Link to actual problem [10387] \[ \boxed {\left (x^{2} a +b \right ) y^{\prime }+y^{2}-2 x y=-\left (1-a \right ) x^{2}+b} \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_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 {a \,x^{2}}{b}+1, \underline {\hspace {1.25 ex}}\eta &= \frac {a \,x^{2}}{b}+1\right ] \\ \left [R &= y-x, S \left (R \right ) &= \frac {b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {1}{x -y}\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}-2 x y +y^{2} \\ \frac {dS}{dR} &= -\frac {1}{R^{2} a +b} \\ \end{align*}