Link to actual problem [10917] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+y x^{n -1} a n=0} \]
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}^{-\frac {a \,x^{n}+\left (b -1\right ) \ln \left (x^{n}\right )}{n}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {a \,x^{n}}{n}} \left (x^{n}\right )^{\frac {b}{n}} \left (x^{n}\right )^{-\frac {1}{n}} y\right ] \\ \end{align*}