Link to actual problem [12232] \[ \boxed {{y^{\prime }}^{2} \sqrt {y}=\sin \left (x \right )} \]
type detected by program
{"exactWithIntegrationFactor"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {1}{\sqrt {\sin \left (x \right )}}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= \int \sqrt {\sin \left (x \right )}d x\right ] \\ \end{align*}