Please, could someone give me some help with this problem? I've struggling for days on it.
Let $a, b_{i j}$ be analytic functions defined in a neighborhood of $0$ in $\mathbb{R}$ and $\varphi ,\psi$ analytic in a neighborhood of $0$ in $\mathbb{R}$. In a neighborhood of $0$ in $\mathbb{R}$, consider $$ \left\{ \begin{array}{ll} u_t + au_x + b_{11}u + b_{12}v = f, \\ v_x + b_{12}u + b_{22}v = g, \end{array} \right. $$
with the condition
$ u(x,0)= \varphi(x) \,\,\ \mathrm{e} \,\,\ v(0,t) = \psi(t) $
$(a)$ Let $(u, v)$ be a smooth solution in a neighborhood of origin. Prove that all derivatives of $u$ and $v$ at $0$ are expressed in terms of those of $a$, $b_{ij}$, $f$, $g$, $\varphi$ e $\psi$ in $0$.
$(b)$ Prove that there is an analytical solution $(u,v)$ in a neighborhood of $0$ in $\mathbb{R}$.
My Attempt: I got the following ideias
1)$u_x(0)$=$\varphi_x(0)$ in any order, cuz I am over the initial condition.
2)$v_t(0)$=$\psi_t(0)$ in any order, cuz I am over the initial condition, as well.
3) When making the other derivatives I need to use the equations given in the system. But I couldn't get it.
I also have no clue about how to solve letter (b).
Can anyone help me? Thanks in advance.