2.14.9.89 problem 889 out of 2993

Link to actual problem [6457] \[ \boxed {2 x^{2} y^{\prime \prime }+y^{\prime } x -y \left (1+x \right )=0} \] With the expansion point for the power series method at \(x = 0\).

type detected by program

{"second order series method. Regular singular point. Difference not integer"}

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 &= {\mathrm e}^{\sqrt {x}\, \sqrt {2}} \sqrt {\frac {\left (2 x -1\right ) \left (\sqrt {x}\, \sqrt {2}-1\right )}{x \left (\sqrt {x}\, \sqrt {2}+1\right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\sqrt {x}\, \sqrt {2}} y}{\sqrt {\frac {\left (2 x -1\right ) \left (\sqrt {x}\, \sqrt {2}-1\right )}{x \left (\sqrt {x}\, \sqrt {2}+1\right )}}}\right ] \\ \end{align*}

\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\sqrt {x}\, \sqrt {2}} \sqrt {\frac {\left (2 x -1\right ) \left (\sqrt {x}\, \sqrt {2}+1\right )}{x \left (\sqrt {x}\, \sqrt {2}-1\right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\sqrt {x}\, \sqrt {2}} y}{\sqrt {\frac {\left (2 x -1\right ) \left (\sqrt {x}\, \sqrt {2}+1\right )}{x \left (\sqrt {x}\, \sqrt {2}-1\right )}}}\right ] \\ \end{align*}