Could anyone help me solve the following system of partial differential equations: $$\frac{\partial u_1(x_1,x_2)}{\partial x_1}=u_1(x_1,x_2)-u_2(x_1,x_2)$$ $$\frac{\partial u_2(x_1,x_2)}{\partial x_2}=u_2(x_1,x_2)-u_1(x_1,x_2)$$
$$u_1(0,x_2)=1$$ $$u_2(x_1,0)=0$$
My attempt:
Applynig the Laplace transform in PDE system, we have
$$\int_0^\infty \frac{\partial u_1(x_1,x_2)}{\partial x_1}e^{-s_1x_1}dx_1=\int_0^\infty u_1(x_1,x_2)e^{-s_1x_1}dx_1-\int_0^\infty u_2(x_1,x_2)e^{-s_1x_1}dx_1$$
$$\int_0^\infty \frac{\partial u_2(x_1,x_2)}{\partial x_2}e^{-s_2x_2}dx_2=\int_0^\infty u_2(x_1,x_2)e^{-s_2x_2}dx_2-\int_0^\infty u_1(x_1,x_2)e^{-s_2x_2}dx_2$$
which results in
$$s_1 \overline u_1(s_1,x_2)-u_1(0,x_2)=\overline u_1(s_1,x_2)-\overline u_2(s_1,x_2)$$ $$s_2 \overline u_2(x_1,s_2)-u_2(x_1,0)=\overline u_2(x_1,s_2)-\overline u_1(x_1,s_2)$$
where $s_1$ and $s_2$ denotes the Laplace variable with respect to $x_1$ and $x_2$, respectively. The over bar stand to variables $u_1$ and $u_2$ in the Laplace domain. However, now I have four unknown ($\overline u_1(s_1,x_2)$, $\overline u_1(x_1,s_2)$, $\overline u_2(s_1,x_2)$ and $\overline u_2(x_1,s_2)$) and only two equations.
Let $\textbf{A} = \langle -u_2,u_1 \rangle$, then
$$ \big|\nabla \times \textbf{A}\big| = \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} = 0 $$
Since the vector field $\textbf{A}$ is irrotational, it must follow that
$$ \textbf{A} = \nabla \phi $$
where $\phi(x_1,x_2)$ is some scalar field. Then we can rewrite
$$ \frac{\partial \phi}{\partial x_1} = -u_2, \quad \frac{\partial \phi}{\partial x_2} = u_1 $$
Thus the system becomes a single PDE
$$ \frac{\partial^2 \phi}{\partial x_1 \partial x_2} = \frac{\partial \phi}{\partial x_1} + \frac{\partial \phi}{\partial x_2} $$
with boundary conditions
$$ \frac{\partial \phi}{\partial x_1}(x_1,0) = 0, \quad \frac{\partial \phi}{\partial x_2}(0,x_2) = 1 $$