Is it true that every PDE is a system of first order quasi-linear PDEs?

Question

As shown in Folland (Introduction to PDE, page 48), a fully non-linear PDE of the form $$ \partial_{x_n}^ku=G(x,(\partial^\beta u)_{|\beta |\leq k, \beta _n<k}) $$ ($G$ smooth) is equivalent to a first order quasi-linear system, ie they have the same regular solutions.

Does this continue to hold without assuming that the term $\partial_{x_n}^ku$ can be isolated? I.e. can we always think of a general fully non-linear PDE $$ F(x,(\partial^ \beta u)_{|\beta|\leq k})=0 $$ as a system of first order quasi-linear PDEs?

Answer 1

Yes. For each multi-index $\beta$ such that $|\beta| < k$, let $p_\beta$ be a function. Then the PDE can be written as \begin{align*} F(x, (p_\beta)_{|\beta|<k}, (\partial_kp_{\alpha})_{1 \le k \le n, |\alpha|=k-1}) &= 0\\ \partial_kp_{\alpha} &= p_{k,\alpha},\ 1 \le k \le n, |\alpha| \le k-2 \end{align*}