Link to actual problem [3501] \[ \boxed {2 \left (x +1\right ) y^{\prime }+2 y+\left (x +1\right )^{4} y^{3}=0} \]
type detected by program
{"bernoulli", "first_order_ode_lie_symmetry_lookup"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (1+x \right )^{2} y^{3}}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1}{\left (1+x \right )^{2} y^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{4} y^{2}+4 x^{3} y^{2}-4 x^{2} y^{2}-16 x \,y^{2}-9 y^{2}-2\right ) y}{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (\frac {1}{2} x^{4}+2 x^{3}-2 x^{2}-8 x -\frac {9}{2}\right ) \ln \left (x^{4} y^{2}+4 x^{3} y^{2}-4 x^{2} y^{2}-16 x y^{2}-9 y^{2}-2\right )}{x^{4}+4 x^{3}-4 x^{2}-16 x -9}-\ln \left (y\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 &=\left (1+x \right )^{2} y^{3} \\ \frac {dS}{dR} &= -\frac {R}{2}-\frac {1}{2} \\ \end{align*}