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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval, let $\\Omega \\subseteq \\R^n \\times \\R^n$ be an open set, and let $F : I \\times \\Omega \\to \\R^n$ be continuously differentiable.\nWrite $F = F(t, x, p)$ with $t \\in I$ and $x, p \\in \\R^n$.\nThe system\n$$F(t, x(t), x'(t)) = 0$$\nis a *differential-algebraic system of equations* if the [[Jacobian matrix|047NDTEPCV]] $\\partial F / \\partial p$ of size $n \\times n$ is singular at every point of $I \\times \\Omega$.\nA solution is a differentiable map $x$ on a subinterval of $I$ with $(x(t), x'(t)) \\in \\Omega$ and $F(t, x(t), x'(t)) = 0$ for each $t$.\n\n## Remarks\n\nExample: for $x = (u, w)$, take the system $u' = -u + w$ and $0 = u - w + \\sin t$.\nHere $F(t, x, p) = (p_1 + u - w,\\ u - w + \\sin t)$ and $\\partial F / \\partial p$ has rows $(1, 0)$ and $(0, 0)$, so it is singular everywhere.\nThe constraint gives $w = u + \\sin t$, hence $u' = \\sin t$, so $u = C - \\cos t$ and $w = C - \\cos t + \\sin t$.\nInitial values must satisfy the constraint: $w(0) = u(0)$.\nNon-example: if $\\partial F / \\partial p$ is invertible at a point, the [[implicit function theorem|0BXBADR94C]] solves for $x'$ near that point, and the system is locally an [[ordinary differential equation|0P97V20RW7]] $x' = f(t, x)$.\nThe form $u' = f(t, u, w)$, $0 = g(t, u, w)$ is called semi-explicit.\nThe 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 $1$.\n",
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