Undetermined coefficient at Frobenius resonance

When two indicial roots differ by a positive integer, the recurrence for the smaller-root Frobenius solution can have a zero coefficient at the index of the larger root. If the corresponding right-hand side also vanishes, that coefficient is free and adds the larger-root solution; if it does not vanish, a pure Frobenius series is obstructed and a logarithmic term may be required. For $x^2{y}^{\prime \prime }-2x{y}^{\prime }-x^{2}y=0$, the recurrence $n(n-3)a_n=a_{n-2}$ has $a_1=0$ and leaves $a_3$ free. Choosing $a_3=0$ selects the even solution, not the only solution with exponent zero.