Link to actual problem [2820] \[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y=8 x^{4}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
type detected by program
{"reduction_of_order", "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"}
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*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{4}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{4}}, S \left (R \right ) &= 4 \ln \left (x \right )\right ] \\ \end{align*}