Can someone give me some help with the following problem? I think it relates with Cauchy-Kovalevskaya Theorem, or maybe some technique from its proof.
Let $u=(u_1, ..., u_N)$ and consider the nonlinear system $$ \dfrac{\partial u_j}{\partial x_n}=G_j\left(x_p, u_q, \dfrac{\partial u_i}{\partial x_k}\right), for \ j=1, ..., k. $$ where $p, k =1, ..., n-1$, $i, q=1, ..., N$ and $G$ is an analytic real function at zero, with the following initial conditions $$ u_j=(x_1, ..., x_{n-1}, 0)=0, for \ j=1, ..., N. $$ Then, show directly, without reducing to an quasilinear system, using the majorants method that this system has an analytic solution near zero.