Differential algebraic equation

In mathematics, differential-algebraic equations (DAEs) are a general form of (systems of) differential equations for vector–valued functions x in one independent variable t,

where ${\displaystyle x:[a,b]\to \mathbb {R} ^{n}}$ is a vector of dependent variables ${\displaystyle x(t)=(x_{1}(t),\dots ,x_{n}(t))}$ and the system has as many equations, ${\displaystyle F=(F_{1},\dots ,F_{n}):\mathbb {R} ^{2n+1}\to \mathbb {R} ^{n}}$. They are distinct from ordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x because these may not all appear (i.e. some equations are algebraic); technically the distinction between an implicit ODE system [that may be rendered explicit] and a DAE system is that the Jacobian matrix ${\displaystyle {\frac {\partial F(u,v,t)}{\partial u}}}$ 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.

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