Link to actual problem [9196] \[ \boxed {y^{\prime }-\frac {y+\sqrt {y^{2}+x^{2}}\, x +x^{3} \sqrt {y^{2}+x^{2}}+x^{4} \sqrt {y^{2}+x^{2}}}{x}=0} \]
type detected by program
{"unknown"}
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}{x^{3}+x^{2}+1}, \underline {\hspace {1.25 ex}}\eta &= \frac {y}{\left (x^{3}+x^{2}+1\right ) x}\right ] \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \frac {x \left (3 x^{3}+4 x^{2}+12\right )}{12}\right ] \\ \end{align*}