# Show that every solution of the following differential equation has an analytic continuation along any path that avoids its singularities .

The problem states as follows. Let $f(z)$ be a meromorphic function on an open set $\Omega$. Consider a differential equation \begin{align} y''+f(z)y=0. \hspace{1cm}(1) \end{align} Show that every solution of $(1)$ has an analytic continuation along any path $\gamma$ on $\Omega$, such that $\gamma$ avoids the poles of $f(z)$.

• What is the independent variable in your ODE? Is it $z$? – avs Aug 30 '16 at 3:09
• Yes, it is. $f$ only depends on $z$. – user364833 Aug 30 '16 at 3:13
• See the ODE section of Cartan's Elementary Theory of Analytic Functions. – avs Aug 30 '16 at 6:04