Link to actual problem [8405] \[ \boxed {y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}}=0} \]
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 {1+x^{2} \left (x^{2} a +b \right )}\, \sqrt {\frac {a \,y^{4}+b \,y^{2}+1}{a \,x^{4}+b \,x^{2}+1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{\sqrt {a \,x^{4}+b \,x^{2}+1}\, \sqrt {\frac {\textit {\_a}^{4} a +\textit {\_a}^{2} b +1}{a \,x^{4}+b \,x^{2}+1}}}d \textit {\_a}\right ] \\ \end{align*}