Link to actual problem [15168] \[ \boxed {y^{\prime }-\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}=0} \]
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 &= \sqrt {x^{2}-2 x +5}\, \sqrt {\frac {9 y^{2}-6 y +2}{x^{2}-2 x +5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (\frac {\left (-\frac {3}{x^{2}-2 x +5}+\frac {9 y}{x^{2}-2 x +5}\right ) \sqrt {9}}{9 \sqrt {\frac {1}{x^{2}-2 x +5}}}+\sqrt {\frac {9 y^{2}}{x^{2}-2 x +5}-\frac {6 y}{x^{2}-2 x +5}+\frac {2}{x^{2}-2 x +5}}\right ) \sqrt {9}}{9 \sqrt {x^{2}-2 x +5}\, \sqrt {\frac {1}{x^{2}-2 x +5}}}\right ] \\ \end{align*}