Link to actual problem [10913] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (a \,x^{2}+b x +c \right ) y=0} \]
type detected by program
{"second_order_change_of_variable_on_y_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 [R &= x, S \left (R \right ) &= \frac {x^{d} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-d +1} \left (\int x^{d -2} {\mathrm e}^{-\frac {x \left (2 x^{2} a +3 b x +6 c \right )}{6}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{d} y}{x \left (\int \frac {x^{d} {\mathrm e}^{-\frac {1}{3} a \,x^{3}-\frac {1}{2} b \,x^{2}-c x}}{x^{2}}d x \right )}\right ] \\ \end{align*}