Link to actual problem [9753] \[ \boxed {y^{\prime \prime } \sin \left (x \right )^{2}-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (-3+2 a \right )^{2}}+3 a +2\right ) y=0} \]
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 &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}}{-12+8 a}} \cos \left (\frac {x}{2}\right )^{\frac {-6+4 a -\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{-6+4 a}} \operatorname {hypergeom}\left (\left [-\frac {-8 a^{2}+\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}-\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}+8 a +6}{4 \left (2 a -3\right )}, -\frac {8 a^{2}+\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}-\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}-16 a +6}{4 \left (2 a -3\right )}\right ], \left [-\frac {6-4 a +\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{2 \left (2 a -3\right )}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {-6+4 a +\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}}{4 \left (2 a -3\right )}} \cos \left (\frac {x}{2}\right )^{\frac {6-4 a +\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{-6+4 a}} \sqrt {\sin \left (x \right )}\, y}{\operatorname {hypergeom}\left (\left [\frac {8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a -\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 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 &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}}{-12+8 a}} \cos \left (\frac {x}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{-6+4 a}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}+\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}-8 a -6}{-12+8 a}, \frac {-8 a^{2}+\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}+\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a +\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (x \right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{-\frac {-6+4 a +\sqrt {16 a^{4}+16 a^{2} b -72 a^{2}-48 a b +4 b^{2}+36 b +81}}{4 \left (2 a -3\right )}} \cos \left (\frac {x}{2}\right )^{-\frac {-6+4 a +\sqrt {16 a^{4}-16 a^{2} b -72 a^{2}+48 a b +4 b^{2}-36 b +81}}{2 \left (2 a -3\right )}} \sqrt {\sin \left (x \right )}\, y}{\operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a +\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )}\right ] \\ \end{align*}