Link to actual problem [9047] \[ \boxed {y^{\prime }-\frac {-y a b +b^{2}+a b +x \,b^{2}-b a \sqrt {x}-a^{2}}{a \left (-y a +b +a +x b -a \sqrt {x}\right )}=0} \]
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 &= \frac {2 a x}{b}, \underline {\hspace {1.25 ex}}\eta &= -1+\frac {a y +b x -a}{b}\right ] \\ \left [R &= \frac {a y-b x -a -b}{a \sqrt {x}}, S \left (R \right ) &= \frac {b \ln \left (x \right )}{2 a}\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}-a^{2} \sqrt {x}-b a \sqrt {x}-2 x \,a^{2}-2 a^{2} y +2 a b x -2 a b y +2 b^{2} x +a^{2}+2 a b +b^{2}}{b a \sqrt {x}+a b y -b^{2} x -a b -b^{2}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}