Types of Implicit ODEs

In the last chapter we discussed the numerical treatment of explicit ordinary differential equations. Here, we will consider the more general case, implicit ordinary differential equations. 

The equations of motion of unconstrained multibody systems are normally given as implicit ODE 

Due to the regularity of E, Eq. (5.1.1) can be transformed into explicit form, though computationally this might not always be advantageous. 

When dealing with constrained multibody systems, the situation is different, because E is singular. Here we have 

In that particular case the system consists of differential equations and some nonlinear equations, often called in this context (somehow sloppy) algebraic equations. Consequently, this type of singularly implicit ODEs is also called a system of differential algebraic equations (DAEs). 

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