How can we know if a solution of differential equation is multivalued without solving it?

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For a differential equation, say $q(x)y''(x)+p(x) y'(x)+s(x)=0$, how could we know if its solutions are multivalued without solving it. Could we study the branch points and branch cut by simply observing $q(x)$, $p(x)$ and $s(x)$.