Link to actual problem [14749] \[ \boxed {4 x^{2} y^{\prime \prime }+y=x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*}
type detected by program
{"kovacic", "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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \ln \left (x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \ln \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{x^{3}}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}