Link to actual problem [9069] \[ \boxed {y^{\prime }-\frac {\left (-1+2 y \ln \left (x \right )\right )^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`], [_Abel, `2nd type`, `class C`]]
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 {x}{2}, \underline {\hspace {1.25 ex}}\eta &= y^{2}\right ] \\ \left [R &= -\frac {-1+2 y \ln \left (x \right )}{y}, S \left (R \right ) &= -2 \ln \left (x \right )\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 &= -\frac {x}{2} \\ \eta &=y^{2} \\ \frac {dS}{dR} &= \frac {2 R +2}{R^{3}+2 R +2} \\ \end{align*}