Link to actual problem [4534] \[ \boxed {y^{\prime }-\left (x^{2}+2 y-1\right )^{\frac {2}{3}}=-x} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y+\frac {x^{2}}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {x^{2}+2 y-1}{2 x^{3}}, S \left (R \right ) &= \frac {\ln \left (x \right )}{2}\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 &=\left (x^{2}+2 y -1\right )^{\frac {2}{3}} \\ \frac {dS}{dR} &= 1 \\ \end{align*}