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Introduction to Differential-Algebraic Equations (DAEs)

The systems of equations that govern certain phenomena (in electrical circuits, chemical kinetics, etc.) contain a combination of differential equations and algebraic equations. The differential equations are responsible for the dynamical evolution of the system, while the algebraic equations serve to constrain the solutions to certain manifolds. It is therefore of some interest to study the solutions of such differential-algebraic equations (DAEs).
Here is a simple example of a DAE. The first equation is an ODE for the function x[t], while the second equation constrains the functions x[t] and y[t] to lie in a submanifold (a straight line) in {x, y} space.
Click for copyable input
This discussion will be restricted to linear DAEs, which are defined as systems of equations of the following type.
Here A and B are matrix functions of the independent variable t, F is a vector function of t, and x (t) is the vector of unknowns. If the matrix A is nonsingular (that is, invertible) then this is a system of ODEs. Thus, the system is a DAE if the matrix A is singular.
If F0, then the system is said to be homogeneous. As for ODEs, the general solution to a DAE is composed of the general solution to the corresponding homogeneous problem and a particular solution to the inhomogeneous system.
DSolve can find the solutions to all DAEs in which the entries of the matrices A and B are constants. Such DAEs are said to have constant coefficients. The algorithm used by DSolve is based on decomposing both A and B into a nonsingular and nilpotent part. This decomposition is used to calculate a generalized inverse for A and B, which effectively reduces the problem to solving a system of ODEs.
It is important to realize that the initial values for a DAE must be prescribed carefully to guarantee a solution for the problem. This can be seen by considering the following system of equations.
This gives
Hence the only solution is
But this solution is inconsistent with the initial condition x1 (0)1.
For examples of DAEs with constant coefficients, which can be solved using DSolve, click here.