Link to actual problem [10169] Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right ) y^{\prime \prime \prime }-3 y^{\prime } \left (y^{\prime \prime }\right )^{2}=0} \end {gather*}
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{x}, S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x^{2}+y^{2}, S \left (R \right ) &= -\arctan \left (\frac {x}{y}\right )\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{2}+y^{2}}{x}, S \left (R \right ) &= -\frac {y^{2}}{\left (x^{2}+y^{2}\right ) \sqrt {y^{2}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= \frac {x^{2}}{2}-\frac {y^{2}}{2}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {y}{x^{2}+y^{2}}, S \left (R \right ) &= \frac {2 \left (\frac {y^{2}}{x^{2}+y^{2}}-1\right ) y}{\left (x^{2}+y^{2}\right ) \sqrt {-\frac {y^{2} \left (\frac {y^{2}}{x^{2}+y^{2}}-1\right )}{x^{2}+y^{2}}}}\right ] \\ \end{align*}