Link to actual problem [12131] \[ \boxed {y^{\prime }-\frac {y}{x +1}+y^{2}=0} \]
type detected by program
{"riccati", "bernoulli", "exactWithIntegrationFactor", "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 {y^{2}}{1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1+x}{y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (x^{2} y +2 x y -2 x -2\right )}{1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (1+x \right ) \left (-\frac {\ln \left (y\right )}{2 x +2}+\frac {x \left (2+x \right ) \ln \left (x^{2} y+2 x y-2 x -2\right )}{\left (2 x +2\right ) \left (x^{2}+2 x \right )}\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 &=\frac {y^{2}}{1+x} \\ \frac {dS}{dR} &= -R -1 \\ \end{align*}