Link to actual problem [9419] \[ \boxed {a^{2} y^{\prime \prime }+a \left (a^{2}-2 b \,{\mathrm e}^{-x a}\right ) y^{\prime }+b^{2} {\mathrm e}^{-2 x a} y=0} \]
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^{3}+2 b \,{\mathrm e}^{-x a}}{2 a^{2}}} \sinh \left (\frac {x a}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \,a^{3}+2 b \,{\mathrm e}^{-x a}}{2 a^{2}}} 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^{3}+2 b \,{\mathrm e}^{-x a}}{2 a^{2}}} \cosh \left (\frac {x a}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \,a^{3}+2 b \,{\mathrm e}^{-x a}}{2 a^{2}}} 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*}