 General solution of an Euler-Cauchy equation

Forthe power ansatz $w=z^{\lambda }$ gives the indicial equation $\lambda (\lambda -1)+a\lambda +b=0$. Distinct roots give $C_1{z}^{{\lambda }_1}+C_2{z}^{{\lambda }_2}$; a repeated root $\lambda$ gives $z^{\lambda }(C_1+C_2\log z)$ on a chosen logarithm branch.