Link to actual problem [10728] \[ \boxed {y y^{\prime }-\frac {y}{\sqrt {a x +b}}=1} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class B`]]
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 +\frac {2 b}{a}, \underline {\hspace {1.25 ex}}\eta &= y\right ] \\ \left [R &= \frac {y}{\sqrt {x a +b}}, S \left (R \right ) &= \frac {\ln \left (x a +b \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 {a \,y^{2}-2 x a -2 b -2 \sqrt {x a +b}\, y}{y a} \\ \frac {dS}{dR} &= 0 \\ \end{align*}