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Let $A\colon \mathbb R^n\to\mathbb R^{n\times n}$ be a matrix valued function, and $v, w\colon \mathbb R^n\to\mathbb R^n$ vector fields. I want to find a function/vector field $u\colon\mathbb R^n\to\mathbb R^n$ that satisfies

$$ u'(x) \cdot v(x) + A(x)\cdot u(x)=w(x) $$

for all $x$. Is this a known differential equation? Does it have a name?

I guess one could call this a first order linear partial differential algebraic equation. Searching for linear PDAE yields some results that seem to deal with similar equations when $A, v, w$ are constant (one example). Anyway, I am still looking for sources and was just hoping that maybe someone has seen something like this before. Ideally I would hope that there is a way to analytically solve it given $v,w$ and $A$.

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