Link to actual problem [8410] \[ \boxed {y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{\frac {2}{3}}=0} \]
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 &= \frac {\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y +a_{0}}\right )^{\frac {2}{3}}}{\left (a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )^{\frac {2}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {\left (a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )^{\frac {2}{3}}}{\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}\right )^{\frac {2}{3}}}d \textit {\_a}\right ] \\ \end{align*}