Link to actual problem [9710] \[ \boxed {y^{\prime \prime }+\frac {\left (\left (\beta +\alpha +1\right ) \left (x -a \right )^{2} \left (x -b \right )+\left (-\beta -\alpha +1\right ) \left (x -b \right )^{2} \left (x -a \right )\right ) y^{\prime }}{\left (x -a \right )^{2} \left (x -b \right )^{2}}+\frac {\alpha \beta \left (a -b \right )^{2} y}{\left (x -a \right )^{2} \left (x -b \right )^{2}}=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_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 &= y, S \left (R \right ) &= \frac {\ln \left (x -a \right )}{-b +a}-\frac {\ln \left (x -b \right )}{-b +a}\right ] \\ \end{align*}