Link to actual problem [14891] \[ \boxed {t \left (y^{\prime \prime } y+{y^{\prime }}^{2}\right )+y^{\prime } y=1} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 1] \end {align*}
type detected by program
{"second_order_integrable_as_is"}
type detected by Maple
[[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
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}{y}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y^{2}}{2}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y}{\sqrt {t}}, S \left (R \right ) &= \frac {\ln \left (t \right )}{2}\right ] \\ \end{align*}