Lemma 12.3.4.label The Cauchy Problem in Proposition 12.3.3 is equivalent to a quasi-linear system of $1^{st}$-order PDEs
\begin{align*}\begin{cases}\pd_{t}Y=\sums{n-1}{j=1}A_{j}(x,t,Y)\pd_{j}Y+B(x,t,Y)\\ Y(x,0)=\Phi(x)\end{cases}\end{align*}
where $Y=\begin{pmatrix}Y_{1}\\ \vdots\\ Y_{N}\end{pmatrix} =(y_{\al j})_{0\leq \abs{\al}+j\leq k,0\leq j\leq k-1},A_{i}=(a^{i}_{\el m})_{1\leq i\leq n-1;m,l=1,...,N},B=\begin{pmatrix}b_{1}\\ \vdots\\ b_{N}\end{pmatrix}$ and $\Phi$ is determined by $\p_{j}$.
Proof. Check Folland. The reason one can go forward is easy to see, simply differentiate. The reason one can go back is because of the holomorphic dependency of Cauchy data $Y(x,0)$ on the $\p_{j}$$\square$
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