= Solution
Apply the <Frobenius method>. For $y=x^\sigma\sum_{n=0}^\infty a_nx^n$, the coefficient of the lowest power $x^\sigma$ comes only from $2x^2y''-xy'+y$; the remaining term $xy$ starts one power higher. As $a_0\ne0$, this lowest coefficient gives the <indicial equation>
$$
2\sigma(\sigma-1)-\sigma+1=2\sigma^2-3\sigma+1=(2\sigma-1)(\sigma-1)=0.
$$
Therefore the two possible <characteristic exponents at a regular singular point> are
$$
\boxed{\sigma=\frac12\quad\text{or}\quad\sigma=1}.
$$
Their difference is not an integer, so the two corresponding <Frobenius solutions> do not require a logarithmic correction.
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