Link to actual problem [9273] \[ \boxed {y^{\prime }-\frac {y \ln \left (x \right ) x +x^{2} \ln \left (x \right )-2 x y-x^{2}-y^{2}-y^{3}+3 x y^{2} \ln \left (x \right )-3 \ln \left (x \right )^{2} y x^{2}+\ln \left (x \right )^{3} x^{3}}{x \left (-y+x \ln \left (x \right )-x \right )}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_Abel, `2nd type`, `class C`], [_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 &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {y}{x}+1\right ] \\ \left [R &= -\frac {-y+x \ln \left (x \right )}{x}, S \left (R \right ) &= x\right ] \\ \end{align*}