|
# |
ODE |
CAS classification |
Solved? |
time (sec) |
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }-\left (-k^{2}+x^{2}\right ) y^{\prime }+\left (x +k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.636 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.612 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a x +b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.695 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.187 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.976 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.096 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.163 |
|
|
\[
{}x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.230 |
|
|
\[
{}x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.977 |
|
|
\[
{}x \left (a \,x^{2}+b \right ) y^{\prime \prime }+2 \left (a \,x^{2}+b \right ) y^{\prime }-2 y a x = 0
\] |
[[_2nd_order, _exact, _linear, _homogeneous]] |
✓ |
5.484 |
|
|
\[
{}x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.671 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (a \lambda +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (c -2 a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.744 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.900 |
|
|
\[
{}x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-n -m \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (n -1\right ) x +b n \left (m +1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.539 |
|
|
\[
{}x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.482 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.125 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.060 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (b +1\right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.188 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.519 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
604.385 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (n \,x^{2}+m x +k \right ) y^{\prime }+\left (-2 \left (a +n \right ) x +1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
60.623 |
|
|
\[
{}\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
19.914 |
|
|
\[
{}2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
115.612 |
|
|
\[
{}x \left (a \,x^{2}+b x +1\right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y^{\prime }+\left (n x +m \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.333 |
|
|
\[
{}x \left (-1+x \right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.803 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }-\left (-\lambda ^{2}+x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
200.094 |
|
|
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\lambda y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
161.744 |
|
|
\[
{}2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
117.592 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
453.848 |
|
|
\[
{}\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\lambda ^{3}+x^{3}\right ) y^{\prime }-\left (\lambda ^{2}-\lambda x +x^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1011.633 |
|
|
\[
{}2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.934 |
|
|
\[
{}x^{4} y^{\prime \prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
3.898 |
|
|
\[
{}x^{4} y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.633 |
|
|
\[
{}x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) x +a b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.474 |
|
|
\[
{}x^{4} y^{\prime \prime }+2 x^{2} \left (x +a \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.821 |
|
|
\[
{}x^{4} y^{\prime \prime }+a \,x^{n} y^{\prime }-\left (a \,x^{n -1}+a b \,x^{n -2}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.323 |
|
|
\[
{}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.579 |
|
|
\[
{}x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y = c \,x^{2} \left (x -a \right )^{2}
\] |
[[_2nd_order, _linear, _nonhomogeneous]] |
✓ |
2.199 |
|
|
\[
{}a \,x^{2} \left (-1+x \right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.897 |
|
|
\[
{}x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.790 |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y = 0
\] |
[_Halm] |
✓ |
1.537 |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.154 |
|
|
\[
{}\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
1.677 |
|
|
\[
{}\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+b^{2} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.570 |
|
|
\[
{}4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (a \,x^{2}+a -3\right ) y = 0
\] |
[_Halm] |
✓ |
1.752 |
|
|
\[
{}\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+2 a x \left (a \,x^{2}+b \right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
2.838 |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.144 |
|
|
\[
{}\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.153 |
|
|
\[
{}a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.467 |
|
|
\[
{}\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (2 a x +c \right ) \left (a \,x^{2}+b \right ) y^{\prime }+k y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
3.280 |
|
|
\[
{}\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +c b \right ) x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.891 |
|
|
\[
{}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-\left (b \,x^{n +1}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.079 |
|
|
\[
{}\left (x^{2}+a \right )^{2} y^{\prime \prime }+b \,x^{n} \left (x^{2}+a \right ) y^{\prime }-m \left (b \,x^{n +1}+\left (m -1\right ) x^{2}+a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.127 |
|
|
\[
{}\left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }-c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
2.461 |
|
|
\[
{}\left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (x -b \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
4.365 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+A y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.502 |
|
|
\[
{}\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }+\left (\left (x^{2}-1\right ) \left (a^{2} x^{2}-\lambda \right )-m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.124 |
|
|
\[
{}\left (x^{2}+1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}+1\right ) y^{\prime }+\left (\left (x^{2}+1\right ) \left (a^{2} x^{2}-\lambda \right )+m^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.148 |
|
|
\[
{}\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+\left (2 a x +k \right ) \left (a \,x^{2}+b x +c \right ) y^{\prime }+m y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
5.054 |
|
|
\[
{}x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y = 0
\] |
[[_Emden, _Fowler]] |
✓ |
2.361 |
|
|
\[
{}x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.103 |
|
|
\[
{}x^{n} y^{\prime \prime }+c \left (a x +b \right )^{n -4} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.411 |
|
|
\[
{}x^{n} y^{\prime \prime }+a x y^{\prime }-\left (b^{2} x^{n}+2 b \,x^{n -1}+a b x +a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.228 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.102 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.230 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (2 x^{n -1}+a \,x^{2}+b x \right ) y^{\prime }+b y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.156 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{n}+b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.108 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{n}-x^{n -1}+a b x +b \right ) y^{\prime }+a^{2} b x y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
3.595 |
|
|
\[
{}x^{n} y^{\prime \prime }+\left (a \,x^{m +n}+1\right ) y^{\prime }+a \,x^{m} \left (1+m \,x^{n -1}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.747 |
|
|
\[
{}\left (a \,x^{n}+b \right ) y^{\prime \prime }+\left (c \,x^{n}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{n}+d -b \lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.543 |
|
|
\[
{}\left (a \,x^{n}+b x +c \right ) y^{\prime \prime } = a n \left (n -1\right ) x^{n -2} y
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
7.152 |
|
|
\[
{}x \left (x^{n}+1\right ) y^{\prime \prime }+\left (\left (a -b \right ) x^{n}+a -n \right ) y^{\prime }+b \left (1-a \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.515 |
|
|
\[
{}x \left (x^{2 n}+a \right ) y^{\prime \prime }+\left (x^{2 n}+a -a n \right ) y^{\prime }-b^{2} x^{2 n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.422 |
|
|
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a^{2} \left (n +1\right ) x^{2 n}+n -1\right ) y^{\prime }-\nu \left (\nu +1\right ) a^{2} n^{2} x^{2 n} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.845 |
|
|
\[
{}x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
11.687 |
|
|
\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }-b \,x^{n -2} \left (\left (b -1\right ) x^{n}+a \left (n -1\right )\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.526 |
|
|
\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-a d +c b \right ) x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.367 |
|
|
\[
{}\left (x^{n}+a \right )^{2} y^{\prime \prime }+b \,x^{m} \left (x^{n}+a \right ) y^{\prime }-x^{n -2} \left (b \,x^{m +1}+a n -a \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.871 |
|
|
\[
{}\left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+c \,x^{m} \left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{m}-a n \,x^{n -1}-1\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
2.118 |
|
|
\[
{}x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
4.094 |
|
|
\[
{}\left (a \,x^{n +1}+b \,x^{n}+c \right )^{2} y^{\prime \prime }+\left (\alpha \,x^{n}+\beta \,x^{n -1}+\gamma \right ) y^{\prime }+\left (n \left (-a n -a +\alpha \right ) x^{n -1}+\left (n -1\right ) \left (-b n +\beta \right ) x^{n -2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
5.357 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda -x \right ) y^{\prime }+y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
1.331 |
|
|
\[
{}\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+\left (\lambda ^{2}-x^{2}\right ) y^{\prime }+\left (x +\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.283 |
|
|
\[
{}2 \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime \prime }+a n \,x^{n -1} b m \,x^{m -1} y^{\prime }+d y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
1.766 |
|
|
\[
{}\left (a \,x^{n}+b \right )^{m +1} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }-a n m \,x^{n -1} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.076 |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.512 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{x}-b \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.523 |
|
|
\[
{}y^{\prime \prime }+a \left (\lambda \,{\mathrm e}^{\lambda x}-a \,{\mathrm e}^{2 \lambda x}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.368 |
|
|
\[
{}y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (2 b +1\right ) {\mathrm e}^{x}+b^{2}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.394 |
|
|
\[
{}y^{\prime \prime }-\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.369 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+c \,{\mathrm e}^{2 \lambda x}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.438 |
|
|
\[
{}y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.493 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{2 a x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
36.723 |
|
|
\[
{}y^{\prime \prime }-a y^{\prime }+b \,{\mathrm e}^{2 a x} y = 0
\] |
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]] |
✓ |
1.361 |
|
|
\[
{}y^{\prime \prime }+a y^{\prime }+\left (b \,{\mathrm e}^{\lambda x}+c \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.862 |
|
|
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.588 |
|
|
\[
{}y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x} \left (b \,{\mathrm e}^{\lambda x}+c \right )^{n}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.673 |
|
|
\[
{}y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✓ |
0.791 |
|
|
\[
{}y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.668 |
|
|
\[
{}y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+\mu \right ) y = 0
\] |
[[_2nd_order, _with_linear_symmetries]] |
✗ |
0.786 |
|