= Solution
Set $t=x-m$ and use the <Frobenius solution> $y=t^\nu(a_0+a_1t+\cdots)$ with $a_0\ne0$. From part (a), the leading coefficients are $P\sim1/t$ and $Q\sim m^2\omega^2/t^2$. The coefficient of the lowest power $t^{\nu-2}$ is therefore
$$
[\nu(\nu-1)+\nu+m^2\omega^2]a_0.
$$
The <indicial equation> reduces to $\nu^2+m^2\omega^2=0$, giving
$$
\boxed{\nu_+=im\omega,\qquad\nu_-=-im\omega.}
$$
Since $m\omega>0$, the two <indicial roots> differ by a nonzero imaginary number, not an integer. Thus the two local <Frobenius solutions> exist independently on a chosen branch of $\log(x-m)$.
Back to article page
