Link to actual problem [12275] \[ \boxed {\left (x^{2}+2 x \right ) y^{\prime \prime }+\left (x^{2}+x +10\right ) y^{\prime }-\left (25-6 x \right ) y=0} \]
type detected by program
{"kovacic"}
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 {{\mathrm e}^{x} y}{\left (2+x \right )^{7}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {88447 x^{4} {\mathrm e}^{-x -2} \left (2+x \right )^{7} \operatorname {expIntegral}_{1}\left (-x -2\right )-11970 x^{4} {\mathrm e}^{-x} \left (2+x \right )^{7} \operatorname {expIntegral}_{1}\left (-x \right )+76477 x^{10}+970261 x^{9}+5171184 x^{8}+14871174 x^{7}+24496796 x^{6}+22249488 x^{5}+9184784 x^{4}+488880 x^{3}-131040 x^{2}+60480 x -40320}{x^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} x^{4} y}{88447 x^{4} {\mathrm e}^{-2} \left (2+x \right )^{7} \operatorname {expIntegral}_{1}\left (-x -2\right )-11970 \,\operatorname {expIntegral}_{1}\left (-x \right ) \left (2+x \right )^{7} x^{4}+\left (76477 x^{10}+970261 x^{9}+5171184 x^{8}+14871174 x^{7}+24496796 x^{6}+22249488 x^{5}+9184784 x^{4}+488880 x^{3}-131040 x^{2}+60480 x -40320\right ) {\mathrm e}^{x}}\right ] \\ \end{align*}