Link to actual problem [5339] \[ \boxed {y^{2} {y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
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 {3 x}{2}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{x^{\frac {2}{3}}}, S \left (R \right ) &= \frac {2 \ln \left (x \right )}{3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {2 y^{3}}{3}+3 x^{2}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {\operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )^{2}}{9 x^{2}+4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )}, S \left (R \right ) &= \int _{}^{x}\frac {1}{\frac {4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )^{2}}{3 \left (9 x^{2}+4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )\right )}+\frac {2 \sqrt {\frac {\operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )^{2} \left (9 \textit {\_a}^{2}+\frac {4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )^{2}}{9 x^{2}+4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )}\right )}{9 x^{2}+4 \operatorname {RootOf}\left (y-\textit {\_Z}^{\frac {1}{3}}\right )}}}{3}+3 \textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}