Numerical methods to solve Differential-Algebraic-Equations

Question

I am new to the topic of differential-algebraic-equations:

$ \dot x = f(x,u,c) $

$0=g(x,u,c) $

where $u$ are control variables and $c$ algebraic variables.In my first literature study i found two approaches:

1. given $x$ and $u$, solve (if non-linear with Newton) $0=g(x,u,c)$ for $c$. Make a step (Euler for instance)
2. Index reduction: derive algebraic equations with respect to time - so that in the end you get a normal ODE wich can be solved with ordinary ODE solver.

Why lead differntial algebraic equations to stiff ODE's?: "From a more theoretical viewpoint, the study of differential-algebraic problems gives insight into the behaviour of numerical methods for stiff ordinary differential equations"

Most of literature that i found talk about method 2; is it method 1 not simpler? Or do I misunderstood something :)? Could you recommend basic literature (simple introduction) to this topic?

Thank you very much!