Link to actual problem [10388] \[ \boxed {\left (x^{2} a +b x +c \right ) y^{\prime }-y^{2}-\left (2 \lambda x +b \right ) y=\lambda \left (\lambda -a \right ) x^{2}+\mu } \]
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 &= 1+\frac {x \left (x a +b \right )}{c}, \underline {\hspace {1.25 ex}}\eta &= -\frac {\lambda \left (x^{2} a +b x +c \right )}{c}\right ] \\ \left [R &= \lambda x +y, S \left (R \right ) &= \frac {2 c \arctan \left (\frac {2 x a +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {2 \arctan \left (\frac {2 \lambda x +b +2 y}{\sqrt {-b^{2}+4 c \lambda +4 \mu }}\right )}{\sqrt {-b^{2}+4 c \lambda +4 \mu }}\right ] \\ \end{align*}
\begin{align*} \\ \operatorname {FAIL} \\ \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 &=\lambda ^{2} x^{2}+b \lambda x +2 \lambda x y +b y +c \lambda +y^{2}+\mu \\ \frac {dS}{dR} &= \frac {1}{R^{2} a +R b +c} \\ \end{align*}