Differential-algebraic system of equations

Remarks

Example: for $x = (u, w)$, take the system $u' = -u + w$ and $0 = u - w + \sin t$. Here $F(t, x, p) = (p_1 + u - w,\ u - w + \sin t)$ and $\partial F / \partial p$ has rows $(1, 0)$ and $(0, 0)$, so it is singular everywhere. The constraint gives $w = u + \sin t$, hence $u' = \sin t$, so $u = C - \cos t$ and $w = C - \cos t + \sin t$. Initial values must satisfy the constraint: $w(0) = u(0)$. Non-example: if $\partial F / \partial p$ is invertible at a point, the implicit function theorem solves for $x'$ near that point, and the system is locally an ordinary differential equation $x' = f(t, x)$. The form $u' = f(t, u, w)$, $0 = g(t, u, w)$ is called semi-explicit. The differentiation index is the least number of differentiations of the system needed to determine $x'$ as a continuous function of $(t, x)$; the example above has index $1$.