Link to actual problem [9268] \[ \boxed {y^{\prime }-y^{2}-\frac {7 x^{2} y}{2}+2 x y-y^{3}-\frac {3 y^{2} x^{2}}{4}+3 y^{2} x -\frac {3 y x^{4}}{16}+\frac {3 x^{3} y}{2}=-\frac {1}{2} x +1+\frac {13}{16} x^{4}-\frac {3}{2} x^{3}+x^{2}+\frac {1}{64} x^{6}-\frac {3}{16} x^{5}} \]
type detected by program
{"abelFirstKind", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Abel]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{2}-4 x +4 y +4\right ) \left (x^{2}-4 x +4 y \right )^{2}}{64}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {4}{x^{2}-4 x +4 y}-\ln \left (x^{2}-4 x +4 y\right )+\ln \left (x^{2}-4 x +4 y+4\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 &=-y^{2}-\frac {7}{2} x^{2} y +2 x y -\frac {13}{16} x^{4}+\frac {3}{2} x^{3}-x^{2}-y^{3}-\frac {3}{4} x^{2} y^{2}+3 x \,y^{2}-\frac {3}{16} x^{4} y +\frac {3}{2} x^{3} y -\frac {1}{64} x^{6}+\frac {3}{16} x^{5} \\ \frac {dS}{dR} &= -1 \\ \end{align*}