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If you write the equation as $(e^{xy}u_y)_x = 0$, then the boundary conditions tell you that $e^{xy} u_y = e^0u_y(0,y) = -\sin y$, so $$ u_y(x,y) = -\sin y\,e^{-xy}, $$ so the solution is $$ u(x,y) = …

First of all, your system seems to uncouple quite strongly. The first and third equations only involve the unknowns $v_1$ and $p_1$ and the second and fourth equations only involve $v_2$ and $p_2$, s …

This isn't a solution, but it's too long for a comment. Before you try to apply Darboux' Method, you might want to clean up your system a bit. First, notice that this is an inhomogeneous linear syste …

Well, assuming that the matrix $\mathbf{B}$ is a real-analytic function of $t$, the local real-analytic theory gives you this result, which may or may not be useful to you: Start with a real-analyt …

When $n=1$, you can always do this, at least near $t=0$, by solving a single inhomogeneous, linear first-order PDE; you can even arrange that $h_2 = h_1$. When $n>1$, there is a geometrical obstructi …

Generally, you want there to be a non-characteristic transversal, i.e., a (let's say, smooth) curve $C$ in your domain $D$ such that each segment of each line $x=x_0$ in $D$ is connected and meets $C$ …

Here is an example for which there is no solution: Let $\Omega_1$ be defined by $x^2+y^2\le 1$ and $\Omega_1$ be defined by $X^2+Y^2\le R^2$, where $R>0$ is large. Take $Z(X,Y) = 0$. Then one is as …

Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but …

It has taken me a while to find time to write a more comprehensive answer to the above question. It turns out that for general dimension $N$, the overdetermined PDE system involved is not involutive, …