Link to actual problem [9620] \[ \boxed {50 x \left (x -1\right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
type detected by Maple
[_Jacobi, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 &= {\mathrm e}^{\frac {i \sqrt {-\frac {1}{x \left (-1+x \right )}}\, x \left (-1+x \right ) \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x}\right )}{5 \sqrt {x \left (-1+x \right )}}}\right ] \\ \left [R &= x, S \left (R \right ) &= 2^{\frac {i \sqrt {-\frac {1}{x \left (-1+x \right )}}\, x^{2}}{5 \sqrt {x^{2}-x}}} \left (-1+2 x +2 \sqrt {x \left (-1+x \right )}\right )^{-\frac {i \sqrt {-\frac {1}{x \left (-1+x \right )}}\, \sqrt {x \left (-1+x \right )}\, x}{5 x -5}} 2^{-\frac {i \sqrt {-\frac {1}{x \left (-1+x \right )}}\, x}{5 \sqrt {x^{2}-x}}} \left (-1+2 x +2 \sqrt {x \left (-1+x \right )}\right )^{\frac {i \sqrt {-\frac {1}{x \left (-1+x \right )}}\, \sqrt {x \left (-1+x \right )}}{5 x -5}} y\right ] \\ \end{align*}