Link to actual problem [4524] \[ \boxed {\left (x +1\right ) y^{\prime }-y-\left (x +1\right ) \sqrt {1+y}=1} \]
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 &= \frac {x}{2}+\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= y +1\right ] \\ \left [R &= \frac {y+1}{\left (1+x \right )^{2}}, S \left (R \right ) &= 2 \ln \left (1+x \right )\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 &=-\sqrt {y +1}\, x -\sqrt {y +1}+y +1 \\ \frac {dS}{dR} &= \frac {1}{R +1} \\ \end{align*}