Link to actual problem [11015] \[ \boxed {x^{2} \left (a x +b \right ) y^{\prime \prime }-2 x \left (a x +2 b \right ) y^{\prime }+2 \left (a x +3 b \right ) y=0} \]
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 {a \,x^{2}}{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 \right )-\ln \left (x a +b \right )\right ] \\ \end{align*}