Link to actual problem [7621] \[ \boxed {36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {1}{6}}}{\left (2 x -1\right )^{\frac {4}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{\frac {4}{3}} y}{x^{\frac {1}{6}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{6}} \left (-\frac {2 \ln \left (1+\left (2 x -1\right )^{\frac {1}{3}}\right )}{3 \left (2 x -1\right )^{\frac {4}{3}}}+\frac {\ln \left (1-\left (2 x -1\right )^{\frac {1}{3}}+\left (2 x -1\right )^{\frac {2}{3}}\right )}{3 \left (2 x -1\right )^{\frac {4}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x -1\right )^{\frac {1}{3}}}{-2+\left (2 x -1\right )^{\frac {1}{3}}}\right )}{3 \left (2 x -1\right )^{\frac {4}{3}}}+\frac {2}{2 x -1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 \left (2 x -1\right )^{\frac {7}{3}} y}{x^{\frac {1}{6}} \left (6 \left (2 x -1\right )^{\frac {4}{3}}+4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x -1\right )^{\frac {1}{3}}}{-2+\left (2 x -1\right )^{\frac {1}{3}}}\right ) x -2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x -1\right )^{\frac {1}{3}}}{-2+\left (2 x -1\right )^{\frac {1}{3}}}\right )+2 \ln \left (1-\left (2 x -1\right )^{\frac {1}{3}}+\left (2 x -1\right )^{\frac {2}{3}}\right ) x -4 \ln \left (1+\left (2 x -1\right )^{\frac {1}{3}}\right ) x -\ln \left (1-\left (2 x -1\right )^{\frac {1}{3}}+\left (2 x -1\right )^{\frac {2}{3}}\right )+2 \ln \left (1+\left (2 x -1\right )^{\frac {1}{3}}\right )\right )}\right ] \\ \end{align*}