Link to actual problem [13682] \[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=10 x +12} \] With initial conditions \begin {align*} [y \left (1\right ) = 6, y^{\prime }\left (1\right ) = 8] \end {align*}
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 [R &= x, S \left (R \right ) &= \frac {y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y-2}{x^{3}}, S \left (R \right ) &= -\frac {1}{x}\right ] \\ \end{align*}