Given a linear PDE of order 1 with $C^ \infty$ real coefficients in a neighborhood of the origin in $\mathbb{R}^n$ :
$$L=\sum_{j=1}^{n}\alpha_j(x)\frac{\partial}{\partial x_j}$$
Suppose that at least one of the coefficients $\alpha_j$ does not vanish at the origin. Show that there is a $C^\infty$ change of variables $x \to y$ in a nbhd of the origin s.t. $L$ in y-coordinates is $w(y)\frac{\partial}{\partial y_1}$ with $w(0)\neq 0$. Is it always possible to choose the coordinates y so as to have $w(y) \equiv 1$ near $y=0$ ?
ATTEMPT:I have solved the above problem for the constant coefficients case that is $\alpha_j$ are constant. For the variable case, how do I proceed ?