Link to actual problem [13772] \[ \boxed {3 x^{2} y^{\prime \prime }-7 y^{\prime } x +3 y=4 x^{3}} \]
type detected by program
{"kovacic", "second_order_euler_ode", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2", "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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{\frac {1}{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{3}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{3}}, S \left (R \right ) &= 3 \ln \left (x \right )\right ] \\ \end{align*}