Link to actual problem [9346] \[ \boxed {y^{\prime \prime }-\left (x^{2} a^{2}+a \right ) y=0} \]
type detected by program
{"kovacic", "second_order_bessel_ode"}
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^{2} a}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {x^{2} a}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{2} a}{2}} \operatorname {erf}\left (\sqrt {a}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2} a}{2}} y}{\operatorname {erf}\left (\sqrt {a}\, x \right )}\right ] \\ \end{align*}