Link to actual problem [9667] \[ \boxed {y^{\prime \prime }-\frac {2 \left (x a +2 b \right ) y^{\prime }}{x \left (x a +b \right )}+\frac {\left (2 x a +6 b \right ) y}{\left (x a +b \right ) x^{2}}=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1", "second_order_change_of_variable_on_y_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 &= x, S \left (R \right ) &= \frac {y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x^{2} a}{b}+x, \underline {\hspace {1.25 ex}}\eta &= \frac {2 a x y}{b}\right ] \\ \left [R &= \frac {y}{\left (x a +b \right )^{2}}, S \left (R \right ) &= -\ln \left (x a +b \right )+\ln \left (x \right )\right ] \\ \end{align*}