Consider the differential equation $\displaystyle z^3{w}^{''}+z{w}^{'}+{\lambda}{w}=0$, where $\displaystyle \lambda$ is a constant.

By considering a solution of the Frobenius form, I have found that their is only one root of the indicial equation, but I cannot show that the series solution terminates, producing a solution with a finite number of terms.

Can anyone help here?