Link to actual problem [8956] \[ \boxed {y^{\prime }-\frac {1}{y+2+\sqrt {3 x +1}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]
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}{3}, \underline {\hspace {1.25 ex}}\eta &= y +2\right ] \\ \left [R &= \frac {y+2}{\sqrt {1+3 x}}, S \left (R \right ) &= \frac {\ln \left (1+3 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 {3 y \sqrt {1+3 x}+3 y^{2}+6 \sqrt {1+3 x}-6 x +12 y +10}{2 y +4+2 \sqrt {1+3 x}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}