Link to actual problem [11061] \[ \boxed {\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+\left (2 a x +k \right ) \left (a \,x^{2}+b x +c \right ) y^{\prime }+y m=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_2"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1+\frac {x \left (x a +b \right )}{c}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= -\frac {2 c \,\operatorname {arctanh}\left (\frac {2 x a +b}{\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {-4 a c +b^{2}}}\right ] \\ \end{align*}