Link to actual problem [10992] \[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -\lambda b \right ) y=0} \]
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 &= \left (x a +\sqrt {-a b}\right )^{\frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}+\frac {1}{2}} \left (-x a +\sqrt {-a b}\right )^{\frac {\sqrt {a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c +4 a^{3} b -a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right )}}{4 a^{2} b}+\frac {1}{2}} {\mathrm e}^{\frac {2 a^{\frac {7}{2}} \sqrt {b}\, \lambda x +2 a^{\frac {5}{2}} \sqrt {-a b}\, \sqrt {b}\, \lambda -2 c x \,a^{\frac {5}{2}} \sqrt {b}-a^{\frac {3}{2}} \sqrt {-a b}\, \sqrt {b}\, c -\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d \,a^{3}+b \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c \,a^{2}}{2 a^{\frac {7}{2}} \sqrt {b}}} \operatorname {HeunC}\left (\frac {2 \left (2 a \lambda -c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c +4 a^{3} b -a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}+\frac {\left (a d -b c \right ) \left (8 a b \lambda -a d -3 b c \right )}{8 a^{3} b}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +\sqrt {-a b}\right )^{-\frac {\sqrt {-4 \sqrt {-a b}\, a^{3} b d +4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \left (-x a +\sqrt {-a b}\right )^{-\frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} {\mathrm e}^{\frac {\left (a^{2} d -a b c \right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )-2 \sqrt {b}\, \left (\left (-\frac {c \sqrt {a}}{2}+\lambda \,a^{\frac {3}{2}}\right ) \sqrt {-a b}+x \left (a^{\frac {5}{2}} \lambda -c \,a^{\frac {3}{2}}\right )\right )}{2 \sqrt {b}\, a^{\frac {5}{2}}}} y}{\sqrt {x a +\sqrt {-a b}}\, \sqrt {-x a +\sqrt {-a b}}\, \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-4 \sqrt {-a b}\, a^{3} b d +4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {4 a^{3} b +\left (8 b d \lambda -d^{2}\right ) a^{2}-8 \left (b \lambda +\frac {d}{4}\right ) b c a +3 b^{2} c^{2}}{8 a^{3} b}, \frac {x a +\sqrt {-a b}}{2 \sqrt {-a b}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunC}\left (\frac {2 \left (2 a \lambda -c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c +4 a^{3} b -a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}+\frac {\left (a d -b c \right ) \left (8 a b \lambda -a d -3 b c \right )}{8 a^{3} b}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) \left (-x a +\sqrt {-a b}\right )^{\frac {\sqrt {a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c +4 a^{3} b -a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right )}}{4 a^{2} b}+\frac {1}{2}} {\mathrm e}^{\frac {8 a^{\frac {7}{2}} b^{\frac {3}{2}} \lambda x +8 a^{\frac {5}{2}} \sqrt {-a b}\, b^{\frac {3}{2}} \lambda -8 c x \,a^{\frac {5}{2}} b^{\frac {3}{2}}+i \pi \,a^{\frac {3}{2}} \sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\, \sqrt {b}-i \pi \,a^{\frac {3}{2}} \sqrt {a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c +4 a^{3} b -a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right )}\, \sqrt {b}-4 a^{\frac {3}{2}} \sqrt {-a b}\, b^{\frac {3}{2}} c -4 \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d \,a^{3} b +4 b^{2} \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c \,a^{2}}{8 a^{\frac {7}{2}} b^{\frac {3}{2}}}} \left (x a +\sqrt {-a b}\right )^{\frac {1}{2}-\frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-x a +\sqrt {-a b}\right )^{-\frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} {\mathrm e}^{-\lambda x} {\mathrm e}^{-\frac {\sqrt {-a b}\, \lambda }{a}} {\mathrm e}^{\frac {c x}{a}} {\mathrm e}^{\frac {\sqrt {-a b}\, c}{2 a^{2}}} {\mathrm e}^{-\frac {i \pi \sqrt {-4 \sqrt {-a b}\, a^{3} b d +4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{8 a^{2} b}} {\mathrm e}^{\frac {i \pi \sqrt {4 \sqrt {-a b}\, a^{3} b d -4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{8 a^{2} b}} {\mathrm e}^{\frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d}{2 \sqrt {a}\, \sqrt {b}}} {\mathrm e}^{-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c}{2 a^{\frac {3}{2}}}} \left (x a +\sqrt {-a b}\right )^{\frac {\sqrt {-4 \sqrt {-a b}\, a^{3} b d +4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} y}{\operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-4 \sqrt {-a b}\, a^{3} b d +4 \sqrt {-a b}\, a^{2} b^{2} c +4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {4 a^{3} b +\left (8 b d \lambda -d^{2}\right ) a^{2}-8 \left (b \lambda +\frac {d}{4}\right ) b c a +3 b^{2} c^{2}}{8 a^{3} b}, \frac {x a +\sqrt {-a b}}{2 \sqrt {-a b}}\right ) \sqrt {-x a +\sqrt {-a b}}\, \sqrt {x a +\sqrt {-a b}}}\right ] \\ \end{align*}