Link to actual problem [5630] \[ \boxed {y^{\prime \prime }+y \left (x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second_order_bessel_ode", "second order series method. Ordinary point", "second order series method. Taylor series method"}
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 &= \frac {\operatorname {WhittakerM}\left (-\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerM}\left (-\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {WhittakerW}\left (-\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{\operatorname {WhittakerW}\left (-\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}\right ] \\ \end{align*}