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In mathematics, a differential-algebraic system of equations is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Such systems occur as the general form of differential equations for vector–valued functions x in one independent variable t,
where x : → R n {\displaystyle x:\to \mathbb {R} ^{n}} is a vector of dependent variables x = , … , x n ] {\displaystyle x=,\dots ,x_{n}]} and the system has as many equations, F = : R 2 n + 1 → R n {\displaystyle F=:\mathbb {R} ^{2n+1}\to \mathbb {R} ^{n}}.They are distinct from ordinary differential equation in that a DAE is not completely solvable for the derivatives of all components of the function x because these may not all appear ; technically the distinction between an implicit ODE system and a DAE system is that the Jacobian matrix ∂ F ∂ u {\displaystyle {\frac {\partial F}{\partial u}}} is a singular matrix for a DAE system. This distinction between ODEs and DAEs is made because DAEs have different characteristics and are generally more difficult to solve.
In practical terms, the distinction between DAEs and ODEs is often that the solution of a DAE system depends on the derivatives of the input signal and not just the signal itself as in the case of ODEs; this issue is commonly encountered in nonlinear systems with hysteresis, such as the Schmitt trigger.
This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair {\displaystyle } of vectors of dependent variables and the DAE has the form