Link to actual problem [13552] \[ \boxed {x^{2} y^{\prime \prime }-20 y=27 x^{5}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{5} \end {align*}
type detected by program
{"reduction_of_order", "second_order_euler_ode"}
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^{5}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{x^{4}}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x}{5}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{5}}, S \left (R \right ) &= 5 \ln \left (x \right )\right ] \\ \end{align*}