Link to actual problem [1192] \[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+\left (-3 x +2\right ) y^{\prime }+4 y=0} \] 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 &= \operatorname {hypergeom}\left (\left [-\frac {5}{4}-\frac {i \sqrt {7}}{4}, -\frac {5}{4}+\frac {i \sqrt {7}}{4}\right ], \left [\frac {\left (-3 \sqrt {2}+4 i\right ) \sqrt {2}}{8}\right ], \frac {1}{2}-\frac {i \sqrt {2}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {5}{4}-\frac {i \sqrt {7}}{4}, -\frac {5}{4}+\frac {i \sqrt {7}}{4}\right ], \left [-\frac {\left (3 \sqrt {2}-4 i\right ) \sqrt {2}}{8}\right ], \frac {1}{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 (2 x +i \sqrt {2}\right )^{-\frac {i \sqrt {2}}{2}+\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {7}}{4}-\frac {i \sqrt {2}}{2}, \frac {1}{2}+\frac {i \sqrt {7}}{4}-\frac {i \sqrt {2}}{2}\right ], \left [-\frac {i \sqrt {2}}{2}+\frac {11}{4}\right ], \frac {1}{2}-\frac {i \sqrt {2}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x +i \sqrt {2}\right )^{\frac {i \sqrt {2}}{2}} y}{\left (2 x +i \sqrt {2}\right )^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {7}}{4}-\frac {i \sqrt {2}}{2}, \frac {1}{2}+\frac {i \sqrt {7}}{4}-\frac {i \sqrt {2}}{2}\right ], \left [-\frac {i \sqrt {2}}{2}+\frac {11}{4}\right ], \frac {1}{2}-\frac {i \sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}