Link to actual problem [728] \[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime }-y^{\prime } x +4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
With the expansion point for the power series method at \(x = 0\).
type detected by program
{"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 &= \left (x^{2}+2\right )^{\frac {3}{4}} \operatorname {LegendreP}\left (i \sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {i \sqrt {2}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x^{2}+2\right )^{\frac {3}{4}} \operatorname {LegendreP}\left (i \sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {i \sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}+2\right )^{\frac {3}{4}} \operatorname {LegendreQ}\left (i \sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {i \sqrt {2}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x^{2}+2\right )^{\frac {3}{4}} \operatorname {LegendreQ}\left (i \sqrt {3}-\frac {1}{2}, \frac {3}{2}, \frac {i \sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}