I need some help:
Let $T>1$, $\Omega_T=\{(x,y)\in\mathbb{R}\>|\>x\in(1,T)\}$, $\Gamma=\{(x,y)\in\mathbb{R}\>|\>x=1\}$ and $g\in C^1(\mathbb{R})$.
Prove that there exists $T>1$ such that the following system has unique solution $u$ that is $C^1$ in $\Omega_T$:
$x\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=e^x(1+u^2)$ in $\Omega_T$
$u=g$ in $\Gamma$
I know how to solve the problem, in fact, I got:
$u(x,y)=\tan[e^y-\frac{e^y}{x}+\arctan[g(y-\log(x))]]$
but I don't know how to use the $T$ condition.