Link to actual problem [556] \[ \boxed {y+\left (x -4 y\right ) y^{\prime }=-9 x^{2}+1} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
type detected by program
{"exact", "differentialType"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]
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 {1}{-4 y +x}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {3 x^{3}+x y -2 y^{2}-x}{-12 y +3 x}\right ] \\ \\ \end{align*}