Link to actual problem [9664] \[ \boxed {y^{\prime \prime }+\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}+\frac {\left (a -b \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )}=0} \]
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
[[_2nd_order, _with_linear_symmetries], [_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 {-b +a}{\left (x +a \right )^{2} \left (x +b \right )}}\, \left (x +a \right ) \sqrt {x +b}\, \arctan \left (\frac {\sqrt {x +b}}{\sqrt {-b +a}}\right )}{\sqrt {-b +a}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {i \sqrt {\frac {-b +a}{\left (x +a \right )^{2} \left (x +b \right )}}\, \left (x +a \right ) \sqrt {x +b}\, \arctan \left (\frac {\sqrt {x +b}}{\sqrt {-b +a}}\right )}{\sqrt {-b +a}}} y\right ] \\ \end{align*}