Link to actual problem [10932] \[ \boxed {\left (x +\gamma \right ) y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y=0} \]
type detected by program
{"exact linear second order ode", "second_order_integrable_as_is"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\int \frac {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x} y\right ] \\ \end{align*}