Link to actual problem [8999] \[ \boxed {y^{\prime }+\frac {\left (x^{4} \sqrt {a}+x^{3} \sqrt {a}-2 \sqrt {a \,x^{4}+8 y}\right ) \sqrt {a}}{2 x +2}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(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 &= 1+x, \underline {\hspace {1.25 ex}}\eta &= -\frac {a \left (1+x \right ) x^{3}}{2}\right ] \\ \left [R &= y+\frac {a \,x^{4}}{8}, S \left (R \right ) &= \ln \left (1+x \right )\right ] \\ \end{align*}