Link to actual problem [8913] \[ \boxed {y^{\prime }-F \left (y+\frac {a \,x^{2}}{4}+\frac {x b}{2}\right )=-\frac {a x}{2}} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {x a}{2}-\frac {b}{2}\right ] \\ \left [R &= y+\frac {x^{2} a}{4}+\frac {b x}{2}, S \left (R \right ) &= x\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 &= 1 \\ \eta &=-\frac {x a}{2}-\frac {b}{2} \\ \frac {dS}{dR} &= \frac {2}{2 F \left (R \right )+b} \\ \end{align*}