Link to actual problem [10993] \[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (\lambda \left (a +c \right ) x^{2}+\left (c -a \right ) x +2 \lambda b \right ) y^{\prime }+\lambda ^{2} \left (c \,x^{2}+b \right ) y=0} \]
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 &= \operatorname {HeunC}\left (\frac {2 \lambda \left (a -c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{2 a^{2}}, \frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{2 a^{2}}, \frac {\left (a -c \right ) \sqrt {-a b}\, \lambda \left (a +c \right )}{a^{3}}, \frac {-\left (a -c \right ) \left (-\lambda \left (a +c \right ) \left (-a b \right )^{\frac {3}{2}}+\left (3 \lambda \left (a +c \right ) \sqrt {-a b}+\left (a -c \right ) \left (-3 b \,\lambda ^{2}+a \right )\right ) b a \right )+4 b \,a^{4}}{8 b \,a^{4}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) \left (x a +\sqrt {-a b}\right )^{\frac {\sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{4 a^{2}}+\frac {1}{2}} \left (-x a +\sqrt {-a b}\right )^{\frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{4 a^{2}}+\frac {1}{2}} \left (x^{2} a +b \right )^{\frac {1}{4}-\frac {c}{4 a}} {\mathrm e}^{\frac {-4 \lambda \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) a^{3}-8 \lambda x c \,a^{\frac {5}{2}}+4 \sqrt {-a b}\, a^{\frac {5}{2}} \lambda -4 \sqrt {-a b}\, a^{\frac {3}{2}} c \lambda -i \sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}\, a^{\frac {3}{2}} \pi +4 \lambda \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c \,a^{2}-i a^{\frac {3}{2}} \pi \sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{8 a^{\frac {7}{2}}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +\sqrt {-a b}\right )^{-\frac {\sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}}{4 a^{2}}} \left (-x a +\sqrt {-a b}\right )^{-\frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{4 a^{2}}} \left (x^{2} a +b \right )^{\frac {c}{4 a}} {\mathrm e}^{\frac {i \sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}\, a^{\frac {3}{2}} \pi +i a^{\frac {3}{2}} \pi \sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}+4 \left (a^{2} \sqrt {b}\, \left (a -c \right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )+\left (c \,a^{\frac {3}{2}}-a^{\frac {5}{2}}\right ) \sqrt {-a b}+2 a^{\frac {5}{2}} c x \right ) \lambda }{8 a^{\frac {7}{2}}}} y}{\operatorname {HeunC}\left (\frac {2 \lambda \left (a -c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}}{2 a^{2}}, \frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{2 a^{2}}, \frac {\left (a -c \right ) \sqrt {-a b}\, \lambda \left (a +c \right )}{a^{3}}, \frac {4 \lambda \left (-a^{2}+c^{2}\right ) \sqrt {-a b}+3 a^{3}+\left (3 b \,\lambda ^{2}+2 c \right ) a^{2}+\left (-6 b c \,\lambda ^{2}-c^{2}\right ) a +3 \lambda ^{2} c^{2} b}{8 a^{3}}, \frac {x a +\sqrt {-a b}}{2 \sqrt {-a b}}\right ) \sqrt {x a +\sqrt {-a b}}\, \sqrt {-x a +\sqrt {-a b}}\, \left (x^{2} a +b \right )^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunC}\left (\frac {2 \lambda \left (a -c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{2 a^{2}}, \frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{2 a^{2}}, \frac {\left (a -c \right ) \sqrt {-a b}\, \lambda \left (a +c \right )}{a^{3}}, \frac {-\left (a -c \right ) \left (-\lambda \left (a +c \right ) \left (-a b \right )^{\frac {3}{2}}+\left (3 \lambda \left (a +c \right ) \sqrt {-a b}+\left (a -c \right ) \left (-3 b \,\lambda ^{2}+a \right )\right ) b a \right )+4 b \,a^{4}}{8 b \,a^{4}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) \left (-x a +\sqrt {-a b}\right )^{\frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{4 a^{2}}+\frac {1}{2}} \left (x^{2} a +b \right )^{\frac {1}{4}-\frac {c}{4 a}} {\mathrm e}^{\frac {-4 \lambda \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) a^{3}-8 \lambda x c \,a^{\frac {5}{2}}+4 \sqrt {-a b}\, a^{\frac {5}{2}} \lambda -4 \sqrt {-a b}\, a^{\frac {3}{2}} c \lambda -i \sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}\, a^{\frac {3}{2}} \pi +4 \lambda \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c \,a^{2}+i a^{\frac {3}{2}} \pi \sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{8 a^{\frac {7}{2}}}} \left (x a +\sqrt {-a b}\right )^{\frac {1}{2}-\frac {\sqrt {-a \left (b \,a^{2} \lambda ^{2}-2 a c \,\lambda ^{2} b +\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda -9 a^{3}+6 a^{2} c -a \,c^{2}\right )}}{4 a^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-x a +\sqrt {-a b}\right )^{-\frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{4 a^{2}}} \left (x^{2} a +b \right )^{\frac {c}{4 a}} {\mathrm e}^{\frac {i \sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}\, a^{\frac {3}{2}} \pi -i a^{\frac {3}{2}} \pi \sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}+4 \left (a^{2} \sqrt {b}\, \left (a -c \right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )+\left (c \,a^{\frac {3}{2}}-a^{\frac {5}{2}}\right ) \sqrt {-a b}+2 a^{\frac {5}{2}} c x \right ) \lambda }{8 a^{\frac {7}{2}}}} \left (x a +\sqrt {-a b}\right )^{\frac {\sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}}{4 a^{2}}} y}{\operatorname {HeunC}\left (\frac {2 \lambda \left (a -c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-a^{3} b \,\lambda ^{2}+2 a^{2} b c \,\lambda ^{2}-a b \,c^{2} \lambda ^{2}-6 \sqrt {-a b}\, a^{3} \lambda +8 \sqrt {-a b}\, a^{2} c \lambda -2 a \sqrt {-a b}\, c^{2} \lambda +9 a^{4}-6 a^{3} c +a^{2} c^{2}}}{2 a^{2}}, \frac {\sqrt {a \left (-b \,a^{2} \lambda ^{2}+2 a c \,\lambda ^{2} b -\lambda ^{2} c^{2} b +6 \sqrt {-a b}\, a^{2} \lambda -8 \sqrt {-a b}\, a c \lambda +2 \sqrt {-a b}\, c^{2} \lambda +9 a^{3}-6 a^{2} c +a \,c^{2}\right )}}{2 a^{2}}, \frac {\left (a -c \right ) \sqrt {-a b}\, \lambda \left (a +c \right )}{a^{3}}, \frac {4 \lambda \left (-a^{2}+c^{2}\right ) \sqrt {-a b}+3 a^{3}+\left (3 b \,\lambda ^{2}+2 c \right ) a^{2}+\left (-6 b c \,\lambda ^{2}-c^{2}\right ) a +3 \lambda ^{2} c^{2} b}{8 a^{3}}, \frac {x a +\sqrt {-a b}}{2 \sqrt {-a b}}\right ) \sqrt {-x a +\sqrt {-a b}}\, \left (x^{2} a +b \right )^{\frac {1}{4}} \sqrt {x a +\sqrt {-a b}}}\right ] \\ \end{align*}