0 \\
y \left (2\right ) &= -1 \\
y^{\prime }\left (2\right ) &= -{\frac {1}{2}} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.657 |
|
| \begin{align*}
\cos \left (x \right ) y^{\prime \prime }&=y^{\prime } \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.785 |
|
| \begin{align*}
-x^{2} y^{\prime }+x^{3} y^{\prime \prime }&=-x^{2}+3 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.967 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+t y^{\prime }-y&=\sqrt {t} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✓ |
4.207 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+\left (t -1\right ) y^{\prime }-y&={\mathrm e}^{-t} t^{2} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
2.110 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.029 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.517 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\
y \left (0\right ) &= 0 \\
y^{\prime }\left (0\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.149 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\
y \left (0\right ) &= 1 \\
y^{\prime }\left (0\right ) &= 0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.109 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\
y \left (0\right ) &= 0 \\
y^{\prime }\left (0\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.121 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=2 t \,{\mathrm e}^{-t} \\
y \left (0\right ) &= a \\
y^{\prime }\left (0\right ) &= b \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
2.085 |
|
| \begin{align*}
\left (t -1\right ) y^{\prime \prime }-t y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.562 |
|
| \begin{align*}
\left (t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.326 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+2 t y^{\prime }-2 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.458 |
|
| \begin{align*}
t^{2} y^{\prime \prime }+2 t y^{\prime }-2 y&=0 \\
y \left (1\right ) &= 0 \\
y^{\prime }\left (1\right ) &= 1 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
1.747 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-2 t y^{\prime }+2 y&=t^{4} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
4.125 |
|
| \begin{align*}
t y^{\prime \prime }-y^{\prime }&=3 t^{2}-1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.438 |
|
| \begin{align*}
t^{2} y^{\prime \prime }-t y^{\prime }+y&=t \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
11.980 |
|
| \begin{align*}
y^{\prime \prime }-\tan \left (t \right ) y^{\prime }-\sec \left (t \right )^{2} y&=t \\
\end{align*} |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
12.079 |
|
| \begin{align*}
t y^{\prime \prime }+\left (t -1\right ) y^{\prime }-y&={\mathrm e}^{-t} t^{2} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.362 |
|
| \begin{align*}
2 y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.119 |
|
| \begin{align*}
-y^{\prime }+x y^{\prime \prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.194 |
|
| \begin{align*}
-2 y^{\prime }+x y^{\prime \prime }&=x^{4} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.115 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.725 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }-x&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.011 |
|
| \begin{align*}
x^{2} \left (1-\ln \left (x \right )\right ) y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.769 |
|
| \begin{align*}
x^{2} y^{\prime \prime }+x y^{\prime }-y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✓ |
26.832 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.948 |
|
| \begin{align*}
\left (x -2\right )^{2} y^{\prime \prime }+3 \left (x -2\right ) y^{\prime }-3 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.112 |
|
| \begin{align*}
\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }+4 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.239 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-3 x y^{\prime }+3 y&=0 \\
\end{align*} |
[[_Emden, _Fowler]] |
✓ |
✓ |
✓ |
✓ |
34.692 |
|
| \begin{align*}
\left (x^{2}-x \right ) y^{\prime \prime }+\left (2 x -3\right ) y^{\prime }-2 y&=0 \\
\end{align*} |
[_Jacobi] |
✓ |
✓ |
✓ |
✗ |
0.957 |
|
| \begin{align*}
\left (2 x^{2}+3 x \right ) y^{\prime \prime }-6 \left (x +1\right ) y^{\prime }+6 y&=6 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.283 |
|
| \begin{align*}
x^{2} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.658 |
|
| \begin{align*}
\left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }-y+1&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
3.456 |
|
| \begin{align*}
\left (4 x^{2}-x \right ) y^{\prime \prime }+2 \left (2 x -1\right ) y^{\prime }-4 y&=12 x^{2}-6 x \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.655 |
|
| \begin{align*}
x \left (x -1\right ) y^{\prime \prime }-\left (2 x -1\right ) y^{\prime }+2 y&=x^{2} \left (2 x -3\right ) \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.453 |
|
| \begin{align*}
y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }&=1 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
1.217 |
|
| \begin{align*}
x \ln \left (x \right ) y^{\prime \prime }-y^{\prime }&=\ln \left (x \right )^{2} \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✗ |
0.666 |
|
| \begin{align*}
\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=\left (x -1\right )^{2} {\mathrm e}^{x} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.611 |
|
| \begin{align*}
\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }+2 \sin \left (x \right ) y^{\prime }-y \left (\cos \left (x \right )+\sin \left (x \right )\right )&={\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )^{2} \\
\end{align*} |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
4.273 |
|
| \begin{align*}
\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y&=-\left (x -1\right )^{2} {\mathrm e}^{x} \\
y \left (-\infty \right ) &= 0 \\
y \left (0\right ) &= 1 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
1.682 |
|
| \begin{align*}
x y^{\prime \prime }+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.744 |
|
| \begin{align*}
y^{\prime \prime } \left (1+{\mathrm e}^{x}\right )+y^{\prime }&=0 \\
\end{align*} |
[[_2nd_order, _missing_y]] |
✓ |
✓ |
✓ |
✓ |
0.391 |
|
| \begin{align*}
x^{2} y^{\prime \prime }-x y^{\prime }+y&=8 x^{3} \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✓ |
1.576 |
|
| \begin{align*}
\left (2 x +1\right ) y^{\prime \prime }+4 x y^{\prime }-4 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.757 |
|
| \begin{align*}
x \left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
✓ |
✓ |
✗ |
0.662 |
|
| \begin{align*}
x^{2} \ln \left (x \right ) y^{\prime \prime }-x y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.489 |
|
| \begin{align*}
x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.490 |
|
| \begin{align*}
\left (2 x +1\right ) x y^{\prime \prime }+2 \left (x +1\right ) y^{\prime }-2 y&=0 \\
\end{align*} |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
✓ |
✓ |
✗ |
0.892 |
|
| \begin{align*}
x \left (x +4\right ) y^{\prime \prime }-\left (2 x +4\right ) y^{\prime }+2 y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.668 |
|
| \begin{align*}
2 x \left (x +2\right ) y^{\prime \prime }+\left (2-x \right ) y^{\prime }+y&=0 \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.655 |
|
| \begin{align*}
x \left (x +1\right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y&=x +\frac {1}{x} \\
\end{align*} |
[[_2nd_order, _exact, _linear, _nonhomogeneous]] |
✓ |
✓ |
✓ |
✗ |
1.265 |
|
| \begin{align*}
\left (2 x +1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-2 y&=x^{2}+x \\
\end{align*} |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
✓ |
✓ |
✗ |
0.744 |
|