Link to actual problem [9726] \[ \boxed {y^{\prime \prime }-\frac {\left (3 x +1\right ) y^{\prime }}{\left (x -1\right ) \left (x +1\right )}+\frac {36 \left (x +1\right )^{2} y}{\left (x -1\right )^{2} \left (3 x +5\right )^{2}}=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {3 x +5}\, \left (-1+x \right )^{\frac {3}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {3 x +5}\, \left (-1+x \right )^{\frac {3}{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {3 x +5}\, \left (-1+x \right )^{\frac {3}{2}} \left (\ln \left (3 x +5\right )+3 \ln \left (-1+x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {3 x +5}\, \left (-1+x \right )^{\frac {3}{2}} \left (\ln \left (3 x +5\right )+3 \ln \left (-1+x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {\left (3 x +5\right ) \left (-1+x \right )}{1+x}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \frac {\ln \left (3 x +5\right )}{12}+\frac {\ln \left (-1+x \right )}{4}\right ] \\ \end{align*}