Link to actual problem [9055] \[ \boxed {y^{\prime }-\frac {\left (18 x^{\frac {3}{2}}+36 y^{2}-12 x^{3} y+x^{6}\right ) \sqrt {x}}{36}=0} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
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 {4 x}{3}, \underline {\hspace {1.25 ex}}\eta &= x^{3}-2 y\right ] \\ \left [R &= -\frac {x^{\frac {3}{2}} \left (x^{3}-6 y\right )}{6}, S \left (R \right ) &= \frac {3 \ln \left (x \right )}{4}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= 0 \\ \eta &=-\frac {x^{\frac {15}{2}}}{27}+\frac {4 x^{\frac {9}{2}} y}{9}+\frac {x^{3}}{3}-\frac {4 x^{\frac {3}{2}} y^{2}}{3}-2 y \\ \frac {dS}{dR} &= -\frac {3}{4 R} \\ \end{align*}