Link to actual problem [14179] \[ \boxed {y^{\prime }-\frac {t^{3}}{y \sqrt {\left (1-y^{2}\right ) \left (t^{4}+9\right )}}=0} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {t^{4}+9}}{y \sqrt {-t^{4} y^{2}+t^{4}-9 y^{2}+9}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {\left (-1+y\right ) \left (y+1\right ) \sqrt {-t^{4} y^{2}+t^{4}-9 y^{2}+9}}{3 \sqrt {t^{4}+9}}\right ] \\ \end{align*}