Link to actual problem [11847] \[ \boxed {\left (x +1\right )^{2} y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y=1} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2", "linear_second_order_ode_solved_by_an_integrating_factor", "second_order_ode_non_constant_coeff_transformation_on_B"}
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 &= x, S \left (R \right ) &= \frac {y}{1+x}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}-1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {1}{2}+y\right ] \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x^{2}}{2}-\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{2}+x y\right ] \\ \left [R &= -\frac {\ln \left (1+x \right ) x^{2}-\ln \left (-1+x \right ) x^{2}-\ln \left (1+x \right )+\ln \left (-1+x \right )-2 x -4 y}{4 \left (x^{2}-1\right )}, S \left (R \right ) &= -2 \,\operatorname {arctanh}\left (x \right )\right ] \\ \end{align*}