Link to actual problem [7185] \[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-y x^{2}=x^{3}+\frac {1}{x}} \]
type detected by program
{"kovacic", "second_order_bessel_ode", "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]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {i x \sqrt {-x^{2}}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {i x \sqrt {-x^{2}}}{2}} y\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}