Link to actual problem [8085] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-3 \left (x +3\right ) y=0} \]
type detected by program
{"kovacic", "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 {y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2+x^{2}-x -\frac {6}{x}+\frac {24}{x^{2}}-\frac {120}{x^{3}}\right ) {\mathrm e}^{-x}-\operatorname {expIntegral}_{1}\left (x \right ) x^{3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} x^{3} y}{-\operatorname {expIntegral}_{1}\left (x \right ) x^{6} {\mathrm e}^{x}+x^{5}-x^{4}+2 x^{3}-6 x^{2}+24 x -120}\right ] \\ \end{align*}