Link to actual problem [9411] \[ \boxed {y^{\prime \prime }-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+2 a \right ) y^{\prime }+\left (\frac {a f^{\prime }\left (x \right )}{f \left (x \right )}+a^{2}-b^{2} f \left (x \right )^{2}\right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\left (\int \frac {\left (f \left (x \right ) b +a \right ) {\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )}+{\mathrm e}^{2 b} b f \left (x \right )-{\mathrm e}^{2 b} a}{{\mathrm e}^{2 b}-{\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )}}d x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\int \frac {\left (f \left (x \right ) b +a \right ) {\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )}-{\mathrm e}^{2 b} \left (-f \left (x \right ) b +a \right )}{{\mathrm e}^{2 b}-{\mathrm e}^{2 b \left (\int f \left (x \right )d x \right )}}d x} y\right ] \\ \end{align*}