Take the ODE $a_2(x)y''(x) + a_1(x)y'(x) + a_0(x)y(x) = L_{a}[y(x)]= 0$ and suppose the there is a common multiple of of $(x-1)$ in all the coefficients s.t. $(x-1)[b_2(x)y''(x) + b_1(x)y'(x) + b_0(x)y(x)] = (x-1)L_b[y(x)]= 0$.
If I solve $L_b[y(x)]= 0$, what does this mean about the solutions to $L_a[y(x)]= 0$?