Link to actual problem [10880] \[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a \left (n +1\right ) x^{n -1}+b \left (1+m \right ) x^{m -1}\right ) y=0} \]
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}^{-\frac {b \,x^{1+m} n +x^{n +1} a m +b \,x^{1+m}+x^{n +1} a}{\left (1+m \right ) \left (n +1\right )}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b \,x^{1+m} n +x^{n +1} a m +b \,x^{1+m}+x^{n +1} a}{\left (1+m \right ) \left (n +1\right )}} \left (\int \frac {{\mathrm e}^{\frac {b \,x^{1+m} n +x^{n +1} a m +b \,x^{1+m}+x^{n +1} a}{\left (1+m \right ) \left (n +1\right )}}}{x^{2}}d x \right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}} y}{\left (\int \frac {{\mathrm e}^{\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}}}{x^{2}}d x \right ) x}\right ] \\ \end{align*}