Lateef Kareem, CypherCrescent Ltd

A DAE is a system that combines differential equations with algebraic constraints. Unlike ordinary differential equations (ODEs), you can’t always solve for the derivatives explicitly. The index of a DAE roughly measures how many times you need to differentiate the algebraic constraints to reduce the system to an ODE.

What makes index-2 DAEs special?

An index-2 DAE means you need to differentiate the algebraic constraints twice to get a system of ODEs. These systems often arise in:

• Mechanical systems with constraints, like a pendulum modelled in Cartesian coordinates.
• Electrical circuits with ideal components.
• Chemical process models with conservation laws.

A classic example

Consider a semi-explicit DAE:

y' = f(t, y, z) 0 = g(t, y)

Since the constraint g(t, y) = 0 is independent of the algebraic variable, the Jacobian ∂g/∂z is singular, and g(t, y) = 0 has to be differentiated twice to express everything in terms of y, in order to convert to index 0 (differential equations).

Why does the index matter?

• Numerical solvers: Many solvers are designed for index-1 systems. Index-2 DAEs often require index reduction techniques before you can apply standard methods.
• Stability and accuracy: Higher-index systems are more sensitive to perturbations and numerical drift, so they need careful handling.

Example 1

Example 2

Example 3