# The differential equation $u'\cdot v + A\cdot u = w$

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 $$A$$.