Link to actual problem [11877] \[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +4 y=-6 x^{3}+4 x^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = 4, y^{\prime }\left (2\right ) = -1] \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_2", "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}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{2}, \underline {\hspace {1.25 ex}}\eta &= x^{2}+\frac {3 y}{2}\right ] \\ \left [R &= \frac {2 x^{2}+y}{x^{3}}, S \left (R \right ) &= 2 \ln \left (x \right )\right ] \\ \end{align*}