Link to actual problem [10973] \[ \boxed {\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+y^{\prime } b -6 y=0} \]
type detected by program
{"unknown"}
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 &= x^{3}+\frac {3}{4} b \,x^{2}-x \,a^{2}+\frac {1}{4} b^{2} x -\frac {5}{12} a^{2} b +\frac {1}{24} b^{3}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {b^{3}}{24}+\frac {b^{2} x}{4}+\frac {\left (-5 a^{2}+9 x^{2}\right ) b}{12}-x \,a^{2}+x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +a \right ) \left (a -x \right ) \left (b -4 x \right ) \left (\frac {x +a}{a -x}\right )^{\frac {b}{2 a}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {x +a}{a -x}\right )^{-\frac {b}{2 a}} y}{\left (x +a \right ) \left (a -x \right ) \left (b -4 x \right )}\right ] \\ \end{align*}