Link to actual problem [8310] \[ \boxed {y^{\prime \prime }-\left (x^{2}+3\right ) y=0} \]
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 &= x \,{\mathrm e}^{\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{2}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{2}}{2}} \operatorname {erf}\left (x \right ) \sqrt {\pi }\, x +{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{{\mathrm e}^{x^{2}} \operatorname {erf}\left (x \right ) \sqrt {\pi }\, x +1}\right ] \\ \end{align*}