= Solution
Substituting $a_n=2^n/n!$ into the $r=0$ recurrence, and $a_n=1/n!$ into the $r=1/2$ recurrence, satisfies every coefficient equation. They give $e^{2x}$ and $\sqrt x\,e^x$. Thus on a positive interval not crossing a singular point,
$$
\boxed{y=A\sqrt x\,e^x+Be^{2x}.}
$$
Their <Wronskian> is $e^{3x}(2x-1)/(2\sqrt x)$, nonzero for $x>0$, $x\ne1/2$, proving independence on each ordinary interval. Both functions are analytic near $1/2$, so this is an <apparent regular singular point>: a pole in normalized coefficients need not force a singular solution. The Wronskian can vanish at that singular point without making the two functions linearly dependent on an ordinary interval.
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