Link to actual problem [10152] \[ \boxed {x^{2} \left (2-9 x \right ) {y^{\prime \prime }}^{2}-6 x \left (-6 x +1\right ) y^{\prime } y^{\prime \prime }+6 y^{\prime \prime } y-36 x {y^{\prime }}^{2}=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (\textit {\_y1} x +3 \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}+3 y \right )}{x \left (9 x -2\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {x \left (9 x -2\right )}{\textit {\_a} \left (\textit {\_y1} x +3 \textit {\_a} +3 \sqrt {\left (-4 x^{3}+x^{2}\right ) \textit {\_y1}^{2}+12 \left (x -\frac {1}{6}\right ) \textit {\_a} x \textit {\_y1} +\textit {\_a}^{2}}\right )}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (-3 x^{2} \textit {\_y1} +\textit {\_y1} x +9 x y +\sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}-y \right )}{9 x -2}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {9 x -2}{\textit {\_a} \left (-3 x^{2} \textit {\_y1} +\left (9 \textit {\_a} +\textit {\_y1} \right ) x +\sqrt {\left (-4 x^{3}+x^{2}\right ) \textit {\_y1}^{2}+12 \left (x -\frac {1}{6}\right ) \textit {\_a} x \textit {\_y1} +\textit {\_a}^{2}}-\textit {\_a} \right )}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {-4 x^{2} \textit {\_y1} +6 x \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}+\textit {\_y1} x +6 x y -\sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}-y}{x^{2} \left (9 x -2\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {9 x^{3}-2 x^{2}}{-4 x^{2} \textit {\_y1} +\left (6 \sqrt {\left (-4 x^{3}+x^{2}\right ) \textit {\_y1}^{2}+12 \left (x -\frac {1}{6}\right ) \textit {\_a} x \textit {\_y1} +\textit {\_a}^{2}}+6 \textit {\_a} +\textit {\_y1} \right ) x -\sqrt {\left (-4 x^{3}+x^{2}\right ) \textit {\_y1}^{2}+12 \left (x -\frac {1}{6}\right ) \textit {\_a} x \textit {\_y1} +\textit {\_a}^{2}}-\textit {\_a}}d \textit {\_a}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {36 \textit {\_y1}^{2} x^{3}+54 x^{2} \textit {\_y1} \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}-17 \textit {\_y1}^{2} x^{2}-3 x \textit {\_y1} \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}+2 \textit {\_y1}^{2} x +6 \textit {\_y1} x y -2 \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}\, \textit {\_y1} -9 \sqrt {-4 \textit {\_y1}^{2} x^{3}+\textit {\_y1}^{2} x^{2}+12 \textit {\_y1} \,x^{2} y -2 \textit {\_y1} x y +y^{2}}\, y -2 y \textit {\_y1} -9 y^{2}}{x^{2} \left (9 x -2\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {9 x^{3}-2 x^{2}}{\left (\left (54 x^{2}-3 x -2\right ) \textit {\_y1} -9 \textit {\_a} \right ) \sqrt {\left (-4 x^{3}+x^{2}\right ) \textit {\_y1}^{2}+12 \left (x -\frac {1}{6}\right ) \textit {\_a} x \textit {\_y1} +\textit {\_a}^{2}}+\left (36 x^{3}-17 x^{2}+2 x \right ) \textit {\_y1}^{2}+2 \textit {\_a} \left (3 x -1\right ) \textit {\_y1} -9 \textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}