Link to actual problem [1140] \[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -2 y=x^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {5}{4}}, y^{\prime }\left (1\right ) = {\frac {3}{2}}\right ] \end {align*}
type detected by program
{"reduction_of_order", "second_order_euler_ode", "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 [R &= x, S \left (R \right ) &= -\frac {\ln \left (x^{2}-4 y\right )}{4}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{x^{2}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}