Link to actual problem [1267] \[ \boxed {\left (\beta \,x^{2}+\alpha x +1\right ) y^{\prime \prime }+\left (\delta x +\gamma \right ) y^{\prime }+\epsilon y=0} \] 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 {1}{2}-\frac {-\delta +\sqrt {\beta ^{2}-2 \beta \delta -4 \beta \epsilon +\delta ^{2}}}{2 \beta }, -\frac {1}{2}+\frac {\delta +\sqrt {\beta ^{2}-2 \beta \delta -4 \beta \epsilon +\delta ^{2}}}{2 \beta }\right ], \left [\frac {\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta +\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \frac {2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}+\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta +\alpha ^{2}-4 \beta }{2 \alpha ^{2}-8 \beta }\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {\beta -\delta +\sqrt {\beta ^{2}+\left (-2 \delta -4 \epsilon \right ) \beta +\delta ^{2}}}{2 \beta }, \frac {-\beta +\delta +\sqrt {\beta ^{2}+\left (-2 \delta -4 \epsilon \right ) \beta +\delta ^{2}}}{2 \beta }\right ], \left [\frac {\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta +\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \frac {-2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}-\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta -\alpha ^{2}+4 \beta }{-2 \alpha ^{2}+8 \beta }\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\left (2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}+\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta +\alpha ^{2}-4 \beta \right )}^{1-\frac {\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta +\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {\beta ^{2}-2 \beta \delta -4 \beta \epsilon +\delta ^{2}}\, \beta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}+\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}, \frac {1}{2}+\frac {\sqrt {\beta ^{2}-2 \beta \delta -4 \beta \epsilon +\delta ^{2}}\, \beta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}-\delta \alpha +2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \left [2-\frac {\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta +\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \frac {2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}+\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta +\alpha ^{2}-4 \beta }{2 \alpha ^{2}-8 \beta }\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\left (2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}+\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta +\alpha ^{2}-4 \beta \right )}^{\frac {\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta +\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}} y}{\left (2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}+\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta +\alpha ^{2}-4 \beta \right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {\beta ^{2}+\left (-2 \delta -4 \epsilon \right ) \beta +\delta ^{2}}\, \beta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}-\beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}+\delta \alpha -2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}, \frac {\beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}+\sqrt {\beta ^{2}+\left (-2 \delta -4 \epsilon \right ) \beta +\delta ^{2}}\, \beta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}-\delta \alpha +2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \left [\frac {4 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}-\delta \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \beta -\delta \alpha +2 \beta \gamma }{2 \beta ^{2} \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}}\right ], \frac {-2 \sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, x \,\beta ^{2}-\sqrt {\frac {\alpha ^{2}-4 \beta }{\beta ^{2}}}\, \alpha \beta -\alpha ^{2}+4 \beta }{-2 \alpha ^{2}+8 \beta }\right )}\right ] \\ \end{align*}