Link to actual problem [9761] \[ \boxed {y^{\prime \prime }+\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}+\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}=0} \]
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 &= \sin \left (x \right )^{-\frac {a +b}{2 a}} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {a^{2}-2 a b -4 a c -4 a d -4 a e +b^{2}}}{4 a}} \cos \left (\frac {x}{2}\right )^{-\frac {-2 a +\sqrt {a^{2}-2 a b -4 a c +4 a d -4 a e +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}, \frac {1}{2}+\frac {2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \sin \left (x \right )^{\frac {a +b}{2 a}} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {\sqrt {a^{2}-2 a b -4 a c -4 a d -4 a e +b^{2}}}{4 a}} \cos \left (\frac {x}{2}\right )^{\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}} y}{\sqrt {2 \cos \left (x \right )-2}\, \operatorname {hypergeom}\left (\left [-\frac {2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}-2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [-\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sin \left (x \right )^{-\frac {a +b}{2 a}} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {a^{2}-2 a b -4 a c -4 a d -4 a e +b^{2}}}{4 a}} \cos \left (\frac {x}{2}\right )^{\frac {2 a +\sqrt {a^{2}-2 a b -4 a c +4 a d -4 a e +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}, \frac {1}{2}+\frac {2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \sin \left (x \right )^{\frac {a +b}{2 a}} \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {\sqrt {a^{2}-2 a b -4 a c -4 a d -4 a e +b^{2}}}{4 a}} \cos \left (\frac {x}{2}\right )^{-\frac {2 a +\sqrt {a^{2}-2 a b -4 a c +4 a d -4 a e +b^{2}}}{2 a}} y}{\sqrt {2 \cos \left (x \right )-2}\, \operatorname {hypergeom}\left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}