Solution
= Solution
Choose a <Riemannian metric> on $X$, using a <partition of unity> if necessary. For each $p$, let
$$
H_p=(\ker D_pF)^\perp.
$$
Since $F$ is a submersion, $D_pF|_{H_p}:H_p\to T_{F(p)}Y$ is an isomorphism. Its inverse depends smoothly on $p$, so
$$
v(p)=\left(D_pF|_{H_p}\right)^{-1}w(F(p))
$$
defines a smooth <vector field> on $X$ satisfying $D_pF(v(p))=w(F(p))$. The formula also gives $v(p)=0$ whenever $w(F(p))=0$.