Link to actual problem [2029] \[ \boxed {y^{\prime }-x \left (1-{\mathrm e}^{2 y-x^{2}}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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}+\ln \left (x \right ), S \left (R \right ) &= \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 &= 0 \\ \eta &=x^{2} {\mathrm e}^{-x^{2}} {\mathrm e}^{2 y}-1 \\ \frac {dS}{dR} &= 0 \\ \end{align*}