Frobenius series

#math
Let us consider the equation $(1)$ again, and say the expansion $$\begin{align} xP(x) &= p_{0} + p_{1}x + p_{2}x^2 \dots \ x^2Q(x) &= q_{0} + q_{1}x + q_{2}x^2 \dots \end{align}$$are valid in $| x | < R; R > 0$ .
Let the indicial equation $$r(r-1) + p_{0}r + q_{0} = 0$$have real roots $m_{1}$ and $m_{2}$ (with $m_{2} \leq m_{1}$ ). Then the DE is guaranteed atleast one solution $$y_{1} = x^{m_{1}}\sum_{m=0}^\infty a_{m}x^m; a_{0} \neq 0$$
The other solution is determined as follows :

If $m_{1} - m_{2}$ is neither 0 nor an integer, then in this case other solution $$y_{2} = x^{m_{2}} \sum_{m=0}^\infty a_{m}x^m; a_{0} \neq 0$$
If $m_{1} = m_{2}$ , then in this case $$y_{2} = y_{1}\ln x + x^{m_{2}}\sum_{m=0}^\infty A_{m}x^m\ ;\ a_{0} \neq 0$$
If $m_{1} - m_{2}$ is an integer, then other solution $$y_{2} = ky_{1}\ln x + x^{m_{2}}\sum_{m=0}^\infty A_{m}x^m\ ;\ a_{0} \neq 0$$ $k$ may be zero.