2.2.1 Algorithm

ode internal name "second_order_series_method_regular_singular_point_repeated_root".

In this case the solution is

\[ y=c_{1}y_{1}+c_{2}y_{2}\]

Where

\begin{align*} y_{1} & =\sum _{n=0}^{\infty }a_{n}x^{n+r_{1}}\\ y_{2} & =y_{1}\ln \left ( x\right ) +\sum _{n=1}^{\infty }b_{n}x^{n+r_{2}}\end{align*}

\(r_{1},r_{2}\) are roots of the indicial equation. \(a_{0},b_{0}\) are set to \(1\) as arbitrary. The coefficients \(b_{n}\) are not found from the recurrence relation but found using using \(b_{n}=\frac {d}{dr}a_{n}\left ( r\right ) \) after finding \(a_{n}\) first, and the result evaluated at root \(r_{2}\). (notice that \(r=r_{1}=r_{2}\) in this case). Notice there is no \(C\) term in from of the \(\ln \) in this case as when root differ by an integer and the sum on \(b_{n}\) starts at \(1\) since \(b_{0}\) is always zero due to \(\frac {d}{dr}a_{0}\left ( r\right ) =0\) always as \(a_{0}=1\) by default.