Link to actual problem [11025] \[ \boxed {2 y^{\prime \prime } x \left (a \,x^{2}+b x +c \right )+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 {i \sqrt {\frac {x \lambda }{x^{2} a +b x +c}}\, \left (x^{2} a +b x +c \right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 x a -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 a x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (2 \sqrt {-4 a c +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) \sqrt {-4 a c +b^{2}}+\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right ) b \right )}{4 \sqrt {x \left (x^{2} a +b x +c \right )}\, a^{2} \sqrt {a \,x^{3}+b \,x^{2}+c x}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {i \sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\frac {\left (b -\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right )}{2}+\sqrt {-4 a c +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 a c +b^{2}}+b}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}}{2}\right )\right ) \sqrt {-\frac {a x}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 x a -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}\, \sqrt {\frac {x \lambda }{x^{2} a +b x +c}}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}{2 a^{2} x}} y\right ] \\ \end{align*}