Link to actual problem [11274] \[ \boxed {\left (x +1\right )^{2} y^{\prime \prime }-\left (x +1\right ) y^{\prime }+6 y=x} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1"}
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*} \\ \left [R &= y-\frac {x}{5}, S \left (R \right ) &= \frac {\ln \left (1+x \right )}{5}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1+x, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{6}+y\right ] \\ \left [R &= \frac {6 y+1}{6+6 x}, S \left (R \right ) &= \ln \left (1+x \right )\right ] \\ \end{align*}