Link to actual problem [3854] \[ \boxed {\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+2 y x=0} \]
type detected by program
{"exact", "differentialType"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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 {1}{a^{2}+x^{2}+y^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= a^{2} y+x^{2} y+\frac {y^{3}}{3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (3 a^{2}+3 x^{2}+y^{2}\right )}{a^{2}+x^{2}+y^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (y \left (3 a^{2}+3 x^{2}+y^{2}\right )\right )}{3}\right ] \\ \end{align*}