Link to actual problem [1385] \[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-4 x^{2}+1\right ) y^{\prime }+y \left (2 x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 {x}{\left (x^{2}+1\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {3}{2}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x}{x^{2}+1}-\frac {x \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )}{\left (x^{2}+1\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (x^{2}+1\right )^{\frac {5}{2}} y}{x \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) x^{2}-\left (x^{2}+1\right )^{\frac {3}{2}}+\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )\right )}\right ] \\ \end{align*}