I am trying to solve a system of first-order linear PDEs. In the best case, I would want to solve it explicitly but proving that there exists a (unique) solution would also be helpful.
Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain with a smooth boundary $\Gamma$. We assume that a smooth function $\phi:\Omega \rightarrow \mathbb{R}$ which is zero at the boundary is given. Then, I am looking for a function $u:\Omega \rightarrow \mathbb{R}$ such that \begin{align} \nabla u = \begin{bmatrix} \phi_{x}^{2} \\ \phi_{x}\phi_{y} \end{bmatrix} \qquad & \text{ in } \Omega, \\ u=g \qquad & \text{ on } \Gamma, \end{align} where $g$ is an arbitrary smooth and bounded function. Note we use the notation $\phi_{x} = \partial_{x}\phi=\frac{\partial \phi}{\partial x}$ and $\phi_{y} = \partial_{y}\phi=\frac{\partial \phi}{\partial y}$ here.
Now let me explain what I have done so far. I tried to solve the problem by integrating the individual equations, $$ \partial_{x}u=\phi_{x}^{2},$$ $$ \partial_{y}u=\phi_{x}\phi_{y}.$$ This way I get \begin{align*} u &= \int \phi_{x}^{2}(x,y)dx + C_{1}(y) \\ u &= \int \phi_{x}(x,y)\phi_{y}(x,y) dy + C_{2}(x). \end{align*}
Using integration by parts I get, \begin{align*} u &= \phi\phi_{x}-\int \phi(x,y)\phi_{xx}(x,y)dx, + C_{1}(y), \\ u &= \phi\phi_{x}-\int \phi(x,y)\phi_{xy}(x,y)dy + C_{2}(x). \end{align*}
Now we would try to find $C_{1}$ and $C_{1}$ based on the boundary conditions.
However, the main conclusion is that the PDE has a solution if for all $(x,y) \in \Omega$ holds $$ \phi\phi_{x}-\int \phi(x,y)\phi_{xx}(x,y)dx, + C_{1}(y) = u = \phi\phi_{x}-\int \phi(x,y)\phi_{xy}(x,y)dy + C_{2}(x)$,$$ or equivalently $$ -\int \phi(x,y)\phi_{xx}(x,y)dx, + C_{1}(y) = u = -\int \phi(x,y)\phi_{xy}(x,y)dy + C_{2}(x).$$
Am I correct that the equality of the two integrals is a necessary condition for the existence of a solution of the PDE?
EDIT: Based on the feedback from the comments I fixed some of my mistakes.