Link to actual problem [10572] \[ \boxed {y^{\prime }-\lambda \arccos \left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}=a m \,x^{m -1}} \]
type detected by program
{"riccati"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \left [R &= y-a \,x^{m}, S \left (R \right ) &= \frac {\operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right ) n x +\sqrt {-x^{2}+1}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right ) x -\operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) n \sqrt {-x^{2}+1}-2 \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) \sqrt {-x^{2}+1}-\sqrt {-x^{2}+1}\, \arccos \left (x \right )^{n +\frac {3}{2}}}{\left (n +2\right ) \sqrt {\arccos \left (x \right )}}\right ] \\ \end{align*}