Let $\alpha \in (0,1)$ and $\eta \in C^{1,\alpha}(-1,1)$ be a solution to
$$
\eta'(x) = w(x,\eta(x)),\qquad \eta(x)=0 \quad\mbox{for }x \in (-1,0],\qquad \eta'(x)=0 \quad\mbox{for }x\in (-1,0],
$$
where $w\in C^{1,\alpha}(\mathbb{R}^2)$ such that $w(0,0)=0$ (the same result can be state by using $C^{k,\alpha}$ function and by imposing that higher order derivatives vanish on $(-1,0]$ as well).
I was wondering if a "unique continuation principle" holds true for solution of those type of equations. More precisely, can we deduce that the only solution vanishing on an open set is the trivial one $\eta \equiv 0$ (eventually by assuming that $w$ has finite vanishing order at the origin)? In this problem existence and uniqueness is not an issue, however it seems non trivial to find some reference in literature for a similar result.
Finally, I was curious about the possibility to extend this question to a system of first order PDE, that is for solutions $\eta \colon B_1\subset \mathbb{R}^n\to \mathbb{R}$ to
$$
\partial_{x_i} \eta(x) = w_i(x,\eta(x))
$$
with $w_i \colon \mathbb{R}^{n+1}\to \mathbb{R}$, for $i=1,\dots,n$ (with same boundary condition as before).