Link to actual problem [10962] \[ \boxed {a_{2} x^{2} y^{\prime \prime }+\left (x^{2} a_{1} +b_{1} x \right ) y^{\prime }+\left (a_{0} x^{2}+b_{0} x +c_{0} \right ) y=0} \]
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 &= {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \operatorname {WhittakerM}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a_{1} x}{2 a_{2}}} x^{\frac {b_{1}}{2 a_{2}}} y}{\operatorname {WhittakerM}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \operatorname {WhittakerW}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}-2 a_{2} b_{1} -4 a_{2} c_{0} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a_{1} x}{2 a_{2}}} x^{\frac {b_{1}}{2 a_{2}}} y}{\operatorname {WhittakerW}\left (-\frac {a_{1} b_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )}\right ] \\ \end{align*}