Link to actual problem [9060] \[ \boxed {y^{\prime }+\frac {y a b -b c +x \,b^{2}+b a \sqrt {x}-a^{2}}{a \left (y a -c +x b +a \sqrt {x}\right )}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
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 &= 2 x, \underline {\hspace {1.25 ex}}\eta &= \frac {a y -b x -c}{a}\right ] \\ \left [R &= \frac {a y+b x -c}{a \sqrt {x}}, S \left (R \right ) &= \frac {\ln \left (x \right )}{2}\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 &=\frac {\sqrt {x}\, a^{2} y +x^{{3}/{2}} a b +a^{2} y^{2}+2 a b x y +b^{2} x^{2}-\sqrt {x}\, a c -2 x \,a^{2}-2 a c y -2 b c x +c^{2}}{a^{2} y -a c +a b x +a^{2} \sqrt {x}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}