Link to actual problem [7184] \[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-y x=x^{2}+\frac {1}{x}} \]
type detected by program
{"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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselI}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselI}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \operatorname {BesselK}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \operatorname {BesselK}\left (\frac {2}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}