= Solution
Move the expansion point to the origin with $z=x+1$. The transformed <ordinary differential equation> is
$$
zy_{zz}+(z-1)y_z+cy=0.
$$
After division by $z$, the functions $z(1-1/z)=z-1$ and $z^2(c/z)=cz$ are <analytic functions> at zero, so this is a <regular singular point>. For a <Frobenius solution> with leading term $a_0z^\rho$, $a_0\ne0$, the lowest power is $z^{\rho-1}$ and its coefficient is
$$
a_0\bigl[\rho(\rho-1)-\rho\bigr]=a_0\rho(\rho-2).
$$
Therefore the <indicial equation> and its <indicial exponents> are
$$
\boxed{\rho(\rho-2)=0,\qquad\rho=0,\ 2.}
$$
The exponent $\rho$ denotes the quantity called $\alpha$ in the paper, and avoids confusion with other uses of that letter.
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