It is easy to generalize the proof tha a differentiable real ODE has differentiable solutions (using a fxed point argument) from the real to the complex case, and deduce that a holomorphic ODE has local holomorphic solutions. Then given a real analytic ODE, use the same power series to look at it as complex holomorphic. Finally go back to the real case by taking real parts. The details of this argument can be read here (see page 32).
http://ode-math.com/PDF_Files/ChapterFirstPages/Chap1Frst35Pages.pdf