Link to actual problem [10828] \[ \boxed {y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y=0} \]
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 {1}{2} a^{2} x^{2}+x a}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {a^{2} x^{2}}{2}} {\mathrm e}^{-x a} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {1}{2} a^{2} x^{2}+x a} \operatorname {erf}\left (i a x -i\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a^{2} x^{2}}{2}} {\mathrm e}^{-x a} y}{\operatorname {erf}\left (i a x -i\right )}\right ] \\ \end{align*}