Link to actual problem [255] \[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +3 y=8 x^{\frac {4}{3}}} \]
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_1", "second_order_change_of_variable_on_y_method_2", "linear_second_order_ode_solved_by_an_integrating_factor"}
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 &= \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {3 x}{4}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{\frac {4}{3}}}, S \left (R \right ) &= \frac {4 \ln \left (x \right )}{3}\right ] \\ \end{align*}