We investigate the structure of general nonlinear index-1 DAEs with properly involved linear derivative term. We show that each solution is actually a somewhat wrapped solution of the inherent explicit ODE. A certain implicitly given decoupling function plays its role when describing both the inherent ODE and the wrapping. With this background, we describe the set of consistent initial values, provide consistent initializations and prove local solvability and perturbation results. In turn, these results prove of value in Chapter 5 to analyze numerical integration methods and in Chapter 6 to investigate the asymptotical flow behavior.
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