Link to actual problem [11104] \[ \boxed {y^{\prime \prime }-\left (a +2 \,{\mathrm e}^{a x} b \right ) y^{\prime }+b^{2} {\mathrm e}^{2 a x} y=0} \]
type detected by program
{"second_order_change_of_variable_on_x_method_2", "second_order_change_of_variable_on_y_method_1"}
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}^{\frac {x \,a^{2}+2 b \,{\mathrm e}^{x a}}{2 a}} \sinh \left (\frac {x a}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x \,a^{2}+2 b \,{\mathrm e}^{x a}}{2 a}} y}{\sinh \left (\frac {x a}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x \,a^{2}+2 b \,{\mathrm e}^{x a}}{2 a}} \cosh \left (\frac {x a}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x \,a^{2}+2 b \,{\mathrm e}^{x a}}{2 a}} y}{\cosh \left (\frac {x a}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y, S \left (R \right ) &= \frac {{\mathrm e}^{x a}}{a}\right ] \\ \end{align*}