Link to actual problem [13564] \[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }-y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 8, y^{\prime }\left (1\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_1", "second_order_change_of_variable_on_y_method_2", "linear_second_order_ode_solved_by_an_integrating_factor"}
type detected by Maple
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {x}}, S \left (R \right ) &= -\frac {1}{2 x}\right ] \\ \end{align*}