= Solution
Consider an analytic <quasilinear partial differential equation>
$$
\sum_{i,j=1}^n a^{ij}(x,u,Du)\,\partial_i\partial_j u
=F(x,u,Du).
$$
Let the analytic initial hypersurface be $\Gamma=\{\phi=0\}$ and prescribe $u=u_0$ and one transverse derivative $\partial_Nu=u_1$ on $\Gamma$. The tangential derivatives of $u_0$ together with $u_1$ determine the full first jet $Du$ on $\Gamma$. The hypersurface is <non-characteristic hypersurface>[non-characteristic] at $x_0$ with respect to these data when
$$
\sum_{i,j}a^{ij}(x_0,u_0(x_0),Du(x_0))
\partial_i\phi(x_0)\partial_j\phi(x_0)\ne0.
$$
This is precisely the <principal symbol of a partial differential equation> evaluated on the conormal $d\phi(x_0)$.
The <Cauchy-Kovalevskaya theorem> then gives a unique real-analytic solution near $x_0$. In coordinates flattening $\Gamma$ to $x_n=0$, non-characteristicity lets the equation solve analytically for $\partial_n^2u$, after which the analytic equation and the two initial jets determine every higher Taylor coefficient.
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