Link to actual problem [11063] \[ \boxed {x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+y b=0} \]
type detected by program
{"second_order_change_of_variable_on_x_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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {-a +\sqrt {a^{2}-4 b}}{4 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {-a +\sqrt {a^{2}-4 b}}{4 x^{2}}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {a +\sqrt {a^{2}-4 b}}{4 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {a +\sqrt {a^{2}-4 b}}{4 x^{2}}} y\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y, S \left (R \right ) &= -\frac {1}{2 x^{2}}\right ] \\ \end{align*}